We introduce a multiparameter deformation of the triply‐graded Khovanov–Rozansky homology of links colored by one‐column Young diagrams, generalizing the “y‐ified” link homology of Gorsky–Hogancamp and work of Cautis–Lauda–Sussan. For each link component, the natural set of deformation parameters corresponds to interpolation coordinates on the Hilbert scheme of the plane. We extend our deformed link homology theory to braids by introducing a monoidal dg 2‐category of curved complexes of type A singular Soergel bimodules. Using this framework, we promote to the curved setting the categorical colored skein relation from our recent joint work and also the notion of splitting map for the colored full twists on two strands. As applications, we compute the invariants of colored Hopf links in terms of ideals generated by Haiman determinants and use these results to establish general link splitting properties for our deformed, colored, triply‐graded link homology. Informed by this, we formulate several conjectures that have implications for the relation between (colored) Khovanov–Rozansky homology and Hilbertschemes.
]]>The last two decades have seen the introduction of powerful homological invariants of knots, links, braids, and tangles, which are connected to classical quantum invariants through a decategorification relationship [Kho00,Rou04,KR08a,KR08b]. These invariants are best understood in the context of di↵erential graded categories: each tangle diagram D is assigned a chain complex Z(D) over an additive category, Reidemeister moves between such diagrams are assigned specific chain maps that are invertible up to homotopy, and movies between diagrams that encode certain braid/tangle cobordisms are assigned (generally non-invertible) chain maps that are natural, up to homotopy [CMW09, Cap08, Bla10, EK10, ETW18]. In the case that D is a knot or link diagram, the homology H Z (L) of Z(D) is therefore an invariant of the link L determined by D. In fact, Z(D) can typically be equipped with 1 additional structure (see below) and determines an invariant Z(L) of the corresponding link L up to quasi-isomorphism. Instances of this higher structure are the subject of this paper.
1.1. Local operators and monodromy. In many cases, a choice of point p 2 D equips Z(D) with an action of a graded-commutative algebra of local operators A Z . In prototypical examples, A Z = Z( ) can be identified with the invariant of the unknot 1 , and the action of A Z at p is induced by the saddle cobordism t D ! D that merges a small unknot with D near the point p. We now mention several specific instances of this setup. For the duration, we work over the rationals Q (see §1.7).
Example 1.1. Let Z(D) = C KR N (D) be the gl N Khovanov-Rozansky complex, whose homology is the gl N Khovanov-Rozansky homology H KR N (L) of the link L determined by D [KR08a]. A choice of p 2 D equips C KR N (D) with an action of the graded algebra A KR N := H ⇤ (CP N 1 ) ⇠ = Q[x]/(x N ). The unknot invariant C KR N ( ) is the free A KR N -module generated by a single element, in degree 1 N , so we may identify A KR N and C KR N ( ) up to shift. Under this identification, the action of C KR N ( ) at p 2 D is induced by the map
where the first part is due to the monoidality of C KR N and the second part is induced by the saddle cobordism.
Remark 1.2. The algebra A Z is sometimes referred to as the sheet algebra of the theory Z. It suggest that elements of A Z should be visualized as the identity cobordism of the trivial (1, 1)-tangle, suitably decorated. See e.g. [MN08,Corollary 2.4], where this terminology appears to have originated.
Example 1.3. Let D be a closed braid diagram of a link L, and let Z(D) = C KR (D) be the triplygraded Khovanov-Rozansky complex, whose homology is the HOMFLYPT homology HHH(L) := H KR (D) [KR08b]. There are two common choices for the sheet algebra in this case: the underived sheet algebra Q[x], or the derived sheet algebra Q
). Here the degrees of the variables, written multiplicatively following Convention 3.6 below, are given by wt(x) = q 2 and wt(⌘) = aq 2 .
The action of this sheet algebra is best understood using Khovanov's formulation of HHH(L) using the Hochschild homology of Soergel bimodules [Kho07], given that this homology theory is not functorial with respect to general link cobordisms. We will refer to the a-degree of the variable ⌘ in the following as the Hochschild-degree.
Example 1.4. In this paper, we are primarily interested in colored link homologies and, more specifically, the ^-colored extension of triply-graded Khovanov-Rozansky homology [WW17]. This homology theory defines invariants of framed oriented links L in which each component c 2 ⇡ 0 (L) is labeled by a non-negative integer b(c), the color, each of which defines a sheet algebra A Z,b(c) . For b 2 N, fix an alphabet X b = {x 1 , . . . , x b }, then the b-colored sheet algebra is given by the ring of symmetric polynomials Sym(X b ) := Q[x 1 , . . . , x b ] S b and the derived sheet algebra by HH • (Sym(X b )).
In addition to the action of sheet algebras, Z(D) is typically also equipped with higher structures stemming from the fact that choices of di↵erent points p 1 , p 2 2 D that lie in the same component 2 c 2 ⇡ 0 (L) should induce homotopic actions of the relevant sheet algebra A Z . To be precise, let ⇢ D be an oriented path from p 1 to p 2 . For each a 2 A Z , let a(p i ) denote the action of a 2 A Z at p i 2 D. Then, the path determines a homotopy (a), with [ D , (a)] = a(p 2 ) a(p 1 ).
1 More precisely, Z( ) is typically a free A Z -module of rank 1, so A Z and Z( ) are isomorphic up to grading shift. 2 Here, and in the following, we slightly abuse terminology and identify components c 2 ⇡ 0 (L) of the link determined by D with the corresponding equivalence class of points in the diagram D. Thus, we can talk about components of D.
Here D is the di↵erential on the complex Z(D), so the super-commutator [ D , ] is the di↵erential on the dg algebra End(Z(D)). Note, however, that a(p 1 ) and a(p 2 ) are homotopic in two di↵erent ways. We can choose two complementary paths , 0 : p 1 ! p 2 in D so that traversing followed by the reverse of 0 yields a loop. It follows that the di↵erence a := (a) 0 (a) is a closed endomorphism of Z(D) of degree wt( a ) = wt(a)t 1 , called the monodromy of a along D. These monodromy endomorphisms, together with certain higher operations, (should) assemble to give an action of the Hochschild homology HH • (A Z ) (itself an algebra, since A Z is graded-commutative) on Z(D). This should not be confused with the passage to a derived sheet algebra as in Example 1.3. Instead, the guiding principle is the following.
Principle 1.5. Each component c 2 ⇡ 0 (L) determines an action of HH • (A Z ) on Z(L), well-defined up to quasi-isomorphism, in such a way that, for a 2 A Z , the Kähler di↵erential d(a) 2 HH 1 (A Z ) acts as the monodromy a along c. The actions along various components (anti-)commute and assemble to give an action of N c2⇡0(L) HH • (A Z ). Remark 1.6. In the above statement, we have invoked the well-known fact that for a commutative Q-algebra A, HH 1 (A) is isomorphic to the A-module of Kähler di↵erentials on A, i. Remark 1.7. In the setting of colored link homologies, the statement of Principle 1.5 should be modified accordingly to account for the fact that the sheet algebras depend on a choice of color.
We thus refer to HH • (A Z ) as the monodromy algebra of A Z . For the duration, we restrict to the setting of (colored) triply-graded Khovanov-Rozansky homology for concreteness, since this will be the invariant we study in this work.
Example 1.8. Continuing Example 1.3, for the (uncolored) triply-graded Khovanov-Rozansky homology with sheet algebra Q[x], the associated monodromy algebra is Q[x] ⌦ ^[⇠], where wt(x) = q 2 and wt(⇠) = q 2 t 1 . Thus, one would expect an action of Q[x 1 , . . . , x r ] ⌦ ^[⇠ 1 , . . . , ⇠ r ] on C KR (D) when D has r components. Such an action was constructed in [GH] in the form of "y-ified" Khovanov-Rozansky homology.
Example 1.9. For colored Khovanov where wt(p i ) = wt(e i ) = q 2i and ⌅ i = d(p i ) and i = (e i ) with wt(⌅ i ) = wt( i ) = q 2i t 1 . If one prefers a more basis independent description, we can identify the monodromy algebra with S b -invariants
1.2. From monodromy to deformation. Now, we show how the action of the monodromy algebra HH • (A Z ) on Z(D) allows us to construct link splitting deformations 3 of Z(D). Again, we work explicitly with triply-graded Khovanov-Rozansky homology, first in the uncolored case (considered in [GH]) and then its extension to the colored case which is achieved in this paper.
Suppose that D is a braid closure diagram for an r-component oriented link L and let C KR (D) be the Khovanov-Rozansky complex associated to D, which carries an action of the monodromy algebra 3 In this paper, the word deformation will always refer to such link splitting deformations, which are based on monodromy data. These are of an entirely di↵erent nature than the deformations of finite-rank type A link homologies based on deformations of underlying Frobenius algebras, that were studied by the second-and third-named authors in [RW16].
Q[x 1 , . . . , x r ] ⌦ ^[⇠ 1 , . . . , ⇠ r ] on C KR (D). The deformed (or "y-ified") complex YC KR (D) is constructed from this action using Koszul duality [BGS96] (see the earlier [BGG78] for the specific case of polynomial/exterior algebras). Explicitly, introduce formal parameters y 1 , . . . , y r with wt(y i ) = q 2 t 2 and form the complex In [GH], it is shown that the homology of YC KR (D) is a well-defined invariant of the oriented link L, up to isomorphism.
The main goal of the present paper is to investigate the colored version of this invariant. For this, suppose that L is a (framed, oriented) colored link, i.e. each component c 2 ⇡ 0 (L) is assigned a color b(c) 0. Recall from Example 1.9 that the monodromy algebra associated to a b-labeled
The following is a consequence of Lemma 5.38; see Remark 5.39.
Proposition 1.10. Let D be a diagram for a framed, oriented, colored link L, presented as a braid closure. The colored Khovanov-Rozansky complex C KR (D) admits an action of the algebra
in which ⌅ c,k is the monodromy of 1 k p k along c. Using this monodromy action (and Koszul duality), we can build the following deformed complex:
(2)
In Theorem 5.30, we establish the following.
Theorem 1.11. The complex YC KR (D) is a well-defined invariant of the framed, oriented, colored link L, up to quasi-isomorphism of modules over Q[p c,i , v c,j ] c2⇡0(L),1i,jb(c) . Consequently, its homology YH KR (L) is an invariant of L up to isomorphism of Q[p c,i , v c,j ]-modules.
Remark 1.12. In this paper, we actually take a di↵erent approach to the definition of the deformed complex from (2). Indeed, rather than first constructing the monodromy morphisms for a link diagram D and then using them to build YC KR (D), we instead build the complex YC KR (D) from local pieces that encode the homotopies associated with paths in D that traverse a single crossing. These take the form of certain curved complexes, that we discuss next. Nonetheless, the two approaches are equivalent; see Remark 5.39.
1.3. Curving Rickard complexes. The undeformed colored Khovanov-Rozansky complex can be studied at the level of braids (without closing up to obtain a link) using the framework of singular Soergel bimodules. It will be quintessential to extend our deformed theory to the level of braids as well. Similar to the considerations in [GH], the notion of curved complexes of singular Soergel bimodules appears naturally.
Recall the monoidal 2-category 4 of singular Soergel bimodules, denoted SSBim. Objects of this category are sequences a = (a 1 , . . . , a m ) of positive integers, and a 1-morphism from b = (b 1 , . . . , b m 0 ) to a = (a 1 , . . . , a m ) is a certain kind of graded bimodule over (R a , R b ). Here, R a denotes the ring of polynomials in P m i=1 a i variables that are invariant with respect to the action of S a1 ⇥ • • • ⇥ S am . Specifically, 1-morphisms SSBim are generated (with respect to direct sum and summands, grading shift, horizontal composition ?, and external tensor product ⇥) by induction and restriction bimodules relating the rings R a,b and R a+b . The 2-morphisms are maps of graded bimodules. See §3.4 for full details.
The 2-category SSBim contains the singular Bott-Samelson bimodules, those bimodules that are constructed from induction and restriction bimodules using only grading shift, ?, and ⇥. These bimodules can be depicted diagrammatically as certain trivalent graphs called webs, e.g.:
c d d k k b k b a .
(Here, the unlabeled edge has label a + b k = c + d k). In diagrams such as these, a trivalent "merge" vertex (when read right-to-left) corresponds to a restriction bimodule R a+b (R a,b ) R a,b and a "split" vertex corresponds to an induction bimodule R a,b (R a,b ) R a+b .
To each colored braid, one can associate a complex of singular Soergel bimodules using the notion of Rickard complexes. For instance, to the (a, b)-colored elementary crossing with a b, one associates a complex of the form
(3) C a,b := s b a { := 0 @ b a 0 b a ! q 1 t b a 1 b a ! • • • ! q b t b b a b b a 1 A .
To each b-labeled edge appearing in such diagrams (either a colored braid diagram, a web depicting a singular Bott-Samelson, or a composition thereof) there is an action of the sheet algebra Sym(X b ) from Example 1.4 on the associated (complex of) 1-morphism(s). We will denote the sheet algebra simply by Sym(X p ), if we wish to emphasize the point where the sheet algebra is acting (thus
It is well-known that the actions on the four endpoints of the Rickard complex are homotopic along the strands, see e.g. [RW16, Proposition 5.7]. For example, choosing points p 1 , p 2 , p 0 1 , p 0 2 on C a,b as follows:
(4)
| we find that the actions of f (X p2 ) and f (X p 0 1 ) and the actions of g(X p1 ) and g(X p 0 2 ) on C a,b are homotopic, for all f 2 Sym(X a ) and g 2 Sym(X b ). Hence, there exists homotopies o (f ) and u (g) so that [ , o (f )] = f (X p2 ) f (X p 0 1 ) , [ , u (g)] = g(X p1 ) g(X p 0 2 ) . Following the recipe from §1.2, we should incorporate the homotopies for a collection of generators of Sym(X a ) into our di↵erential, as in (2). One choice of generating set is the collection of elementary symmetric functions e 1 , . . . , e a 2 Sym(X a ). Considering the corresponding homotopies { o k } a k=1 and { u k } b k=1 (built in Lemma 4.20 below) and extending scalars, we thus can consider C a,b , equipped with the "di↵erential"
(5
where U = {u o k } 1ka [ {u u k } 1kb . As we show, the homotopies o/u i each square to zero and pairwise anti-commute, so we find that (6)
Hence, YC a,b := (C a,b , tot ) is not a chain complex in the traditional sense, but rather a curved complex.
Recall that the latter is a generalization of the notion of chain complex, and consists of a pair (X, tot ) where the curved di↵erential tot squares to a nonzero element F 2 End(X). We will refer to the curvature in (6), and its analogue for more general braids, as e-curvature. More generally, using the operations ? and ⇥ in SSBim, we can associate a Rickard complex C( b ) to any colored braid b that is built from complexes C a,b and C _ a,b assigned to colored positive and negative crossings, as in (3). In §4, we show that these complexes can be deformed in a similar manner to C a,b .
Theorem 1.13. The Rickard complex C( b ) associated to a colored braid b admits a deformation to a curved complex YC( b ) with e-curvature. Such a deformation is unique, up to homotopy equivalence. associated to positive and negative crossings in hand, one need only construct well-defined composition operations to build the curved complexes associated to arbitrary braids. Hence, we establish the following.
Theorem 1.14. There exists a monoidal dg 2-category Y(SSBim) wherein 1-morphisms are curved complexes of singular Soergel bimodules with e-curvature. Appropriate horizontal compositions ? and external tensor products ⇥ of the curved complexes YC a,b and YC _ a,b associated with positive and negative crossings assign a 1-morphism in Y(SSBim) to any colored braid (word), which satisfies the braid relations up to canonical homotopy equivalence.
In our description of YC a,b above (and subsequent statements about YC( b )), we chose the elementary symmetric functions as the generators of the sheet algebra Sym(X b ) of a b-colored strand. Thus our homotopies k encode e-curvature. At times, we will find it beneficial to work with curved complexes built from other homotopies, which similarly identify sheet algebra actions at the ends of braid strands. Specifically, let p and p 0 be points at the left and right ends of a b-colored braid strand, and let N (X p , X p 0 ) Sym(X p |X p 0 ) ⇠ = Sym(X p ) ⌦ Sym(X p 0 ) denote the diagonal ideal, which is generated by all elements of the form f (X p ) f (X p 0 ) with f 2 Sym(X b ). In addition to the set of generators
for N (X p , X p 0 ), we can also work with the generating sets
(See §2 for details on symmetric functions.) By change of variables, it is possible to pass from curved complexes of singular Soergel bimodules with e-curvature (i.e. curvature modeled on N e ) to curved complexes with h -curvature and p-curvature, modeled on N h and N p , respectively. See §4.5 and §5.5. The choice of generators for N (X p , X p 0 ) is conceptually immaterial, but each of the above leads to a notion of curved complex of singular Soergel bimodules that is useful in particular instances. For example, complexes with h -curvature appear most often "in the wild," and a straightforward change of variables leads to complexes with e-curvature that are well-behaved 2-categorically. Passing to p-curvature requires working over a field of characteristic zero, but such curvature is best adapted to establishing Markov invariance. In particular, Proposition 1.10 and Theorem 1.11 proceed by passing from curved Rickard complexes in Y(SSBim) to curved complexes with (appropriate) p-curvature, before taking braid closure to obtain (uncurved) complexes associated to the corresponding link diagram.
Remark 1.15 (y-variables vs. u-variables vs. v-variables). In defining the curved di↵erential tot in (5), we used the variable name u to distinguish from the y variables appearing in the uncolored case (1). Similarly, we will use the variable names v and v in the setting of h -and p-curvature, respectively. In each of these settings, these deformation parameters play the role of (but are not literally equal to) a generating set of symmetric functions in the uncolored y-variables. The relation between the uncolored y deformation parameters and the v (and u) parameters is outlined below and discussed in detail in §4.6 and §4.7. It is best understood in the context of interpolation theory and is related to the geometry of the Hilbert scheme Hilb n (C 2 ).
1.4. Relation to previous work. A notion of curved Rickard complexes, and its application to link homology, has been studied by Cautis-Lauda-Sussan (CLS) in [CLS20]. Their construction starts at the level of a categorified quantum group U Q (sl m ), where m corresponds to the number of braid strands. As such, their construction does not see one alphabet X i per (left) braid boundary point 1 i m, but only the formal di↵erence alphabets X i X i+1 . Related to this, CLS consider only one deformation parameter u of weight q 2 t 2 . On the other hand, our construction starts at the level of complexes of singular Soergel bimodules and considers a family of deformation parameters for each strand, whose size is given by the strand label. These parameters account for higher degree homotopies of Rickard complexes, as proposed in [CLS20, Section 1.3]. We expect that our constructions can be lifted from singular Soergel bimodules to a suitable version of the categorified quantum group U Q (gl m ). For a basic comparison of the constructions in [CLS20] to our construction, consider the curved Rickard complex associated to the positive crossing (3). Retaining the notation from (4), the curvature considered by CLS takes the form (7) e 1 (X p2 ) e 1 (X p 0 1 ) z o u e 1 (X p1 ) e 1 (X p 0 2 ) z u u This expression has only a single deformation parameter u, that is weighted by scalars z o and z u , and curved complexes with this curvature encode homotopic actions of e 1 (X p2 ) e 1 (X p1 ) and e 1 (X p 0 1 ) e 1 (X p 0 2 ). By contrast, our curvature (6) has a family of deformation parameters {u o k } a k=1 associated to the over strand and a family {u u k } b k=1 associated to the under strand. Curved complexes with this curvature encode the homotopic actions of all e k (X) along each strand. Note that by specializing u o 1 = z o u, u u 1 = z u u and u o k = 0 = u u k for all k > 0 in (6), we recover (7). Hence, our link homologies encodes the CLS invariant as a special case, see §5.6 and 10.1.
Our multi-parameter curved Rickard complexes give rise to a variety of deformations of colored, triply-graded Khovanov-Rozansky link homology that appear closely related to the geometry of the Hilbert scheme Hilb m (C 2 ), and its isospectral analogue X m . Indeed, recall that the aforementioned work of Gorsky-Hogancamp [GH] uses the y-ified (uncolored) triply-graded homology to establish a precise relation between YH KR (L) and the isospectral Hilbert scheme X m . There, the variables {x 1 , . . . , x m , y 1 , . . . , y m } occurring in the curved Rickard complex assigned to an m-strand braid correspond to local coordinates on an open subset of X m . In particular, in lowest Hochschild degree the y-ified Khovanov-Rozansky homology of an uncolored unknot is
y] which (after extending scalars) is the coordinate ring of C 2 = Hilb 1 (C 2 ). In the colored case, we have
. . , e b (X b ), v 1 , . . . , v b ] after changing from the u to v variables, as in Remark 1.15. As explained in §4.7, these v-variables are interpolation coordinates for y-variables in terms of x-variables:
Comparing to [Hai01, Proposition 3.6.3], we see that the generators {e 1 (X b ), . . . , e b (X b ), v 1 , . . . , v b } of YH KR ( b ) can be understood as coordinates on an open a ne subset of Hilb b (C 2 ). We will comment further on relations between colored YH KR (L) and Hilbert schemes in the following section.
1.5. Curved colored skein relation, link splitting, and the Hopf link. Important applications of deformed link homologies derive from their controlled behavior under unlinking, i.e. their link splitting properties, see e.g. Batson-Seed [BS15]. Closer to the present paper, the connection between (uncolored) YH KR (L) and Hilbert schemes mentioned above relies on the link splitting map from the full twist braid to the identity braid. This map identifies the lowest Hochschild-degree summand of the homology of the (m, m)-torus link with an ideal
The (uncolored) link splitting map is determined by a deformation of the categorified HOMFLYPT skein relation. In order to understand link splitting behavior in the colored, triply-graded context, we develop a curved version of the colored skein relation that we introduced in the companion paper [HRW21]. The culmination of §6 is the following result. Herein, the brackets J K Y denote curved lifts of the Rickard complexes associated to shown tangled webs, and tw D denotes a twist in the di↵erential by a curved Maurer-Cartan element D (see §4.1 for a review of this terminology).
Theorem 1.16 (Corollary 6.20). Let ⌅ := {⇠ 1 , . . . , ⇠ b } be a set of exterior variables with wt(⇠ i ) = t 1 q 2i
, then there exists a homotopy equivalence (10)
of curved twisted complexes. Further, the curved Maurer-Cartan element D 1 is one-sided with respect to the partial order by the index s, and the right-hand side is a certain curved Koszul complex.
For obvious reasons, we will sometimes refer to the left-hand side of (10) as the complex of threaded digons, denoted by TD b (a). As the notation suggests, it is useful to view this complex as a function of the threading a-colored strand. One consequence of Theorem 1.16 is that TD b (a) ' 0 when a < b (in that case, the webs on the right-hand side correspond to the zero bimodule).
Remark 1.17. An equivalent formulation of the curved colored skein relation (10) that more closely resembles the usual HOMFLYPT skein relation (relating positive and negative crossings to their oriented resolution) is as follows:
Note that the left-hand side involves complexes that interpolate between a positive and negative crossing. The twists here have similar properties to those in Theorem 1.16.
Example 1.18. The Rickard complex for a crossing between a pair of 2-colored strands has the form
The subquotients with respect to the filtration indicated by the dotted lines (and colored black, blue, and green) are homotopy equivalent to the indicated complexes that appear on the left-hand side of (10). (Compare to the corresponding figure in [HRW21, Section 1], where the dashed components of the di↵erential do not appear.) Additional details can be found in Example 6.14.
In §7 we use the curved colored skein relation (10) to study splitting properties of the deformed, colored, triply-graded link homology. In particular, we obtain an explicit model for the colored link splitting map from the colored full twist on two strands to the identity braid in §7.2. In Theorem 7.14 we obtain a simultaneous splitting of the complex of threaded digons.
In §8, we begin the study of the colored full twist. Let us briefly recall the uncolored case, restricting to the lowest Hochschild degrees for brevity.
). There is a canonical map (the splitting map) Y FT m ! 1 m relating the y-ified Rouquier complexes of the full-twist and trivial braids. This map induces an injective map
of y-ified triply graded homologies, where T (m, l) denotes the (m, l) torus link (in particular T (m, 0) is the m-component unlink and T (m, m) is the closure of the full twist braid). Moreover, the image of the above map coincides with the ideal
A key component of this result is the parity property [EH19] enjoyed by T (m, m) (see also [Mel17,HM19]).
Paralleling the uncolored case, we conjecture that the parity property holds in the colored setting as well; see Conjecture 8.5. Under this assumption, we show that, in lowest Hochschild degree, the deformed, colored homology of the b-colored (m, m)-torus link embeds as an ideal J b in the algebra
In the uncolored case, the ideal J 1 m is precisely the ideal I 1 m mentioned above. Generalizing this, we pose the following (restated in the main text as Conjecture 8.15):
via the appropriate analogue of (9). Let T (m, m; b) denote the b-colored (m, m)-torus link, then the colored splitting map identifies the lowest Hochschild degree summand of YH KR T (m, m; b) with the ideal
where (X bi ) denotes the Vandermonde determinant.
(To see that I b is well-defined, we refer the reader to Lemma 8.13.)
The 1-strand case b = (b) of Conjecture 1.20 is trivial and the m-strand uncolored case b = 1 m was proven in [GH]. In §9, we prove this conjecture in the 2-strand case, i.e. for the (a, b)-colored Hopf link. First, in Proposition 9.4 we confirm that the colored Hopf link is indeed parity. We then embark on a complicated inductive journey, using the simultaneous colored skein splitting referenced above, which culminates in Theorem 9.33 with the verification of Conjecture 1.20 when m = 2. Along the way, we encounter specific Haiman determinants (reviewed in §2.4) and in Corollary 9.40 we give an explicit set of generators for I a,b using these elements.
Finally, in §10, we use Theorem 9.33 to extend our link splitting results from §7 to the case of arbitrary colored links. For certain specializations, this generalizes results obtained in [BS15,GH] to our setting, and recovers the colored link splitting result from [CLS20]. We also speculate on the interpretation of more-interesting specializations, making contact with Conjecture 1.25, which is stated in the following section.
1.6. Further conjectures. To conclude this (extended) introduction, we collect two particularly enticing conjectures. 1.6.1. Homology of cables. We propose a precise relation between deformed, colored, triply-graded homology and the deformed (uncolored) triply-graded homology of cables, focusing on the case of cabled knots, for ease of exposition. Recall the following result in the undeformed setting.
Theorem 1.21 ([GW19, Theorem 6.1 and Corollary 6.5]). Let K be a framed, oriented knot, and let K b denote the b-cable of K (a b-component link). Both the triply-graded Khovanov-Rozansky complex C KR (K b ) and the gl N Khovanov-Rozansky complex C KR N (K b ) carry an action of the symmetric group S b , up to homotopy, induced by braiding components of the cable. In the gl N case, the isotypic component corresponding to the trivial representation is equivalent to the b-colored gl
It is natural to consider the extension of this result for deformed triply-graded link homology. Interestingly, the most naïve extension of this result is false.
Example 1.22. Let U be the 0-framed unknot, and U b its b-cable. Then,
Remark 1.23. Note that there is an injective algebra map
By Lemma 4.33, the variables v k can be expressed as certain ratios of the form f (X,Y) (X) in which f (X, Y) is S b -antisymmetric, and the map (12) extends to an isomorphism
To generalize this pattern to general knots, we must reinterpret the process of adjoining ratios f/ (X) in which f 2 I 1 b . Motivated by the relation between I 1 b and full twists (Theorem 1.19), we formulate the following.
Conjecture 1.24. Let K be a framed, oriented knot, and let K denote K with framing increased by one. For each integer b 0 we consider a directed system of complexes (13)
in which the maps are inherited from the "bottom eigenmap" for the curved Rickard complex YC(FT b ) associated to the b-strand full-twist FT b . The (homotopy) colimit of this directed system is quasiisomorphic to the b-colored curved Rickard complex YC KR (K, b), as complexes of modules over
Let us comment on this conjecture. First, note that the b-cable of K and the b-cable of K di↵er by the insertion of a full twist braid FT b . Thus, the directed system (13) is inherited from a directed system [GH]. There are many choices one can make for the connecting maps. Indeed, results in [GH] show that
Among all such morphisms, the one corresponding to the Vandermonde determinant (X) is distinguished as the generator of cohomological degree zero. The associated morphism 1 b ! FT b in YSBim b (or in the undeformed category of complexes or Soergel bimodules) is referred to as the "bottom eigenmap", adopting terminology from [EH17]. Conjecture 1.24 is supported by our Theorem 9.33, i.e. our verification of Conjecture 1.20 in the case of colored Hopf links, since taking colimits of directed systems of the form (13) is akin to inverting the Vandermonde (X). We save explorations along these lines for future work. 1.6.2. Threaded digons and the Hilbert scheme. Finally, in a di↵erent direction, we propose that a generalization of the complex TD b (a) of threaded digons from Theorem 1.16 constructs a family of complexes desired in the Gorsky-Negut ¸-Rasmussen (GNR) conjecture [GNR21]. The m = 1 case of Conjecture 1.25 follows from Theorem 1.16. Indeed, we noted above that TD b (a) ' 0 when a < b, and Tr 1 (TD 1 (1) ⌦ Q[v 1 ]) can be explicitly identified with the identity braid on one strand using the right-hand side of (10). For item (ii), observe that the elementary symmetric function e b ( ) has similar behavior to the (conjectured) behavior of TD b ( ): it vanishes when the number of variables is smaller than b. Less-heuristically, results of Morton [Mor02] show that the complex Tr 1 (TD 1 (1, . . . , 1) ⌦ Q[v 1 ]) would categorify the sum (i.e. 1 st elementary symmetric function) of the Jucys-Murphy elements. The extension to general b is suggested by this, and Conjecture 1.24. Lastly, we mention that work of Elias [Eli] proposes a di↵erent approach to constructing E 1 . 1.7. Coe cient conventions. Throughout, we work over the field Q of rational numbers for simplicity. All of our results remain true over an arbitrary field of characteristic zero. We believe that all results should hold over the integers; however, the proof we present for Markov invariance of YH KR (L) uses the power-sum symmetric functions 1 k p k (X), thus requires working over a field of characteristic zero. Nonetheless, our 2-category Y(SSBim) is defined using e-curvature (equivalently, h -curvature), which allows for an integral version of this 2-category.
1.8. Organization of the paper. In Section 2 we recall background on symmetric functions, including the formalism of symmetric functions in the di↵erence of two alphabets and Haiman determinants. Section 3 introduces categorical background and sets up conventions for gradings and homological algebra necessary to introduce curved Rickard complexes of singular Soergel bimodules, which happens in Section 4. This section also includes a thorough discussion of di↵erent choices of deformation parameters with their relative advantages and relations between them. Section 5 develops the deformed, colored triply-graded link homology. Proceeding towards a study of link splitting, in Section 6 we obtain a curved colored skein relation, which we use in Section 7 to construct splitting maps. Section 8 introduces conjectures relating the deformed homology of colored torus links to Hilbert schemes, which we prove in the case of two strands in Section 9 by computing the homology of the colored Hopf link using the curved colored skein relation. In the final Section 10 we study the link splitting properties of the deformed colored triply-graded link homology.
Acknowledgements. This project was conceived during the conference "Categorification and Higher Representation Theory" at the Institute Mittag-Le✏er, and began in earnest during the workshop "Categorified Hecke algebras, link homology, and Hilbert schemes" at the American Institute for Mathematics. We thank the organizers and hosts for a productive working atmosphere. We would also thank Eugene Gorsky and Lev Rozansky for many useful discussions.
Funding. M.H. was supported by NSF grant DMS-2034516. D.R. and P.W. were supported in part by the National Science Foundation under Grant No. NSF PHY-1748958 during a visit to the program "Quantum Knot Invariants and Supersymmetric Gauge Theories" at the Kavli Institute for Theoretical Physics. D.R. was partially supported by NSF CAREER grant DMS-2144463: "Link homology-in Type A and beyond" and Simons Collaboration Grant 523992: "Research on knot invariants, representation theory, and categorification." P.W. was partially supported by the Australian Research Council grants 'Braid groups and higher representation theory' DP140103821 and 'Low dimensional categories' DP160103479 while at the Australian National University during early stages of this project. P.W. was also supported by the National Science Foundation under Grant No. DMS-1440140, while in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2020 semester.
In this section, we collect assorted background material on symmetric functions. The reader can safely skip §2.1, §2.2, and §2.3 and return whenever results or formulas from here are used in the later parts of the paper. We do, however, recommend a look at §2.4 for readers unfamiliar with Haiman determinants.
2.1. Symmetric functions. Symmetric functions play an important role throughout this paper. A more detailed exposition appears in [HRW21, Section 2.1].
Alphabets are finite or countably infinite sets that we denote by blackboard letters, such as X, Y etc. Given an alphabet X, the Q-algebra of symmetric functions on X will be denoted Sym(X). Symmetric functions on a finite alphabet will also be called symmetric polynomials. For pairwise disjoint alphabets X 1 , . . . , X r , we write Sym(X
are separately symmetric in each of the alphabets X i .
Definition 2.1. The elementary symmetric functions e j (X), complete symmetric functions h j (X), and power sum symmetric functions p j (X) are each defined via their generating functions as follows:
By convention e 0 (X) = h 0 (X) = 1 and p 0 (X) is undefined. In the case of countably infinite alphabets, we sometimes drop the alphabet from the notation and write E(t), H(t), and P (t), and e j , h j , and p j , for the functions introduced above.
The elementary and complete symmetric functions are related by the identity (14) H(X, t)E(X, t) = 1 , i.e.
( 1) j h i (X)e j (X) = k,0 8k 0 , and each are related to the power sum symmetric functions by the Newton identity:
We will work with the highly useful formalism of linear combinations of alphabets, see [HRW21, Definition 2.3]. In particular, for the generating functions in Definition 2.1, we have
for any a 1 , a 2 2 Q. A good illustration of how this formalism works is given by the following computation:
( 1) j h i (X)e j (X 0 ).
If A is a commutative algebra, we let N A ⌦ A be the kernel of the multiplication map A ⌦ A ! A. Equivalently, N is the ideal inside A ⌦ A generated by di↵erences a ⌦ 1 1 ⌦ a.
Remark 2.2. The Hochschild homology HH • (A) of a commutative algebra A is itself a gradedcommutative algebra. We have HH 0 (A) ⇠ = A and HH 1 (A) ⇠ = N/N 2 . More generally, each HH k (A) is a module over HH 0 (A) ⇠ = A.
We will focus on the case A = Sym(X) for a finite alphabet X and identify Sym(X) ⌦2 = Sym(X|X 0 ). Let N (X, X 0 ) Sym(X|X 0 ) be the ideal generated by elements of the form f (X) f (X 0 ). In this section, we will record various relationships between generating sets of N (X, X 0 ).
) is generated by any of the following:
Proof. Lemma 2.4 below shows that the families of elements e k (X) e k (X 0 ), h k (X X 0 ), h k (X) h k (X 0 ), and e k (X X 0 ) for 1 k a generate the same ideal. Since any symmetric function f can be written as a polynomial in the e k for various k, it is straightforward to check that this ideal is N (X, X 0 ). Indeed, one direction of containment is immediate, so it su ces to prove that any f (X) f (X 0 ) is in the ideal generated by the e k (X) e k (X 0 ). To see this, we may restrict to the case where f is a monomial in the e k and then induct on the degree of the monomial. If f = e k , the statement is trivial. Otherwise write f = e k0 f 0 for some k 0 and expand
which is in the ideal generated by the e k (X) e k (X 0 ) by the induction hypothesis. Finally, observe that p k (X) p k (X 0 ) = p k (X X 0 ), so each of the above symmetric functions involves the virtual alphabet X X 0 . The relations ( 16) and (19) then imply the second and third statement. ⇤ Lemma 2.4. Let X, X 0 be alphabets. For k 1, we have
and their inverse rewriting formulas
Note that we obtain another collection of identities by applying the algebra involution p k 7 ! p k , h k $ ( 1) k e k , or by swapping the roles of X and X 0 .
Proof. These equations are e ciently proved by the following manipulations of generating functions, i.e.
and
establish ( 16) -( 19). Equation ( 20) is proved by the computation
3. Hook Schur functions and h-reduction. Let X be an alphabet (or a formal linear combination of alphabets). If Y is an alphabet with |Y| c, then complete symmetric functions h N (X) can be expressed as Sym(X + Y)-linear combinations of complete symmetric functions h n (X) for n c. We refer to this process as h-reduction and describe it explicitly in Lemma 2.9.
Example 2.5. Consider the following identity in the polynomial ring
This allows us to write x a i as a Q[X] Sa -linear combination of monomials x k i with 0 k a 1.
The general case of h-reduction requires Schur functions associated to hook shapes.
Definition 2.6. For i, j 0 we write s (i|j) := s (i+1,1 j ) for the hook Schur functions, which can be described as the family of symmetric functions satisfying:
By convention s (i|j) = 0 if i < 0 or j < 0. We denote the two-parameter generating function of the hook Schur functions in an alphabet X by
Lemma 2.7. The two-parameter generating function of the hook Schur functions satisfies
Proof. The rearrangement H(t)E(u) = 1 + (t + u)S(t, u) is a generating function restatement of the characterizing identity
provided we interpret s (i| 1) and s ( 1|j) as zero. ⇤
In particular the algebra automorphism Sym(X) ! Sym(X) sending h k $ e k also sends S(t, u) 7 ! S(u, t), hence s (i|j) 7 ! s (j|i) . The following description of hook Schur functions is also useful.
Lemma 2.8. For i, j 0 we have
Proof. We will prove the first identity; the second follows by symmetry (apply the involution h k $ e k ). First, note that we have the generating function identity
Then we rewrite the hook Schur generating function as follows
which gives rise to the identity in the statement. ⇤ Lemma 2.9 (h-reduction). If Y has cardinality |Y| c, then for any X and r 1 we have
Before proving this lemma, we note the following special cases.
Corollary 2.10 (Reducing monomials). For all m a 1 and 1 i a, we have
( 1) a j s (m a|a j) (x 1 , . . . , x a )x j 1 i .
Proof. Take X = {x i }, Y = {x 1 , . . . , b x i , . . . , x a }, and c = a 1 in Lemma 2.9. ⇤ Corollary 2.11. If X and X 0 are alphabets of cardinality c, then we have
Proof. Take X 7 ! X X 0 and Y 7 ! X 0 in Lemma 2.9. Note that the i = 0 summand in (21) is zero in this case since the Young diagram for the hook (r 1|c) has c + 1 rows, which exceeds the cardinality of X. ⇤ Proof of Lemma 2.9. We begin by proving the Y = ? case. Lemma 2.8 gives us the identity ( 1
Summing up over all indices i, j 0 with i + j = c yields
= h c+r (X) , which is the |Y| = 0 case of the lemma. Now, we deduce the general identity. First, we compute
( 1) i h c i+r (X + Y)e i (Y) .
We have used the fact that e i (Y) = 0 for i > c |Y|. Next, we apply the Y = ? case of the lemma to rewrite h c i+r (X + Y). The resulting identity is
Finally, we fix k and sum over all indices i, j with i + j = c k to obtain
2.4. Haiman determinants. Our description of the deformed, colored homology of (m, m)-torus knots (conjectural, when m > 2) in §8 and §9 relies on certain (anti)symmetric polynomials constructed using determinants.
Let R be a commutative Q-algebra and consider a tuple (f 1 , . . . , f N ) of elements f i 2 R. Let f i,j denote the element of R ⌦N given by f
where f i occurs in the j-th position, and set
Remark 2.12. We will consider this construction in the special cases
In such cases, we will identify R ⌦N with the polynomial ring Q
respectively, where X = {x 1 , . . . , x N } and Y = {y 1 , . . . , y N }. When the f i 2 R are monic monomials, we refer to the elements constructed in (22) as Haiman determinants, due to their appearance in [Hai01, Section 2.2].
It will be useful to introduce the following short-hand.
Definition 2.13. Let 1 • • • N 0 be a weakly decreasing sequence of non-negative integers of length N , then we associate to it an N -tuple of monomials in x as follows:
Convention 2.14. Given a positive integers N l and a partition = ( 1 , . . . , l ) with l parts, we will sometimes view as a weakly decreasing sequence of length N by appending to it N l zeros.
Example 2.15. We have M N (;) = (x N 1 , . . . , x, 1) and thus hdet(M N (;)) = (X), where (X) = Q 1i<jN (x i x j ) is the usual Vandermonde determinant. More generally, for a partition = ( 1 , . . . , l ) with l N , Jacobi's bialternant formula implies
where s (X) denotes the Schur polynomial associated to in the alphabet X of cardinality N . Example 2.17. For S = {x 2 , x, 1, y}, the Haiman determinant is
In this section we review background on homological algebra and singular Soergel bimodules.
3.1. Categories of coe cients. We will assume familiarity with the notion of a category enriched in a symmetric monoidal category R. In particular, if B is R-enriched, then Hom B (X, Y ) is an object of R for all X, Y 2 B, and the composition of morphisms in B is given by morphisms in R
where ⌦ is the monoidal structure in R. We view R as the category of coe cients for B. In this section, we discuss the various categories of coe cients that will appear in this paper.
We let K denote the category of finite-dimensional Q-vector spaces and let K denote the category of all Q-vector spaces. A category is Q-linear if it is enriched in K.
Let be an abelian group and let K[ ] denote the category of finite-dimensional -graded Q-vector spaces. An object of this category is a Q-vector space of the form L 2 M where each M is finitedimensional, and M = 0 for all but finitely many 2 . Morphism spaces in this category are the -graded Q-vector spaces
Hom K (M 0 , N 0 + ) . Now, suppose is equipped with a symmetric bilinear form h , i : ⇥ ! Z/2Z. This determines a monoidal structure on K[ ], given on objects by
The monoidal structure just defined is symmetric, with braiding given by
At times, we will relax the finiteness conditions that define K[ ], and let KJ K denote the category of -graded Q-vector spaces. This category is again symmetric monoidal, via the same formulae. Now, if is additionally equipped with an element d 2 satisfying hd, di = 1 2 Z/2Z, then we can consider the category K[ ] dg of -graded complexes with di↵erentials of degree d. (Note that we suppress h , i and d from this notation, as they will be clear in context.) Objects in K[ ] dg are pairs (X, ) where X 2 K[ ] and 2 End d K[ ] (X) satisfies 2 = 0, and morphism spaces are the complexes
This category is the prototypical di↵erential -graded category. It comes equipped with a tensor product
The category KJ K dg is defined similarly.
The most important for our uses are introduced in the following examples.
Example 3.2. Let = Z with trivial bilinear form hi, ji = 0. We will identify the group algebra
with the algebra of Laurent polynomials
and similarly indicate the choice of coordinate q in Z by writing Z = Z q . Paralleling this notation, we denote
In both cases, we also let q denote the grading shift functor, defined by (qM ) i = M i 1 . These category appear when we consider graded rings, modules, and bimodules. Note that the dg categories K[ ] dg and KJZ q K dg are trivial for this choice of h , i, since there is no element d 2 Z with hd, di = 1.
Example 3.3. Let = Z t with hj, j 0 i = jj 0 and d = 1, then K[ ] dg is the usual category of bounded complexes of finite-dimensional Q-vector spaces (with the cohomological convention for complexes, i.e. di↵erentials have degree +1).
Example 3.4. Combining Examples 3.2 and 3.3, let = Z q ⇥ Z t with h(i, j), (i 0 , j 0 )i = jj 0 . On the level of group algebras, we have Q[Z q ⇥ Z t ] = Q[q ± , t ± ], and we denote the categories of graded Q-vector spaces similarly:
As above, we regard monomials q i t j as a grading shift functors, via (q i t j M ) k,l = M k i,l j . Taking d = (0, 1) gives the dg categories
which appear when we consider complexes of graded modules or bimodules over Z q -graded rings.
Finally, triply-graded Khovanov-Rozansky homology takes values in the following symmetric monoidal category of triply-graded Q-vector spaces.
Example 3.5. Let = Z a ⇥ Z q ⇥ Z t . As in Examples 3.3 and 3.4, the t-grading is cohomological, so we take d = (0, 0, 1). The a-grading also has a cohomological flavor, but is independent from t. This is reflected in our choice of symmetric bilinear form:
The resulting categories of triply-graded Q-vector spaces will be denoted K[a ± , q ± , t ± ] and KJa ± , q ± , t ± K, and complexes therein by K[a ± , q ± , t ± ] dg and KJa ± , q ± , t ± K dg . These categories occur when we consider (co)homological functors, such as Hochschild (co)homology, applied to complexes of graded bimodules.
Convention 3.6. In (di↵erential) -graded categories for as in Examples 3.2, 3.4, and 3.5, we will typically indicate the degree of a morphism multiplicatively, by indicating its weight. For example, wt(f ) = a i q j t k means that the Z a ⇥ Z q ⇥ Z t -degree of f is (i, j, k).
Retaining notation from the previous section, the most important di↵erential -graded categories are categories of (curved) complexes. To begin, suppose A is a Q-linear category. Let A[t ± ] denote the category whose objects are sequences (X i ) i2Z with X i 2 A and X i = 0 for all but finitely many i, and morphisms
In other words Hom A[t ± ] (X, Y ) is the Z-graded Q-vector space spanned by homogeneous multimaps
The dg category of bounded chain complexes C(A) can be built from A[t ± ] in a standard way. Objects of C(A) are complexes: pairs (X, ) where X 2 A[t ± ] and 2 End 1 A[t ± ] (X) with 2 = 0. The morphism spaces in C(A) are the complexes
Here, we equip the latter with the Z/2Z-valued symmetric bilinear form h( , j), ( 0 , j 0 )i ⇥Z = h , 0 i + jj 0 and let d = (0, 1). In other words, when forming categories of complexes over -graded categories, our di↵erentials have degree (0, 1) 2 ⇥ Z t . In this way, C(A) is a di↵erential ⇥ Z t -graded category.
We will define the category of curved complexes in a similar fashion. Informally, the basic idea is to replace the equation 2 = 0 with the equation 2 = F , where F is an element of the center of A[t ± ]. (We will see below that this does not work sensu stricto, but that there is any easy fix.) Definition 3.8. The center of a Z t -graded category B is the Z-graded algebra Z(B) of natural transformations from the identity functor of B to itself. Precisely, a degree k element of Z(B) is an assignment
Proof. Each object X 2 A[t ± ] comes equipped with a family of idempotent endomorphisms e i which project on to the i-th object X i . Any central element must commute with these e i , hence must be degree zero. Now, a degree zero central element F acting on X is completely determined by how it acts on X 0 , since such an F must commute with the degree k map relating X and its shift t k X. ⇤
We now run into a slight snag: 2 has degree two, yet the only degree two element of Z(A) is zero. In order to remedy this, the two standard approaches are to collapse the grading from Z to Z/2Z or formally extend scalars from Q to an appropriate graded ring. We prefer to preserve the Z-grading, hence extend scalars as follows.
Definition 3.10. If B is Z t -graded category and R is a Z t -graded ring, then we let B ⌦ R denote the Z t -graded category with the same objects as B, and morphisms
Definition 3.12 (Curved complexes). Let A be a Q-linear category, R a Z t -graded ring, and F a degree two element of Z(A) ⌦ R. Let C F (A; R) denote the dg category whose objects are pairs (X, ) with X 2 A[t ± ] and 2 End 1
When the ring R is clear from context, we will omit it from the notation and denote the category C F (A; R) simply by C F (A).
Remark 3.13. If A is -graded, then both C(A) = C 0 (A) and C F (A) are dg ⇥ Z t -graded categories.
3.3. Frobenius extensions. Frobenius extensions between rings of partially symmetric polynomial rings will play an important role in defining morphisms between singular Soergel bimodules. Definition 3.14. A Frobenius extension is an inclusion of commutative rings ◆ : A ,! B such that B is free and finitely generated as an A-module, together with a non-degenerate A-linear map @ : B ! A, called the trace. Here, non-degeneracy asserts the existence of A-linear dual bases {x ↵ } and {x 0 ↵ } for B such that @(x ↵ x 0 ) = ↵, . For a graded Frobenius extension between graded rings, we require ◆ to be grading preserving and @ and the dual bases to be homogeneous.
Fix N > 0, and let R := Q[x 1 , . . . , x N ] be the polynomial ring, Z q -graded by declaring deg q (x i ) = 2. Given a parabolic subgroup S a = S a1 ⇥ • • • ⇥ S am of the symmetric group S N , we let R a ✓ R denote the ring of polynomials invariant under the action of S a . Note that R b ⇢ R a if and only if S b S a .
Here, the transposition s i acts by swapping variables x i and x i+1 in g.
Proof. See [Wil08, Theorem 3.1.1 and Corollary 2.14]. ⇤ More specifically, the following two examples will be used throughout.
Example 3.16. Let X = {x 1 , . . . , x N } be an alphabet with deg q (x i ) = 2. Then Sym(X) ,! Q[X] is a graded Frobenius extension of rank N ! with non-degenerate trace given by:
Here, Alt denotes the antisymmetrizer
by the permutation of variables and (X) = Q 1i<jN (x i x j ) denotes the Vandermonde determinant. A Sym(X)-linear basis of Q[X] is given by the monomials
N 1 where 0 n i N i. We have
The basis dual to the monomial basis above has elements
Example 3.17. Let X 1 = {x 1 , . . . , x a } and X 2 = {x a+1 , . . . , x a+b } be alphabets with
is a graded Frobenius extension of rank ab with trace given by the Sylvester operator:
A Sym(X 1 +X 2 )-linear basis of Sym(X 1 |X 2 ) is given by the Schur functions s (X 1 ) indexed by partitions with 1 b having at most a parts (i.e. the Young diagram for fits inside the a ⇥ b box). We denote the set of such partitions by P (a, b). The dual basis is then given by the signed Schur functions ( 1) | b | s b (X 2 ) where b 2 P (b, a) denotes the dual complementary partition. The sum of products of pairs of basis and dual-basis elements appears when expanding the Schur function of the a ⇥ b box in the di↵erence of alphabets:
3.4. Singular Soergel bimodules and webs. The Q-linear monoidal 2-category of singular Soergel bimodules can be assembled from certain full subcategories of the categories of graded bimodules between rings of partially symmetric polynomials over Q in variables of degree two. These subcategories are generated by certain induction and restriction bimodules under bimodule composition ?, direct sum, taking direct summands, and grading shifts. We now unpack this description, using the notation established in §3.3.
Consider the 2-category Bim N wherein • Objects are tuples a = (a 1 , . . . , a m ) with a i 1 and
• 2-morphisms are homomorphisms of graded bimodules. Horizontal composition is given by tensor product over the rings R a , and will be denoted by ?. The composition of 2-morphisms is the usual composition of bimodule homomorphisms. We will write 1 a := R a for the identity bimodule, saving the notation R a for the rings themselves. We denote the number of entries in a by #(a), i.e. for a = (a 1 , . . . , a m ) we have #(a) = m. The collection Bim := {Bim N } N 0 is a monoidal 2-category, with (external) tensor product
given on objects by concatenation of tuples: (a 1 , . . . , a m1 )⇥(b 1 , . . . , b m2 ) := (a 1 , . . . , a m1 , b 1 , . . . , b m2 ). and on 1-and 2-morphisms by tensor product over Q.
A singular Bott-Samelson bimodule is, by definition, any (R a0 , R ar )-bimodule of the form
for some sequence of rings and subrings R a0 R b1 ⇢ • • • R br ⇢ R ar , or a grading shift thereof. In particular, we have the merge and split bimodules (terminology explained below) given by
Here, q k denotes a shift up in degree by k, and `(a) = P i ai 2 denotes the length of the longest element in S a . Any singular Bott-Samelson bimodule is isomorphic to a shift of the horizontal composition of some sequence of merges and splits.
Definition 3.18. The 2-category SSBim N of singular Soergel bimodules is the smallest full 2-subcategory of Bim N that contains the singular Bott-Samelson bimodules and is closed under taking shifts, direct sums, and direct summands. The collection {SSBim N } N 0 assembles to form a full monoidal 2subcategory of Bim that we denote SSBim. We will denote the 1-morphism category in SSBim All other singular Bott-Samelson bimodules can be obtained from these merge and split bimodules using grading shift, horizontal composition ?, and tensor product ⇥. Graphically, ? corresponds to the horizontal glueing of diagrams along a common boundary, and ⇥ corresponds to vertical stacking of diagrams. We will refer to the graphs built from the diagrams in (25) via ? and ⇥ as webs, which we always understand as mapping from the labels at their right endpoints to those at their left.
All maps between singular Bott-Samelson bimodules can be built using ? and ⇥ from the following elemental maps 5 (which encode the Frobenius extension structures discussed in §3.3):
(1) Decoration endomorphisms
(3) Digon collapse morphisms:
of weight q ab . Here @ a,b is the Sylvester operator from Example 3.17. (4) Zip morphisms:
of weight q ab . Here, the latter is viewed as an element in Sym(X
In the cases (2)-(5), the degree/weight of the morphism is determined by the shift present in the definition of the merge bimodule in (24).
The following webs will play an important role in the following:
(26)
In the second diagram, we establish conventions for the alphabets associated with each edge; they have cardinalities as given by the corresponding labels in the first diagram. This will aid in specifying decoration endomorphisms of the corresponding bimodules. The notation F (l) E (k) 1 a,b is borrowed from the theory of the categorified quantum group for sl 2 , whose extended graphical calculus [KLMS12] can also be used to encode 2-morphisms between certain singular Bott-Samelson bimodules, see [HRW21, Proposition 2.18].
5 This follows e.g. from the results in [Web17], which combine with [QR16] to show that the n ! 1 limit of the gl n foam 2-category defined in the latter is equivalent to the 2-category of singular Bott-Samelson bimodules.
For example, we will use morphisms:
+ r := ( 1) b k l k • r : a+k l b k+l l a+k b k k b a ! a+k l b k+l l 1 a+k 1 b k+1 k 1 b a (28) r := ( 1) a+b+k+l 1 l k • r : a+k l b k+l l a+k b k k b a ! a+k l b k+l l+1 a+k+1 b k 1 k+1 b a
each of which has degree a b + k l + 1 + 2r. Here, the signedfoot_0 extended graphical calculus diagrams can be interpreted as encoding a horizontal (dashed) slice through a movie of webs that describes the morphism of singular Soergel bimodules:
The morphism ( 27) is given by reading left-to-right with the top arrow labels and (28) is given by reading right-to-left with the bottom arrow labels. The unlabeled isomorphisms are composites of (co)associativity isomorphisms for merge (split) bimodules. The variable x is associated with the 1labeled edge, and the extended graphical calculus diagram encodes the intersection with the dashed slice.
We will occasionally wish to encode such morphisms using perpendicular graphical calculus, which corresponds instead to taking a vertical slice, e.g.
cr?cr ! col?col 1 1 ⇠ = 1 1 un ! zip 1 x r ! x r 1 col ! cr and in this calculus the morphisms (27) and (28) are given by (29) + r := a+k b k 1 • r and r := a+k b k 1 • r
.
Note that some of the web edges are not visible in this calculus. We will denote decoration endomorphisms on such edges by drawing the endomorphism in an appropriate region, e.g. using the conventions in (26) we have
All of the relations used in the sequel between perpendicular graphical calculus diagrams can be deduced either from the corresponding relations in extended graphical calculus, or from relations in the gl n foam 2-category defined in [QR16]. As mentioned above, the latter is known to describe the 2-category of singular Bott-Samelson bimodules in the n ! 1 limit (see e.g. [QRS18, Section 5.2], [Wed19, Proposition 3.4], or [HRW21, Appendix A]). See Figure 1 for a graphical depiction of such a foam, together with the slices giving ( 27) and (29).
, Figure 1. The foam corresponding to + 0 and its slices that yield the corresponding extended and perpendicular graphical calculus diagrams, respectively.
There exist two ?-and -contravariant duality functors on SSBim:
which satisfy the adjunctions
Since the bimodules a+b M a,b and a,b S a+b generate SSBim as a monoidal 2-category, this duality can be succinctly recorded as follows:
Further, the relevant (co)unit morphisms are given by the digon creation/collapse and (un)zip morphisms. ⇤
We will be interested in complexes of singular Soergel bimodules. The natural setting for their study is the dg 2-category of singular Soergel bimodules, which is obtained by taking the dg category of complexes is each Hom-category of SSBim. Definition 3.20. Let C(SSBim) be the monoidal dg 2-category with the same objects as SSBim, and wherein the 1-morphism category a ! b equals C( b SSBim a ).
In other words, 1-morphisms in C(SSBim) are complexes of singular Soergel bimodules and 2morphism spaces in C(SSBim) are Hom-complexes of bimodule maps. Horizontal composition and external tensor product of 1-morphisms is defined as usual, e.g.
The components of the horizontal composition and external tensor product of 2-morphisms are defined using the Koszul sign rule. For example, if f 2 Hom C(SSBim) (X, X 0 ) and g 2 Hom C(SSBim) (Y, Y 0 ) are given, then f ? g is defined component-wise by:
Note that the (graded) middle interchange law:
holds in C(SSBim).
Convention 3.21. Since the 1-morphism categories of SSBim are Z q -graded, the 1-morphism category C( b SSBim a ) is enriched in KJq ± , t ± K dg . We will use the convention that deg(f ) = (i, j) means f has q-degree (or "Soergel degree") i and cohomological degree j. Further, the singlyindexed Hom-space Hom k C(SSBim) (X, Y ) always refers to cohomological degree, while the doubly-indexed Hom
As in Convention 3.6, we will typically indicate these degrees multiplicatively by writing wt(f ) = q i t j , and will also use the variables q, t to denote the corresponding shift functors. Thus, for example, wt( X ) = q 0 t 1 = t.
3.5. Colored braids and Rickard complexes. We next recall the complexes of singular Soergel bimodules assigned to colored braids. In this paper, the set S of colors we will be Z 1 . Let Br m denote the m-strand braid group, which acts on S m by permuting coordinates (this action factors through the symmetric group S m ).
Definition 3.22. The S-colored braid groupoid Br(S) is the category wherein
• objects are sequences (a 1 , . . . , a m ) with a i 2 S, m 1, and • morphisms are given by
Morphisms in Br(S) are called S-colored braids and elements in Hom Br (a, b) will be denoted by b a , or occasionally by b or a since the domain/codomain determine one another. We will write Br m (S) for the full subcategory of Br(S) with objects having exactly m entries.
The colored braid groupoid is generated by the colored Artin generators i : (a 1 , . . . , a i , a i+1 , . . . , a m ) ! (a 1 , . . . , a i+1 , a i , . . . , a m ) which, when composable, satisfy relations analogous to the usual (type A) braid relations. A colored braid word is a sequence of colored Artin generators and their inverses. We say that a colored braid word ( ) a represents the corresponding product of colored Artin generators in Br(S).
We now use the colored Artin generators to associate complexes C( b a ) in SSBim to Z 1 -colored braid words b a . Here, it is convenient to abuse notation by writing:
(Note that C( ) alone does not denote a well-defined complex.)
depending on whether a b or a b, respectively. Analogously, we also have:
As graded objects, we identify
where
For the Artin generator i of the braid group Br m and a = (a 1 , . . . , a m ), we set:
. This assignment extends to arbitrary colored braid words using horizontal composition. Given a braid word = "r ir
i1 )1 a the Rickard complex assigned to the colored braid a . This terminology is justified by the following proposition.
Proposition 3.25 ([HRW21, Proposition 2.25]). The complexes C( "1 i1 • • • "r ir )1 a satisfy the (colored) braid relations, up to canonical homotopy equivalence. ⇤ Rickard complexes of colored braids extend to invariants of braided webs (using horizontal composition and external tensor product), since they satisfy the following fork-slide and twist-zipper relations.
In this section, we introduce a dg 2-category of curved complexes of singular Soergel bimodules, and define curved Rickard complexes as certain special 1-morphisms. 4.1. Perturbation theory for (curved) complexes. In the following, we will need the notion of a twist. Suppose that
(As a special case, we could have
) is an object of C F1 (A), and the object (X, + ↵) 2 C F1+F2 (A) is said to be a twist of (X, ). We set tw ↵ ((X, )) := (X, + ↵)
and will often simply write the former as tw ↵ (X) when the di↵erential on X is understood. Note that the element ↵ satisfies the Maurer-Cartan equation with curvature:
We will refer to ↵ as a (curved) Maurer-Cartan element, or, by abuse of terminology, as a twist.
In various places, we will need to promote a homotopy equivalence X ' Y to a homotopy equivalence between twists tw ↵ (X) ' tw (Y ). This is the subject of homological perturbation theory. For our purposes, the following result su ces; see e.g. [Mar01,Hog] for a more-thorough discussion.
determine a homotopy equivalence tw ↵ (X) ' tw (Y ) in C F1+F2 (A), where := f ↵ e g. ⇤ Remark 4.2. In all of our applications of Proposition 4.1, invertibility of id X + ↵ k X will follow since ↵ k X is nilpotent. For example, this holds when k X acts summand-wise on a finite one-sided twisted complex tw ↵ (X). Recall that the latter means that
with X i = 0 for all but finitely many i 2 Z, and the components ↵ i,j : X j ! X i of the twist ↵ satisfy ↵ i,j = 0 for i j. Further, invertibility of id X + ↵ k X implies that id X + k X ↵ is also invertible, so no further assumptions are necessary to define e g. Although the homotopy k Y does not appear in the definition of e f or e g, it would appear in the formula for the perturbed homotopy f
A preliminary discussion on curvature. Our curved complexes of singular Soergel bimodules will have curvatures modeled on strand-wise curvature 7 of the form
which we refer to as h -curvature. Here X, X 0 are alphabets of cardinality a and v 1 , . . . , v a are deformation parameters with wt(v k ) = q 2k t 2 . However, in order to define horizontal composition in our 2-category of curved complexes of singular Soergel bimodules, it will be auspicious to work with a di↵erent curvature that is modeled on strand-wise e-curvature, which is of the form
As it turns out, we can regard these curvatures as equivalent after an appropriate change of variables.
Definition 4.3. Let X be an alphabet of cardinality a and consider collections of deformation parameters
It is an easy exercise using (15) to verify that the formulae in (37) are mutually inverse.
for all 1 k a, where X 0 is the alphabet 1 ⌦ X.
Proof. We compute
Note that the k = l term in this sum is zero. On the other hand, Lemma 4.4 gives
Note that Lemma 4.4 and Corollary 4.5 remain true (with the same proof) if we extend U and V to include deformation parameters u 0 , v 0 , with weights q 0 t 2 , which we assume are related via the obvious extension of (37) to the case k = 0. Thus we also have the following identity, which is the "k = 0 case" of Corollary 4.5. (Alternatively, this could be established by a straightforward computation.) Corollary 4.6. We have
Curved complexes over SSBim. We now introduce the monoidal dg 2-category of curved complexes of singular Soergel bimodules. Informally, this 2-category is formed via the following procedure. First, we consider a 2-subcategory consisting of 1-morphisms X : a ! b in C(SSBim) where b is obtained by permuting the indices of a. In particular, all Rickard complexes give 1-morphisms in this 2-category. Next, in each Hom-category we adjoin #(a) alphabets
i=1 of formal variables via Definition 3.10. Finally, we pass to a certain category of curved complexes in each Hom-category, for a choice of curvature that e.g. encodes the homotopies that "slide" the action of symmetric polynomials on the left boundary along a strand of a Rickard complex to the right.
We now make this informal description precise.
Definition 4.7. Fix an integer m 0 and let Y(SSBim, m) be the 2-category wherein:
• objects are pairs (a, ) where a = (a 1 , . . . , a m ) is an object in SSBim and 2 S m .
For each such object ((a 1 , . . . , a m ), ), we introduce an alphabet
with wt(u i,r (a, )) = q 2r t 2 . We also write U i (a, ) for the subalphabet with fixed i 2 {1, . . . , m}. Further, we place an equivalence relation on objects by declaring (a, )
, then we will identify the associated U-variables u i,r (a, ) = u i,r (b, ⌧) and drop the dependence on (a, ) from the notation.
Here we identify the rings acting on the left and right (boundary
In such expressions, we will sometimes suppress the summation limits depending on colors a (i) by summing over all r 1 and declaring u i,r := 0 for r > a (i) . We let Y(SSBim)
Remark 4.8. The appearance of permutations in the objects of Y(SSBim) may appear surprising. Below, when we assign 1-morphisms in Y(SSBim) to braids, the permutations in the (co)domain objects will encode a numbering of the strands in the braid. E.g. if the domain object is (a, ), then (i) = j tells us that the strand meeting the j th boundary point on the right of the braid is the i th strand in this numbering.
We have written the di↵erential on a 1-morphism (X, tot X ) in Y(SSBim) using the superscript "tot" in order to emphasize the fact that tot X decomposes canonically (and uniquely) into a sum of terms:
where X lives in End C(SSBim) (X) and X lives in the ideal End C(SSBim) (X) ⌦ Q[U] >0 generated by polynomials in U with zero constant term. Hence, (X, X ) defines a complex of singular Soergel bimodules, and (X, X + X ) = tw X ((X, X )). It will frequently be useful to decompose 2-morphisms in Y(SSBim) according to their U-degree zero parts and their "strictly positive U-degree" parts, according to the following definition.
Convention 4.10. Henceforth, we will write 1-morphisms in Y(SSBim) in the form tw X (X) where
In this language, the composition of 1-morphisms takes the form
(We will show in Lemma 4.12 that it is well-defined.) Further, we will sometime embellish this notation as tw X ( b,⌧ X a, ) or tw X ( b X a ) when we wish to emphasize the data specifying the objects. Remark 4.11. If tw X ( b,⌧ X a, ) is a 1-morphism in Y(SSBim), then the linear part of X is a sum of terms of the form i,r u i,r , where i,r 2 End 2r, 1 (X) satisfies
. This implies that the endomorphisms i,r necessarily square to zero and pairwise anti-commute, so
e r (X 0 (i) ) is null-homotopic on tw X (X). This conclusion holds for non-strict morphisms as well, as can be seen by di↵erentiating the equation ( X + X ) 2 = F with respect to the variable u i,r .
over finitely many triples (i, r, k) where
If f is homogeneous of weight q l1 t l2 , then we have (42) wt(f i,r,k ) = q l1+2 P j kj rj t l2 2 P j kj , since u k i,r has weight q 2kr t 2k . In particular, since the curved complexes tw X (X) in Y(SSBim) are bounded, this implies that the curved Maurer-Cartan element X is nilpotent.
We now elaborate on the 2-categorical structure on Y(SSBim). First, we check the following.
Lemma 4.12. Horizontal composition is well-defined on Y(SSBim).
Proof. It su ces to show that given composable 1-morphisms tw X ( a, X a 0 , 0 ) and tw X ( a 0 , 0 X a 00 , 00 ), the twist X?Y = X ? id Y + id X ? Y satisfies the conditions in Definition 4.7. The only non-immediate check is to compute its curvature. Let m = #(a) = #(a 0 ) = #(a 00 ), and note that a (i) = a 0 0 (i) = a 00 00 (i) for all 1 i m. We then have
as desired. Here, we use that [( X + X ) ? id Y , id X ? ( Y + Y )] = 0 by (31). ⇤
Next, we note that the external tensor product ⇥ can be extended from C(SSBim) to Y(SSBim) in a straightforward manner. On the level of objects, we have
i.e. using the standard inclusion S #(a1) ⇥ S #(a2) ,! S #(a1)+#(a2) .
On the level of 1-morphisms, if tw
where, analogous to (30), we have
The verification that the twist 1 ⇥id X2 + id X1 ⇥ 2 satisfies the curvature condition in Definition 4.7 is similar to the verification in Lemma 4.12, thus we omit it.
Finally, the external tensor product of 2-morphisms in Y(SSBim) is defined via the Q[U]-linear extension of the external tensor product in C(SSBim). Explicitly, for homogeneous f :
Thus defined, ⇥ endows Y(SSBim) with the structure of a monoidal dg 2-category.
Remark 4.13. There is a monoidal dg 2-functor Y(SSBim) ! C(SSBim) that forgets the permutations, sets all variables u i,r equal to zero, and sends tw X (X) 7 ! X. We refer to tw X (X) as a curved lift or curved deformation of the complex X, and similarly for 2-morphisms in Y(SSBim).
Our next result allows us to upgrade homotopy equivalences between 1-morphisms in C(SSBim) to equivalences in Y(SSBim), provided we are given a curved lift of one of the 1-morphisms.
Proposition 4.14. Suppose that tw X (X) is a 1-morphism in Y(SSBim), and f : X ! Y is a homotopy equivalence in C(SSBim), then there exists a curved lift tw Y (Y ) of Y and an induced homotopy equivalence e f : tw
Proof. We use Proposition 4.1, and its notation. The present result is an immediate consequence, applied to X 2 C(SSBim) with ↵ = X , once we have confirmed that X k X 2 End C(SSBim)[U] (X) is nilpotent. To see the latter, note that since X = 0 mod hu i,r i, the same holds for X k X . This implies that ( X k X ) `= 0 mod hu i,r i `. Writing ( X k X ) `in components as in (41), this in turn implies that there exists ` 0 so that ( X k X ) `= 0 by equation ( 42), since X is bounded. ⇤ Lemma 4.15. Suppose that (a, ) and (b, ⌧) are objects in Y(SSBim) such that #(a) = #(b) and
and all symmetric functions f , then there exists a curved lift tw L ( b,⌧ L a, ) of L in Y(SSBim) which is unique up to homotopy equivalence.
Proof. The existence statement follows via obstruction-theoretic arguments analogous to those in [GH, Section 2.10]. We elaborate on the details, since such constructions are crucial for us.
Abbreviate by writing E(L)
. We first claim that E(L) is complete (in the graded sense) with respect to the filtration by ideals
In other words, given a sequence of elements f k 2 m k (L), each of which is homogeneous of cohomological degree `(not depending on k), we claim that the infinite sum
Indeed, each f k may be written as a sum of terms h⌦g where g 2 Q[U] is a polynomial in which each monomial has total U-degree k (hence cohomological degree 2k) and where f 2 End C(SSBim) (L), which necessarily has cohomological degree ` 2k. By the definition of C(SSBim), L is a bounded complex in C(SSBim), so such terms are zero for k 0. Thus P 1 i=0 f i is in fact a finite sum, establishing the claim. Now, establishing the existence of a curved lift of L amounts to constructing an element
is the di↵erential on the dg algebra E(L) and F is the curvature element in (38). Without loss of generality, we assume that k is U-homogeneous of degree exactly k, and then write equation (44) in terms of its U-homogeneous components, obtaining the family of equations ( 45)
We construct such k by induction. For the base case k = 1, the hypotheses on L guarantee that F ' 0 in E(L), which gives the existence of 1 . Now, let k > 1 and assume that 1 , . . . , k 1 that satisfy (45) have been constructed. Consider the element
, which is the obstruction to defining k . A straightforward computation using (45) shows that O k is closed. Additionally, we can write O k = P g h g ⌦ g in which g runs over all monomials of total U-degree k, and therefore each h g 2 End C(SSBim) (L) is a closed element of cohomological degree 1 2k. Since L is invertible, we have a quasi-isomorphism of dg algebras
In particular, neither of these algebras have cohomology in negative degrees, thus the obstruction O k vanishes up to homotopy. This gives the existence of k and completes the proof of the existence of the curved lift tw (L).
To show uniqueness of the lift up to homotopy equivalence, let X = tw (L) and Y = tw 0 (L) be two curved lifts of L and X _ = tw 00 (L _ ) a curved lift of an (up to homotopy) inverse of L. It follows that X _ ? Y is a curved lift of a complex that is homotopy equivalent to 1 a in C(SSBim). Hence, Proposition 4.14 implies the existence of a homotopy equivalence
for some twist 000 . However, since End k C(SSBim) (1 a ) = 0 for k < 0, (42) implies that 000 = 0. Thus X _ ? Y ' 1 a, , which shows that Y is a two-sided inverse to X _ up to homotopy. The same argument shows that X is a two-sided inverse to X _ up to homotopy. Thus X ' Y by uniqueness of two-sided inverses. ⇤ Remark 4.16. Using obstruction-theoretic arguments, it is possible to strengthen the uniqueness statement in Lemma 4.15 as follows. Suppose, that X 2 C(SSBim) is invertible and that tw (X) and tw 0 (X) are two curved lifts of X in Y(SSBim). Then, in fact, there is a (closed) isomorphism ' : tw (X) ! tw 0 (X) of curved complexes of singular Soergel bimodules such that
4.4. Curved Rickard complexes. Our first goal is to define a lift of the two-strand Rickard complex C a,b 2 C(SSBim) from Definition 3.23 to a curved Rickard complex YC a,b 2 Y(SSBim). More precisely, writing a := (a, b), a 0 := (b, a), and t for the transposition in S 2 , we define lifts of C a,b to 1-morphisms in b,t Y(SSBim) a, for both possible permutations 2 S 2 . Pictorially, we denote the 2-strand Rickard complex and its curved analogue by
As in Definition 3.23, we will abbreviate by writing
. This is the singular Soergel bimodule corresponding to the web
where the labels on the edges are given by the sizes of the alphabets (46)
Here , are given componentwise by = + 0 and
(following Definition 4.7, u (i),m = 0 for m > a i ). An analogous construction defines a 1-morphism YC _ a,b 2 Y(SSBim) from (a, ) to (a 0 , t ), which is a curved lift of the inverse Rickard complex C _ a,b . The proof of this proposition is found below, but first we make some observations. The rightward di↵erential is the usual di↵erential on the Rickard complex C a,b . Meanwhile the leftward di↵erential is linear in the variables u (1),m , u (2),m . Thus, (46) defines a strict curved deformation of C a,b , in the sense of Remarks 4.11 and 4.13. Definition 4.18. For each integer r 1, let ⇥ r denote the degree q 2r t 1 endomorphism of C a,b which is given component-wise on C k a,b by the morphisms ( 1) k r 1 , where r 1 is the morphism from (28). Lemma 4.19. The endomorphisms ⇥ r 2 End C(SSBim) (C a,b ) satisfy
Proof. The first two relations follow from the definition of ⇥ r and an easy computation in the U(gl 2 ) thick calculus [KLMS12]. The proof of the third relation is a computation analogous to the one appearing in the proof of [RW16, Proposition 5.7]. We omit the details here (they appear in [HRW21, Lemma 2.30], where a more-general result is established). ⇤
We now pass from the homotopies ⇥ r , which are related to h -curvature (35), to homotopies related to the e-curvature (36).
Lemma 4.20. For each m 1, let m , 0 m 2 End C(SSBim) (C a,b ) be given by
These endomorphisms satisfy
2 ) e m (X 1 ) .
Proof. The first four relations are immediate from Lemma 4.19. For the penultimate relation, we compute
= e m (X 2 ) e m (X 0 1 ) .
For the last relation, we first note that
Here, we use that X 0 2 X 1 = X 2 X 0 1 when acting on b,a SSBim a,b . We then have
2 ) e m (X 1 ) . ⇤ Proposition 4.22. For r > 0, we have a+r = 0 , b+r = 0 and
Proof. Let us distinguish the alphabets B (k) and B (k+1) (of cardinality k and k + 1), living on webs
and C k+1 a,b . Note that we may regard Hom
). We also have actions of symmetric functions in the alphabets
)-submodule generated by the elements { r } r 0 and let D = {x} denote the alphabet on the cup of 0 . Since r 1 = x r 1 • 0 = h r 1 (D)• 0 , we may regard M as a module over Sym(X 2 |X 0 2 |B (k) |B (k+1) |D). It follows (e.g. from the "movie" following (28), or the corresponding foam) that
Finally, the identities for ⇥ a+r and ⇥ b+r follow from h-reduction, i.e. Lemma 2.9. For instance:
where we have used ), the elements ⇥ r and r can all be written as Sym(X 2 |X 0 2 )-linear combinations of ⇥ 1 , . . . , ⇥ min{a,b} .
Proof of Proposition 4.17. The fact that = 0 is well-known (the Rickard complex is a complex); see e.g. [WW17, Corollary 6.5]. We write
), where we again set u (i),m = 0 for m > a i . Lemma 4.20 implies that = 0 and that
which is as desired. ⇤
We now use Proposition 4.17 to assign a curved Rickard complex to any colored braid. The input for this construction is a Z 1 -colored braid word a (in the sense of §3.5) and a numbering of the strands 8 in the braid. If 2 Br m , then we require that a has strands numbered from 1 to m. Reading the sequence of strand numbers on the incoming and outgoing boundaries (respectively) defines two permutations , ⌧ 2 S m such that ⌧ = . (Here, and in the following, we abuse notation and write for the permutation induced by the braid under the canonical homomorphism Br k ! S k .) We will denote the data of a Z 1 -colored braid a = b a with numbered strands by b,⌧ a, or by a, (since ⌧ =
). Further, we will occasionally omit the numbering from our notation when it is not locally relevant.
Theorem 4.24. For each braid 2 Br m , each a 2 Z m 1 , and each 2 S m , the Rickard complex
that is unique up to homotopy equivalence. In particular, the assignment a, 7 ! YC( a, ) satisfies the (colored) braid relations up to homotopy.
Before the proof, we remark that the curvature equation for YC( a, ) is
for ⌧ = .
Proof. Since C( a ) 2 C(SSBim) is an invertible 1-morphism and the Rickard complexes assigned to colored braids satisfy the (colored) braid relations up to homotopy, this is an immediate consequence of Lemma 4.15. However, since we only sketched the existence portion of that proof, we give an explicit construction here. First, we prove the theorem for the identity braid 1 a and any chosen 2 S m . In this case,
is supported in even cohomological degrees, so any (degree 1) di↵erential on 1 a must be zero for degree reasons. In particular, 1a = 0, so the curvature equation for 1 a becomes
Here, we use that f (X (i) ) f (X 0 (i) ) acts by zero on the identity bimodule 1 a for any symmetric function f . This equation has the unique solution 1a = 0, so the curved Rickard complex associated to the identity braid is just 1 a with tot = + = 0, regardless of our choice of .
We next give a constructive proof of the existence of the curved Rickard complex associated to a non-trivial braid (word). For the Artin generator i and the identity permutation id 2 S m , we define YC(( i ) a,id ) := 1 (a1,...ai 1 ),id ⇥ YC (ai,ai+1),id ⇥ 1 (ai+2,...am),id 8 Thus, one can think of such braids as being
where the middle tensor factor is the 2-strand curved Rickard complex from Proposition 4.17. For any other 2 S m , we obtain YC(( i ) a, ) from YC(( i ) a,id ) by the substitution u i,r 7 ! u 1 (i),r . Analogously, we define YC(( 1 i ) a, ) using YC _ (ai,ai+1),id . For a general braid word = "r ir • • • "1 i1 , we define YC( a, ) by taking the horizontal composition of the curved Rickard complexes assigned to its constituent Artin generators YC(( "j ij ) ⌧j (a),⌧j ), in analogy to (32). Here, ⌧ j is the permutation associated with the braid
ir . Lemma 4.12 guarantees that YC( a, ) is a well-defined 1morphism in Y(SSBim). It is the unique curved lift of C( a ) associated with the permutation by Lemma 4.15, since Proposition 3.25 gives that C( a ) 2 C(SSBim) is an invertible 1-morphism.
Finally, we show that the assignment a, 7 ! YC( a, ) satisfies the braid relations, up to homotopy equivalence. Suppose that a, and a, are two braid words representing the same colored braid. By Proposition 3.25, we have a homotopy equivalence C( a, ) ' C( a, ). Proposition 4.14 provides a lift to a homotopy equivalence between YC( a, ) and some curved lift of C( a, ). However, by Lemma 4.15, such lifts are unique up to homotopy equivalence, and so YC( a, ) ' YC( a, ). ⇤
We pause to record a useful observation, which will help to establish a well-defined module structure on our deformed colored link homology. By construction, the complex YC( a, ) built in Theorem 4.24 is a strict 1-morphism, in the sense of Remark 4.11. As such, the linear part of the curved Maurer-Cartan element gives homotopies i,r , now considered as elements of End Y(SSBim) YC( a, ) , for 1 i m and 1 r a (i) that square to zero, pairwise anti-commute, and satisfy
) is a well-defined dg module over the (i th ) two point dg algebra (48)
, and further that these actions assemble to give a dg ( N i A i L,R )-module structure on YC( a, ). Let W i be an alphabet with
By restricting the action of (48) to the left and right alphabets, we obtain two (possibly) distinct actions of the algebra Sym(W i ) using the identifications
. Our next result shows that these actions on YC( a, ) give quasi-isomorphic dg Sym(W i )-modules. We state the result in slightly greater generality. Proposition 4.25. Let X be a dg ( N i=1 m A i L,R )-module (e.g. a strict 1-morphism in Y(SSBim)), then the induced left and right actions of Sym(W i ) give quasi-isomorphic dg Sym(W i )-modules (thus quasi-isomorphic dg (
i , the corresponding left and right Q[e 1 , . . . , e a ]-modules are quasi-isomorphic. Correspondingly, this immediately reduces to the 1-variable case.
Thus, let C be a dg module over
Now, view M as a dg Q[x]-module, where x acts via z M , and let C i denote the dg Q[x]-module C wherein x acts via x i , for i = L, R. The deformation retract M ! C L determined by
-linear; however, its homotopy inverse is not. Thus it (only) gives a quasi-isomorphism of
As discussed in §4.2 (and implicitly used in Lemma 4.20), we will find it convenient to translate back-and-forth between e-and h -curvatures, which are modeled on P (e k (X) e k (X 0 ))u k and P h k (X X 0 )v k respectively. Indeed, equation (39) and Lemma 4.12 show that complexes with e-curvature possess a straightforward horizontal composition, while Lemma 4.19 suggests that complexes with h -curvature appear more regularly "in the wild."
where the curvature element is
We will abbreviate by writing
which we understand simply as a disjoint union of categories.
Proposition 4.27. Retaining the setup from Definition 4.26, there is an isomorphism
of dg categories determined by the mutually inverse assignments
Proof. The functor b,⌧ V(SSBim) a, ! b,⌧ Y(SSBim) a, is defined on curved complexes tw X (X) = (X, X + X ) by sending X to itself, and defined on morphisms
using the first substitution rule in (50). In particular, this determines the image of the curved Maurer-Cartan element X . This is well-defined since Corollary 4.6 implies that it takes curved complexes with h -curvature (49) to those with e-curvature (38). The functor in the other direction is defined analogously using the second substitution rule in (50). As for (37), a computation using (15) shows that the substitutions (50), and thus the associated functors, are mutually inverse. ⇤ Remark 4.28. By pulling back structure from Y(SSBim), the categories b,⌧ V(SSBim) a, assemble to give a monoidal dg 2-category. We will not record the precise formulae for the various operations, but do wish to point out a subtlety regarding the action of the deformation parameters
inherits an action of the latter by acting on the left. Specifically, given g 2 Q[V], we let g • id X denote the endomorphism
However, we could also consider the action of such g act on the right, via
These actions do not necessarily agree. Rather, they are related by an appropriate analogue of Lemma 4.4. Specifically, on b,⌧ X a, we have that
Moreover, given another 1-morphism a, Y c,⇢ , we can consider the composite b,⌧ X ? Y c,⇢ and use the above formula to change from the action of v i,k ?id X ?id Y to id X ?v i,l ?id Y and further to id X ?id Y ?v i,r or directly in one step. The consistency of these changes is a consequence of the formula
In any case, the action of h l r (X ⌧ (i) X 0 (i) ) on any 1-morphism in V(SSBim) is null-homotopic (see e.g. Remark 4.11), thus the left and right actions of V are always homotopic.
Convention 4.29. Henceforth, we will not distinguish between the dg categories b,⌧ V(SSBim) a, and b,⌧ Y(SSBim) a, , and in each instance will use the notation coinciding with the relevant deformation parameters U or V.
In §6 - §9, we will be particularly interested in the = id = ⌧ case of Definition 4.26. To simplify notation, we will use the shorthand
for the category of curved complexes of singular Soergel bimodules with curvature
4.6. Alphabet soup II: from y's to u's and v's. In the special (uncolored) case of a = 1 m = b, the categories 1 m ,⌧ Y(SSBim) 1 m , recover (a version of) the category Y(SBim) of curved complexes of Soergel bimodules from [GH]. Setting X = {x 1 , . . . , x m } and
for deformation parameters Y = {y 1 , . . . , y m }. These parameters can be understood as equaling either the u's from Definition 4.7 or the v's from Definition 4.26 (since h 1 (x i x 0 i ) = x i x 0 i = e 1 (x i ) e 1 (x 0 i )). Note that Q[X, X 0 ]-modules can be viewed as Sym(X|X 0 )-modules, by restricting the left and right actions. In terms of SSBim, this corresponds to horizontally composing Soergel bimodules X by appropriate merge and split bimodules which, on each side of X, merge the boundaries to a single m-colored strand, e.g.
Since colored links can be interpreted as the "(anti)symmetric part" of appropriate cables of links in a similar manner (recall Theorem 1.21 and Conjecture 1.24), we will find it fortuitous to relate the thin curvature in (52) to the h -curvature from (35) and the e-curvature from (36). Indeed, certain endomorphisms that appear naturally in this story (see §4.7 below) are crucial in our investigation of the colored link splitting map in §7.
To this end, let a 1 and fix alphabets
-modules and consider the following categories of curved complexes
) and e k (X) e k (X 0 ) as Q[X, X 0 ]-linear combinations of the elements x i x 0 i produces dg functors from YA to VA and YA, respectively. Precisely, we have the following.
Proposition 4.30. The dg categories VA and YA are isomorphic via the mutually inverse substitutions
Moreover, the substitutions
e l 1 (x 1 , . . . , x i 1 , x 0 i+1 , . . . , x 0 a )u l determine dg functors YA ! VA and YA ! YA that are compatible with the isomorphism VA ⇠ = YA.
Proof. The isomorphism between VA and UA follows from the discussion in §4.2, which is the 1-strand case of Proposition 4.27.
Similarly, the substitutions (53) and (54
that determine dg functors which are the identity on objects and are given on morphism complexes by extension of scalars
To confirm that these functors are indeed well-defined, it su ces to show that the algebra maps preserve curvature. We first confirm this for the functor YA ! YA. To begin, we explicitly write e k (X) e k (X 0 ) as an element of the ideal generated by
Consider the di↵erence of monomials:
Summing over all sequences with 1
Hence, we compute
Alphabet soup III: interpolation coordinates. Retain the notation from §4.6 and consider the identity bimodule 1 (1,...,1) 2 YA. In End YA (1 (1,...,1) ), we have that x i = x 0 i , thus we find that the dg functor
This motivates the following. Definition 4.31. Set (55)
In other words, y i is defined to be the polynomial of degree |X| 1 in x i with coe cients v r . We call these coe cients interpolation coordinates and highlight that they are independent of i.
This inclusion is equivariant for the action of S a that (simultaneously) permutes the x i (and y i ). Note, however, that (55) is a special case of (53), which is only compatible with Proposition 4.30 when X = X 0 . Nevertheless, symmetric functions in the alphabet Y give well-defined elements of Sym(X)[V] ⇠ = End V(SSBim) (1 (a) ), thus we can consider them as operators acting (on the left or right) on suitable 1-morphisms in V(SSBim). For the duration of the paper, the variables y i will always be understood in this context. (See Remark 4.28 above, which addresses a subtle point concerning these actions.)
Our terminology in Definition 4.31 is chosen since we can express the v i in terms of x i , y i by formulae familiar from interpolation theory. We now make this precise, and establish further identities involving the interpolation coordinates. We will make use of identities from §2.3 and the Haiman determinants from §2.4.
Example 4.32. When a = 2, the elements
for M 1 = {1, y} and M 2 = {x, y}.
Generalizing Example 4.32, the following gives the general rule to recover the interpolation coordinates from the x i and y i . (See also Lemma 7.7 below for another formulation.) Lemma 4.33. Let X = {x 1 , . . . , x a } and Y = {y 1 , . . . , y a }, then we have
where M k = {x a 1 , . . . , [ x a k , . . . , 1, y}.
Proof. The second equation follows from the definitions and reordering rows. For the first equation, consider the matrix defining the determinant hdet(x a 1 , . . . , x a k+1 , y, x a k 1 , . . . x 0 ) and rewrite its kth row as:
The only summand with a nonzero contribution is the one for r = a k + 1, which yields v a k+1 in the quotient.
In Definition 4.31 we have defined the variables y i in terms of x i and a family of interpolation coordinates v r which depends on the cardinality of the alphabet X. Sometimes, however, it is useful to take the opposite viewpoint and start with alphabets X and Y and the assumption that the variables y i can be interpolated by polynomials in x i (with coe cients then determined by Lemma 4.33). For example, in later parts of the paper, we will need to understand the behavior of the interpolation coordinates v r under inclusions of the alphabets X and Y.
Proposition 4.34. Consider integers 1 c d and alphabets
Proof. We first prove uniqueness. Suppose ', :
] to be the algebra map defined by '(x i ) = x i and the substitution rule (56). It su ces to verify that '(y i ) = y i for 1 i c. Indeed:
There is an interesting generalization of Proposition 4.34 that we will use (in various forms) throughout this paper.
where X S ⇢ X (c) and X 0 S ⇢ X 0(c) denote the corresponding subalphabets. For c d, the algebra map
determined by x i 7 ! x i , x 0 i 7 ! x 0 i , and the rule (56) sends Z (c)
Proof. We compute:
In passing to the last line, we have used Lemma 2.9 with X = X S X 0 S and Y = (X (c) X S ) + X 0 S (which has cardinality c). ⇤ Remark 4.36. Lemma 4.35 establishes a crucial "stability" property for the expressions Z
S under the inclusion (57). The fact that y i 7 ! y i in Proposition 4.34 is essentially the special case of Lemma 4.35 corresponding to S = {i} and X 0(c) = 0 (hence Z S = x i y i ).
In this section, we have discussed the variables y i associated with a single a-colored strand. Indeed, as mentioned above, any symmetric polynomial in the alphabet Y gives a well-defined element in Sym(X)[V] = End V(SSBim) (1 (a) ). Paralleling the passage from §4.2 to §4.5, it is possible to pass from the one-strand case to the general case, since curvature in Y(SSBim) is modeled on the strand-wise h -and e-curvatures. See §4.8 and §8.1 for aspects of the (pure) 2-strand and m-strand cases, respectively.
Remark 4.37. The reader familiar with the Hilbert scheme Hilb a (C 2 ) should note that interpolation coordinates arise naturally in its study. Compare the generators {e 1 (X), . . . , e a (X), v 1 , . . . , v a } of Sym(X)[V] = End V(SSBim) (1 (a) ) to the coordinates on the open a ne set U x ⇢ Hilb a (C 2 ) in [Hai01, Proposition 3.6.3]. 4.8. Alphabet soup IV: the two-strand categories. In this section we set up some framework and notation for working with the categories V a,b , i.e. the m = 2 case of (51). This section can be skipped on first reading, and referred to as needed.
First, we will use the abbreviation
for the relevant dg category of (uncurved) complexes of singular Soergel bimodules. We fix alphabets as follows:
We denote 9 the deformation parameters associated with the first and second entries of a = (a, b) by ( 58)
L,a } , V
R,b } respectively, and let
which is the (pure) 2-strand analogue of (55).
We now observe that it is possible to specify objects in V a,b using a reduced collection of deformation parameters. R,j = vj for all i, j. The functor in the other direction is more interesting. The starting point for its constructing is the observation that since
2 )| X = 0 for any X 2 a,b SSBim a,b and any (positive degree) symmetric function f , we have that
When a b, an application of Lemma 4.35 shows that if we set
L,b+i 9 Our notation here indicates that V
(a) L and V (b)
R should be viewed as associated to the "left" a-colored and "right" b-colored strands of a 2-strand pure braid, when drawn vertically. They would be called V 1 and V 2 in the language of §4.5.
for 1 j b, then
Further, this remains true (trivially) when a < b, provided we let v (a)
L,j = 0 for j > 0, since in this case (60) gives
It follows that (61)
This suggests the following.
Proposition 4.39. There is a functor V a,b ! V a,b which is the identity on objects and morphisms in C a,b , and which sends
Proof. It need only be checked that (62) is compatible with curvatures, i.e. that R,i to the expression on the right-hand side of (62). We call ⇡ the reduction functor. We say that an object We conclude this section by establishing further notation for working with the reduced categories V a,b . At times, we will need to consider the relation between such categories as we allow b to vary. For each ` 0, introduce interpolation coordinates V (`) = {v (`) 1 , . . . , v(`) `} and elements (64)
k for all a + 1 i a + `. A priori, the definition of y i depends on `. However, if we let (65)
X [1,...,c] = {x 1 , . . . , x c } , X 0 [1,...,c] = {x 0 1 , . . . , x 0 c } then for each pair of integers ` b we have an inclusion of algebras (66)
Crucially, Lemma 4.34 implies that this sends y i 7 ! y i . Furthermore, Lemma 4.35 implies that the inclusion (66) is compatible with curvature, in the sense that (68
i .
In this section, we use curved complexes of singular Soergel bimodules to construct our deformed, colored, triply-graded link homology. We begin by recalling the construction of colored, triply-graded Khovanov-Rozansky homology, which is an invariant of framed, oriented, colored links taking values in the symmetric monoidal category KJa ± , q ± , t ± K from §3.1. Khovanov-Rozansky homology is defined by applying the Hochschild homology functor to the Rickard complex of a braid representative of a link L, and our deformation is defined by replacing the Rickard complex with its curved analogue from §4.4. In this section (and those following) we will typically denote our (co)domain objects in SSBim by b and c, since our notation for the object a is easily confused with the Hochschild degree a.
5.1. Hochschild (co)homology. We begin by recalling the basics of Hochschild homology and cohomology. Suppose that R and S are Z q -graded algebras and let R B S denote the category of Z q -graded (R, S)-bimodules. When R = S, a graded (R, R)-bimodule M 2 R B R may be viewed as a module over the enveloping algebra R e = R ⌦ R op , and the i th Hochschild homology and cohomology of M are defined as: , M ) for i 0, respectively. Note that both inherit a Z q -grading from R B R , thus are objects in KJq ± K. The total Hochschild (co)homology functors are defined by
which we therefore view as functors R B R ! KJa ± , q ± K (see §3.1).
Remark 5.1. We view Hochschild homology as supported in negative a-degree, while Hochschild cohomology is supported in positive a-degrees.
The following is a standard fact about Hochschild homology; see e.g. [Rou17].
Proposition 5.2. Let R and S be Z q -graded algebras. Suppose that R M S 2 R B S and S N R 2 S B R are Z q -graded bimodules that are projective as R-modules and S-modules, then there is an isomorphism of Z a ⇥ Z q -graded Q-vector spaces
that is natural in M and N . ⇤
The total Hochschild (co)homology functors can be extended from bimodules to complexes of bimodules term-wise. Explicitly, if X 2 C( R B R ) is a complex of graded (R, R)-bimodules, then we may write X = tw (
which is an object in KJa ± , q ± , t ± K dg . In other words, HH • (X) equals the Z a ⇥ Z q ⇥ Z t -graded Qvector space L i,k a i t k HH i (X k ), with appropriate di↵erential. Proposition 5.2 extends to this setting as follows.
Proposition 5.3. Let R and S be Z q -graded algebras. Suppose that R X S 2 C( R B S ) and S Y R 2 C( S B R ) are complexes of Z q -graded bimodules, that are projective as R-modules or S-modules, then there is an isomorphism of
that is natural in X, Y (in the dg sense).
Proof. Denote the natural isomorphism from Proposition 5.2 by ⌧ M,N : HH
This induces an isomorphism on the level of Z a ⇥ Z q ⇥ Z t -graded Q-vector spaces:
1 (z)) using naturality of the isomorphism from Proposition 5.2. The sign here is equal to ( 1) |f ||g|+kl+|f |l . On the other hand, we have
as desired. Indeed, this is the appropriate (signed) version of naturality in this context. For example, it guarantees that the isomorphism ⌧ X,Y : HH
Remark 5.4. The hypotheses of Propositions 5.2 and 5.3 hold for (complexes of) singular Soergel bimodules. Indeed, any X 2 b SSBim c is free as either a left R b -module or right R c -module.
From this point forward, we focus on the case when R is a polynomial ring, since this is the relevant setting for singular Soergel bimodules. In this case, the Koszul resolution provides a direct means for computing Hochschild (co)homology and gives an explicit relation between these two invariants. Let 10 R = Q[z 1 , . . . , z N ] with deg q (z i ) = d i for 1 i N . The Koszul resolution of R is the complex of graded (R, R)-bimodules given by ( 72)
It follows from (69) that HH • (M ) is the homology of K ⌦ R⌦R M , while HH • (M ) is the homology of Hom R⌦R (K, M). This implies that HH i (M )
by (70). Moreover,
The latter acts on the former by identifying ⌘ ⇤ i with the derivation sending ⌘ i 7 ! 1 and ⌘ j 7 ! 0 for j 6 = i.
We now recall the partial Hochschild (co)homology functors from [RT21], which generalize the (uncolored) partial trace functors first introduced in [Hog18]. These functors refine the Hochschild (co)homology functors from §5.1, and allow them to be applied to the complex C( b ) "one strand at a time." As such, they are useful in proving the invariance of colored, triply-graded link homology (and its deformation defined in §5.4 below) under the second Markov move. We will refer to these functors collectively as the colored partial trace functors.
Recall that SSBim is a full 2-subcategory of the 2-category Bim from §3.4, and that in these 2categories the Hom-categories are enriched in the symmetric monoidal category KJq ± K of Z q -graded Q-vector spaces. We now consider the bounded derived category of Bim, denoted D(Bim), which is enriched in the category KJa ± , q ± K of Z a ⇥ Z q -graded Q-vector spaces. This 2-category D(Bim) is the natural setting for Hochschild (co)homology of singular Soergel bimodules, via the inclusion (74) SSBim ,! Bim ,! D(Bim) .
We emphasize to the reader that the cohomological grading in D(Bim) is the a-degree, which is independent from the cohomological grading used in the dg category of (curved) complexes C(SSBim), which is the t-degree. 10 Here, we call our variables {z i }, since in general they will not be the variables X = {x i }, but rather symmetric functions in subalphabets of the alphabet X.
Since singular Soergel bimodules in b SSBim c are free as either left R b -modules or right R c -modules, horizontal composition of such bimodules is exact. Thus, the inclusion in (74) is indeed a 2-functor (i.e. derived tensor product over the rings R b equals the usual tensor product for such bimodules). Since ⇥ is given in both settings by tensor product over Q, (74) is a monoidal 2-functor.
For X, Y 2 b SSBim c , we have
In particular, for X 2 b SSBim b this gives
Here, X and X 0 denote the alphabets corresponding to the last (c-labeled) boundary points. The complex Tr c (X) is regarded as an object in the 1-morphism category b 0 D(Bim) c 0 ; in particular, its cohomological degree is the a-degree. Paralleling the relation in (73), we also define the partial Hochschild cohomology for such X to be:
We now recall various properties of the colored partial trace functors. Further details are provided in [RT21, Section 4.C] and [Hog18, Sections 3.2 and 3.3]; hence, our treatment is concise.
We first record an adjunction that will be used to compute various Hom-spaces. Using the Koszul resolution of Sym(X) as a bimodule over itself, it is easy to see the following.
Proposition 5.9. Let X 2 D(Bim) and c 1, then
where |X| = c, wt(e i (X)) = q 2i , and wt(⌘ i ) = a 1 q 2i . ⇤
The next result establishes a certain "bilinearity" of the colored partial trace. There is an isomorphism
naturally in Z 1 and Z 2 . ⇤ Our next result describes the behavior of Rickard complexes under the second Markov move. In order to state it, we first need the following. Lemma 5.12. Let b 1, then we have homotopy equivalences
Proof. This appears in [RT21, Lemma 4.12] in terms of partial Hochschild cohomology, namely:
After shifting the grading by a b q b(b+1) according to Remark 5.7, we recover the shown expressions. ⇤ Proposition 5.13. For any 1-morphism X in c⇥b C(SSBim) c⇥b , we have equivalences
that are natural in X.
Proof. This is an immediate consequence of Remark 5.7, Proposition 5.10, and Lemma 5.12. ⇤ 5.3. Colored, triply-graded homology. In [Kho07], Khovanov showed that triply-graded Khovanov-Rozansky homology [KR08b] can be reformulated using the Hochschild homology of Soergel bimodules. We now recall the extension of this result to colored, triply-graded link homology via singular Soergel bimodules, as described in [MSV11, Wed19,Cau17]. See also [WW17] for a geometric analogue of this construction. We say that a Z 1 -colored braid b c is balanced if it is an endomorphism in Br(Z 1 ), i.e. if c = b. In this case, the (standard) braid closure c b represents a colored link. The classical Alexander theorem implies that every colored link in S 3 arises in this way, and Markov's theorem classifies the redundancy in presenting links by braid closures. Following the references above, we will define the colored, triply-graded link homology as the homology of a properly normalized version of the Hochschild homology of Rickard complexes. The result will be an invariant of colored, framed, oriented links. First, we need some setup. where x ranges over all crossings in the braid , {a(x), b(x)} is the multiset of colors meeting x, and s(x) 2 {±1} is the sign of x. Furthermore, when b is balanced we set:
These quantities give the sum of the colors of the components of the closure of b , the colors on the strands of b , and the sum of the q-degrees of all the generators of the partially symmetric polynomial ring R b . Definition 5.16. Let L be a colored link, presented as the closure of a balanced, colored braid b . The colored, triply-graded Khovanov-Rozansky complex of b is
This is an object 11 of KJa ± , q ± , t ± K dg (see §3.1). The colored, triply-graded Khovanov-Rozansky homology of b is defined by H KR ( b ) := H(C KR ( b )), which is an object of KJa ± , q ± , t ± K. Remark 5.17. It can be seen that "( b )+N ( b ) n( b ) is even, so the shift (at 1 )
is well-defined. We leave verification of this as an exercise.
Remark 5.18. One can also express C KR ( b ) in terms of Hochschild cohomology:
Theorem 5.19. A change of braid representative b for a framed, oriented, colored link L induces a homotopy equivalence between the corresponding complexes C KR ( b ). Consequently, H KR (L) := H KR ( b ) 2 KJa ± , q ± , t ± K is a well-defined invariant of the framed, oriented, colored link L, up to isomorphism. Furthermore, both C KR ( b ) and H KR (L) are monoidal under split (disjoint) union t: Proof. Following the classical Markov theorem, the invariance statement is a consequence of Proposition 3.25, conjugacy invariance of Hochschild homology (Proposition 5.3), and the behavior of Hochschild homology of Rickard complexes under the second Markov move (Lemma 5.12). See also [WW17, Theorem 1.1] or [Cau17, Theorem 4.1]. The monoidality follows since split links can be represented by split braids, the fact that the Hochschild homology of Rickard complexes is monoidal under the external tensor product ⇥, and the observation that the numerical invariants "( ), N ( ), and n( ) are additive under ⇥. The framing behavior follows from Lemma 5.12. The module structure on C KR ( b ) is inherited from the module structure of singular Soergel bimodules. Homotopies relating the module structures specified by points on a single strand on two sides of a crossing were studied in §4.4, and the last statement therefore follows from Proposition 4.25. ⇤
Example 5.20. Let U(b) be the 0-framed b-colored unknot, presented as the closure of a single blabeled strand. In this case, no di↵erentials or grading shifts enter into Definition 5.16. To compute the unknot invariant, let X be the size b alphabet associated to the b-labeled strand. Then, we have:
11 In fact, the homogeneous components of C KR ( b ) are finite-dimensional, the a-and t-grading are bounded and the q-grading is bounded from below. The same holds for H KR ( b ).
where wt(e i (X)) = q 2i and wt(⌘ i ) = a 1 q 2i . The Poincaré series of this homology is:
As is typical in link homology theory, what is usually called "the unknot invariant" actually serves two distinct roles: first literally as the invariant of the unknot, and second as the algebra that acts on the invariant of any link upon specifying a chosen point. The latter is the derived sheet algebra from Example 1.4, so-named because it computes endomorphisms of a single strand in D(Bim). In the colored, triply-graded Khovanov-Rozansky theory, the derived sheet algebra for a b-labeled strand is given by the dg algebra HH • (Sym(X)), whose Poincaré series is:
◆ .
The action of the derived sheet algebra on the unknot invariant coincides with the classical action of Hochschild cohomology on Hochschild homology. By (73), the underlying triply-graded vector spaces of the unknot invariant and the derived sheet algebra only di↵er by a grading shift.
Remark 5.21. We expect that Theorem 5.19 can be strengthened as follows. If the labels on the components of L are b 1 , . . . , b r , then we expect that C KR (L) is well-defined up to quasi-isomorphism as a dg-module over the dg algebra given as the tensor product of the derived sheet algebras of all b i . We will not need this stronger statement in the present paper.
Remark 5.22. It is sometimes desirable to use a renormalized version of Definition 5.16 that favors Hochschild cohomology over Hochschild homology, and in which the unknot invariant is identified with the derived sheet algebra. For this one sets:
where n 2 ( b ) :=
). Remark 5.23. The invariant H KR (L) decategorifies to a version of the HOMFLYPT invariant P (L) of links colored by one-column Young diagrams by specializing the three-variable Poincaré series at t = 1. More specifically, the uncolored part of the invariant (i.e. where all colors are single box Young diagrams) is determined by the unknot invariant that we read o↵ Example 5.20 and the following skein relation:
where r = 0 if all shown strands on the left-hand side belong to the same link component, and r = 1 otherwise.
5.4. The deformed, colored, triply-graded homology. We now define our deformed, colored, triply-graded homology. This proceeds in parallel to §5.3 by replacing the Rickard complex C( b ) with an appropriate curved analogue constructed from YC( b,! 1 ) and the set ⌦(!) of cycles of . We will use the V-variables description of the Hom-categories in Y(SSBim), as it is more convenient for our present considerations. Suppose we are given a 1-morphism tw (X) 2 b,id Y(SSBim) b,! 1 , where b = (b 1 , . . . , b m ) and ! 2 S m . If ! = id, then the h -curvature on tw (X) (given by (49)) takes the form
k , which becomes zero after applying HH • . Thus, (77) tw HH•( )
⌘ is a well-defined complex (with zero curvature), and we obtain our sought after deformation of (5.16) by replacing HH • with (77). If ! 6 = id, then we will need to "prepare" tw (X) before taking Hochschild homology.
Hence, we introduce X-alphabets parametrized by elements of ⌦(!) (i.e. the cycles of !) by setting
and we introduce deformation parameters indexed by ⌦(!), denoted (78)
We now change variables from V to V ⌦(!) so that the resulting complex has curvature (79)
Note that if tw (X) = YC( b ) is the curved complex associated to a braid, then (79) has one alphabet
,bi } of deformation parameters for each link component in c b . We refer to curvature of the form (79) as bundled curvature, since the deformation parameters on various braid strands are bundled according to the corresponding link components.
The following gives a method for obtaining bundled curvature.
Definition 5.24. Define a surjective algebra map
Lemma 5.25. Suppose tw (X) 2 b,1 Y(SSBim) b,! 1 . The substitution (80) yields a complex
⌘ with curvature (79).
Proof. For simplicity, we consider the case when ! has only a single cycle
where the second line is obtained by iterating identities of the form
,l . This completes the proof when ! has one cycle. The proof for general ! is accomplished by applying the above computation to each cycle of !. It di↵ers only in more-tedious bookkeeping, so we omit the details. ⇤
There may be other changes of variables that obtain the curvature (79) from the curvature
The following says that, for curved Rickard complexes YC( b ), any two choices are equivalent.
Lemma 5.26. Let b 2 Br m (Z 1 ) be balanced and let ! 2 S m be the permutation represented by . !) ] are two Maurer-Cartan elements with the same curvature (79),
Proof. Similar to Lemma 4.15. ⇤ Note that the curvature (79) vanishes upon identifying X [i] and X 0 [i] . Since Hochschild homology factors through the quotient
), we now arrive at the definition of our link invariant.
Definition 5.27. Let L be a colored link which is presented as the closure of a balanced colored m-strand braid b and let ! 2 S m be the permutation represented by . Let
be the curved Maurer-Cartan element with curvature (79) from Lemma 5.25. Let
⌘ then the deformed, colored, triply-graded link homology YH KR (L) is the homology of the chain complex Remark 5.28. Although we have used the specific Maurer-Cartan element from Lemma 5.25 to define YH KR (L), Lemma 5.26 shows that we could have used any Maurer-Cartan element with bundled curvature (79).
Remark 5.29. We will sometimes abuse notation by writing YC KR (L) instead of YC KR ( b ). This is justified by Theorem 5.30 below. As noted above, the set of cycles ⌦(!) of the permutation ! determined by can be identified with the set ⇡ 0 (L) of components of the link L. We will thus also denote V ⇡0(L) := V ⌦(!) in this context, and further write
As defined, YH KR (L) is an object of KJa ± , q ± , t ± K, and (as with H KR (L)) the homogeneous components are finite-dimensional, the a-and t-grading are bounded, and the q-grading is bounded from below. In fact, the module structure on YC KR ( b ) allows us to endow YH KR (L) with additional structure, that we now describe.
Let L be the closure of a colored braid b . For each component c 2 ⇡ 0 (L) of color b(c), introduce an alphabet X c of cardinality b(c) and set
Given a point p 2 b (away from a crossing), the 2-categorical structure of Y(SSBim) endows YC KR ( b ) with the structure of a dg module over Sym(X c ), where c is the component of L containing p. If we choose one such point for each component of L, this endows YC KR ( b ) with a dg A L -module structure. We call such a choice of points a pointing of L.
We can now state precisely in what sense YH KR (L) is a colored link invariant.
Theorem 5.30. Choose a pointing of c b , then the renormalized complex YC KR ( b ) from (82) depends only on the framed, oriented, colored link L := c b , up to quasi-isomorphism of A L -modules. Consequently, YH KR (L) is a well-defined A L -module, up to isomorphism. The proof of Theorem 5.30 (i.e. the Markov invariance of YC KR ( b )) is established in the following section. There, we show that YC KR ( b ) can equivalently be described using curved Rickard complexes with strand-wise curvature P a k=1
)) vk ; this curvature is the most-straightforwardly adapted to the Markov moves.
Before doing so, however, we first establish some easy consequences of Definition 5.27.
Example 5.31. Let U(b) be the 0-framed b-colored unknot. Since U(b) can be presented as the closure of a single b-labeled strand, no di↵erentials and grading shifts enter into Definition 5.27. To compute the unknot invariant, let X and V be the size b alphabets associated to the b-labeled strand. Then, we have:
where wt(e i (X)) = q 2i , wt(⌘ i ) = a 1 q 2i , wt(v i ) = q 2i t 2 . The Poincaré series of the unknot homology is thus:
Remark 5.32. As for the undeformed invariant, we expect that Theorem 5.30 can be strengthened to exhibit YC KR (L) as a dg-module over the tensor product of derived sheet algebras, well defined up to quasi-isomorphism; see Remark 5.21.
and therefore an isomorphism
Proof. Let b m be a braid representative of K. Since K is a knot, the associated permutation
The uniqueness of curved lifts with bundled curvature (i.e. Lemma 5.26) implies that we may take ¯ = 0 when forming YC KR (K). This implies the first statement (after identifying v k = v [1],k ), and taking homology gives the second. ⇤ 5.5. Alphabet soup V: power sums. In our considerations thus far, we have worked with strandwise curvature modeled on h -curvature (or, equivalently, e-curvature). In order to most easily establish invariance of the complex YC KR ( b ) under the Markov moves, we find it beneficial to also consider strand-wise curvature modeled on p-curvature:
The following lemma will allow us to translate between such curvature and those previously considered.
Lemma 5.34. We have
Proof. This is a straightforward application of Newton's identity relating the power sum and complete symmetric functions, as manifest in ( 16) and (19). For example, after the stated change of variables, we have
) we obtain the identity in the statement. ⇤
Remark 5.35. The above substitutions send vk $ v k modulo N (X, X 0 ). In practice, the alphabets X and X 0 will be associated to the left and right endpoints of a strand of a braid. The above substitution is not compatible with horizontal composition of braids and will only be applied immediately before closing the braid, during which we identify X and X 0 .
For each b = (b 1 , . . . , b m ) of SSBim, introduce alphabets of deformation parameters V1 , . . . , Vm where Vi = { vi,1 , . . . , vi,bi } and wt( vi,r ) = q 2r t 2 . Set Vb := V1 [• • •[ Vm . Given b and a permutation ! 2 S m , it is convenient to consider two copies of the alphabets Vb , which will act on two sides of a b-colored braid with underlying permutation !. As a bookkeeping tool, we introduce:
)-bimodule and a commutative algebra with multiplication
Paralleling our notation for the alphabets X i and X 0 i , we will write vi,k := vi,k ⌦ 1 and v0 i,k := 1 ⌦ vi,k . We will sometimes denote (S ! ) b by either
Finally, if S is any Q-algebra and B is an (S, S)-bimodule, then we will let [B] denote the S-
Note that [B] = HH 0 (B), but we wish to not confuse the reader with this occurrence of Hochschild homology and the functor HH • , which (in this paper) we apply exclusively to singular Soergel bimodules. Observe that
Lemma 5.36. Let b be a balanced, colored braid, and let ! 2 S m be the permutation represented by . There exists a Maurer-Cartan element
This twist is unique up to homotopy equivalence in the sense of Lemma 5.26.
Proof. By (84), the curvature element (86) can also be written as
Thus a Maurer-Cartan element with curvature (86) can be constructed from the Maurer-Cartan element on the curved complex YC( b,! 1 ), which has curvature P
We next show how to obtain bundled p-curvature, establishing the analogue of Lemma 5.25 in this context. In the following two results, the Maurer-Cartan element is understood to be the one constructed in the proof of Lemma 5.36.
Lemma 5.37. Let b be a balanced, colored braid, and let ! 2 S m be the permutation represented by . There exists a Maurer
gives us an algebra map
Taking [ ] to be the image of under this algebra map produces a Maurer-Cartan element with curvature (87). ⇤
The twist in Lemma 5.37 is unique, in the sense of Lemma 5.26. Using this [ ], we can recover the complex YHH • ( b ) from (81), up to homotopy equivalence.
Lemma 5.38. Let b be a balanced, colored braid, and let ! 2 S m be the permutation represented by
where HH • ([ ]) denotes the image of [ ] from Lemma 5.37 under the algebra map
Proof. Note that a substitution as in Lemma 5.34 will convert the curvature (87) into (79). Further, since we work with bundled curvature and Hochschild homology identifies the alphabets X [i] and X 0 [i] , Remark 5.35 shows that the relevant substitution simply sets v[i],k = v [i],k . Our uniqueness statement (Lemma 5.26) then establishes the lemma. ⇤
Remark 5.39. The Maurer-Cartan element from Lemma 5.36 is strict, in the sense of Remark 4.11, thus its linear part determines null-homotopies e
⌘ . Applying HH • then produces the monodromy maps ⌅ c,k from Proposition 1.10. This pairs with the discussion preceding Theorem 5.30 to establish Proposition 1.10.
Using this "power sum model" for YHH • ( b ) established in Lemma 5.38, we now prove Markov invariance of YC KR ( b ).
Proof of Theorem 5.30. It su ces to show that YC KR ( b ) is invariant under the Markov moves, up to quasi-isomorphism of A L -modules. We will establish the Markov moves in turn, and then observe that all maps used are A L -module quasi-isomorphisms. A straightforward computation shows that the Maurer-Cartan elements (89)
) c have curvature as in Lemma 5.36. Now, using Proposition 5.3, we have an isomorphism of dg algebras
Thus, by Lemma 5.38, YHH • ( 0 00 b ) ' YHH • ( 00 0 c ). Since the numerical invariants "( ), n( ), and N ( ) from Definition 5.15 agree for agree for 0 00 b and 00 0 c , this establishes invariance of YC KR ( ) under the first Markov move.
Markov II: Let b = c ⇥ b and suppose that b is a balanced, colored m-strand braid that can be written as b = b ( 0 ⇥1 b ) m 1 . (Recall that m 1 denotes the (m 1) st Artin generator.) Let ! 2 S m be the permutation represented by and let ! 0 2 S m 1 be the permutation represented by 0 . Observe that 0 c is necessarily also balanced. For 1 i m 1, the cycles [i] ! 0 and [i] ! are related by
Thus, there is a canonical bijection between the cycles of ! and ! 0 given by sending
! by our hypotheses on the braid .) Henceforth we will identify the algebras
and we will use the notation v[i],k without specifying whether [i] is regarded as a cycle of ! or ! 0 .
Introduce alphabets X i , X 0 i , X 00 i which act by left-, middle-, and right-multiplication (respectively) on
In particular we have X 0 i = X 00 i for i = 1, . . . , m 2 and X m = X 0 m when acting on C( b ). The bundled curvature (87) equals
Since X m = X 0 m , the i = m term of the first summation on the right is zero for all k. Furthermore, since X 0 i = X 00 i for i = 1, . . . , m 2, we have
Here, we have also used the fact that X 0 m 1 + X 0 m = X 00 m 1 + X 00 m when acting on C( b m 1 ). Therefore, the curvature element (87) in the present setting reduces to
which coincides with the bundled curvature (87) for 0 . Thus
satisfies the conditions of Lemma 5.36 for b . Lemma 5.38 now gives that tw HH•(([ 0 ]⇥id)?id)
Recall that Proposition 5.13 gives a homotopy equivalence
of undeformed Rickard complexes, which intertwines the actions of End C(SSBim) (C( 0 c )) on both sides. This implies that the induced map
intertwines the actions of Maurer-Cartan elements HH • (([ 0 ] ⇥ id) ? id) and HH • ([ 0 ]). Hence, Lemma 5.38 gives that
) , and a similar argument gives that
) . Comparing the shifts in (82), this establishes the requisite behavior of YC KR ( ) under the second Markov move.
Module structure: First, note that all homotopy equivalences used above are Q[V ⇡0(L) ]-linear, so it su ces to show that they are quasi-isomorphisms of N c2⇡0(L) Sym(X c )-modules. This follows from Proposition 4.25. Indeed, therein it is shown that, up to quasi-isomorphism, we can assume that the Sym(X c ) action is given at any point p on the corresponding component. In particular, all homotopy equivalences following from our uniqueness results are quasi-isomorphisms, since we can assume Sym(X c ) is acting via alphabets on the left, and these homotopy equivalences are equivalences in categories of curved complexes of bimodules. This similarly shows that the maps establishing Markov II invariant are quasi-isomorphisms: we can assume that the Sym(X c ) action is given on the left, and does not act via X m . It remains to show that (91) is a quasi-isomorphism. For this, we can use Proposition 4.25 to assume that the Sym(X c ) actions occur in the "middle" of the left-hand side (i.e. via the action of End SSBim (1 c ) on C( 0 c ) ? 1 c ? C( 00 b )), and on the left on the right-hand side. This map is a Sym(X c )-linear isomorphism for these actions, thus a quasi-isomorphism. ⇤ Remark 5.40. Theorem 5.30 describes the link invariant YH KR (L) as a module over the algebra
i.e. the deformation parameters acting here are { v[i],k } [i]2⌦(!),k2{1,...,bi} . However, it is straightforward to describe the action on YH KR (L) of the various other deformation parameters. By definition (i.e. by ( 85)), the parameters v[i],k act as the parameters vi,k . Further, it follows from (47) that elements of N (X i , X 0 ! 1 (i) ) acts null-homotopically on YC KR ( b ), and thus by zero on YH KR (L). Remark 5.35 then implies that the parameters vi,k and v i,k act identically on YH KR (L), and the assignment (80) implies that the parameters v i,k and v [i],k acts identically as well. Lastly, the assignment (50) implies that the parameters u i,k act on YH KR (L) by ( 1)
5.6. Coe cients and spectral sequences. We now collect some straightforward results on homology with coe cients and spectral sequences that we need for our link splitting results in §8 -10 below. First, we will use the following common generalization of Definitions 5.27 and 5.16. Definition 5.41. Let b be a balanced, colored braid. We consider two types of homology with coe cients. If M is a Q[V ⇡0(L) ]-module, we define
If instead M 0 is an A L -module, we define
If either case, if L is the colored link obtained as the closure of b , then the deformed, colored, triplygraded Khovanov-Rozansky homology of L with coe cients in M is defined by YH KR (L, M) := H(YC KR ( b , M)) .
Remark 5.42. Since YC KR ( b ) is free when regarded as a module over Q[V ⇡0(L) ], the tensor product in (93) coincides with the derived tensor product 93) can (and often will) be thought of as a special case of (94), with
M . Moreover, we expect that YC KR ( b ) is free as an A L -module, so that the derived tensor product in (94) may be replaced with the ordinary tensor product. This amounts to showing that Hochschild homology of any singular Soergel bimodule 1 b B1 b is free as a module over the appropriate symmetric polynomial ring End SSBim (1 b ). (Note that this holds in the uncolored case.) Remark 5.43. As in Remark 5.29, we will sometimes write YC KR (L, M) instead of YC KR ( b , M). Strictly speaking, this complex depends on a choice of braid representative of L, but the resulting complex depends only on the framed, oriented, colored link L up to quasi-isomorphism of A L -modules. Thus, the homology with coe cients YH KR (L, M) is a well-defined module over A L /Ann(M ), up to isomorphism (here Ann(M ) ⇢ A L denotes the annihilator of M ).
Remark 5.44. We do not require that M be a doubly graded module over A L , so tensoring with M may involve a collapse of gradings.
Example 5.45. We consider the following examples:
(1) For M = Q[V ⇡0(L) ], we have YC KR (L, M) = YC KR (L).
(2) For the trivial module M = Q (on which all variables v [i],k act by zero), we have YC KR ( b , M) = C KR ( b ).
(3) Fix a scalar z c 2 Q for each link component c 2 ⇡ 0 (L) and let u be a formal variable of weight wt(u) = q 2 t 2 . Consider the L) acts as multiplication by z c u, and v c,r acts by zero when r > 1. In this case, YH KR (L, M z ) recovers the deformed triply-graded homology of Cautis-Lauda-Sussan [CLS20, Theorem 6.3], which satisfies splitting properties between components labeled by distinct scalars z c after inverting u.
Remark 5.46. The bigraded Khovanov-Rozansky gl N link homologies (as well as certain deformations thereof) can be computed from the Rickard complexes of colored braids by applying a functor that is trace-like up to homotopy, and which induces homotopy equivalences for the second Markov move similar as in Lemma 5.12; see [Wed19, Theorem 3.21] or [QR18,Section 6]. This implies the existence of link splitting deformations of bigraded colored Khovanov-Rozansky gl N link homologies. Specifically, for coe cients in M z as in Example 5.45 (3), one obtains bigraded colored homologies that satisfy link splitting properties and agree with the invariants from [CLS20, Theorem 5.4] (modulo conventions).
Finally, we collect various spectral sequences associated with YC KR (L). Let L be a colored link. For each c 2 ⇡ 0 (L), introduce an alphabet of (odd) variables
r=1 with wt(⇠ c,r ) = q 2r t 1 and where b(c) denotes the color of the component c. Set ⌅ ⇡0(L) := S c2⇡0(L) ⌅ c . Proposition 5.47. We have
where the twist D is Q[⌅ ⇡0(L) ]-linear and V ⇡0(L) -irrelevant. We also have
where the twist D 0 equals
Proof. The first statement is true by construction.The second statement follows from the first since the additional polynomial variables can be cancelled against extra exterior variables via the twist D 0 , as in standard Koszul duality (relating complexes of modules over polynomial and exterior algebras). ⇤
The twist D in Proposition 5.47 strictly increases V-degree, and D 0 strictly decreases ⌅-degree. Taking the spectral sequence associated to these filtered complexes thus yields the following.
Corollary 5.48. There are spectral sequences
Remark 5.49. The filtration by ⌅-degree on the complex tw D 0 (YC KR (L) ⌦ ^[⌅ ⇡0(L) ]) has finitely many steps, so the associated spectral sequence converges after finitely many steps. On the other hand, the filtration by V-degree on tw (C KR (L) ⌦ Q[V ⇡0(L) ]) is infinite. Nonetheless, the associated spectral sequence is a first quadrant spectral sequence (bounded below in both cohomological degree and V-degree), hence is reasonably well-behaved.
The relation between C KR (L) and YC KR (L) can be reformulated in terms of homological perturbation theory as follows.
Lemma 5.50. There is a homotopy equivalence
. Here, L) ] is viewed as a complex with zero di↵erential.
Proof. Since we are working with field coe cients, we can choose a homotopy equivalence in KJa ± , q ± , t ± K dg relating C KR (L) and its homology:
Homological perturbation (Proposition 4.1) now gives us a homotopy equivalence
for some twist D 00 . Here, we use Remark 4.2 (and the analogue of (42) to see the necessary nilpotence). ⇤ Lemma 5.50 is particularly useful in the following context.
Definition 5.51. A colored link L is parity if its (undeformed) triply-graded link homology H KR (L) is supported in purely even (or purely odd) cohomological degrees.
Theorem 5.52. Suppose that L is parity, then
Proof. The twist D 00 from Lemma 5.50 is necessarily zero for degree reasons: deg t (D 00 ) = 1 and L) ] is supported in exclusively even (or exclusively odd) cohomological degrees. ⇤
In [HRW21], we proved the colored skein relation for singular Soergel bimodules, which takes the form of a homotopy equivalence:
, where K( ) denotes the Koszul complex associated to the action of h i (X 2 X 0 2 ) for 1 i b. This is a homotopy equivalence of filtered complexes, where the filtration on the left-hand side is given by the index of summation s, and the filtration on the right-hand side requires some preparation to describe. We will use the shorthand q s(b 1) t s MCCS s a,b for the summands on the left-hand side, because, reading left-to-right, the associated diagram is the horizontal composition of "Merge-Crossing-Crossing-Split." We denote the term on the right-hand side (without the shift) by KMCS a,b , since it is a Koszul complex built on the complex The purpose of this section is to lift the colored skein relation, which is a homotopy equivalence in C a,b , to the curved setting. We begin in §6.1 with a quick discussion of (curved) Koszul complexes, and then in §6.2 use this language to describe the homotopy equivalence (103). In §6.3 and 6.4, we curve both sides of the skein relation and promote (103) to a homotopy equivalence in V a,b . 6.1. Koszul complexes with curvature. We will use the following construction of (curved) Koszul complexes. To this end, recall the reduced 2-strand categories V a,b from Definiton 4.38. Definition 6.1. Fix integers a, b 0 and let ⇠ 1 , . . . , ⇠ b be formal odd variables with wt(⇠ i ) = q 2i t 1 . For X 2 C a,b , let K(X) 2 C a,b denote the twisted complex
Remark 6.2. We can write K(X) and VK(X) as one-sided twisted complexes constructed from K(X k ) and VK(X k ), where X = ( L k X k , X ). More precisely:
and similarly for VK(X).
Proposition 6.3. K and VK( )
Proof. This follows since we may describe
The formation of curved Koszul complexes such as VK(X) is one of the most basic methods for constructing objects of V a,b . 6.2. The uncurved skein relation. We now recall the uncurved version of the skein relation, which was established in the companion paper [HRW21].
Consider the singular Soergel bimodules W k 2 a,b SSBim a,b indicated by the following webs (with relevant alphabets specified in the second diagram):
We identify the alphabets with subalphabets of X = {x 1 , . . . , x a+b } and X 0 = {x 0 1 , . . . , x 0 a+b } as follows:
(98) The relevant mnemonic is that MCS a,b is the result of certain Gaussian eliminations on MCS a,b , i.e. a "slimmer" version thereof. We can write KMCS a,b as a one-sided twisted complex as in Remark 6.2:
where
. Next, we change basis within each K(W k ) by declaring ( 100)
The e↵ect of this change of basis is captured by the following.
Proposition 6.6 ([HRW21, Proposition 3.10]). We have
With respect to this isomorphism, the di↵erential
if and only if i p j p 2 {0, 1} for all 1 p r, in which case it equals + m (see (27)) where m = P r p=1 (i p j p ). ⇤
A priori, the complex q b(a b 1) t b KMCS a,b (which is homotopy equivalent to the right-hand side of the skein relation) is graded both by cohomological degree in MCS a,b and the exterior algebra degree in ^[⇠ 1 , . . . , ⇠ b ]. According to [HRW21], the key to understanding the colored skein relation is a refinement of the exterior grading into two independent gradings. This is accomplished with the following. Definition 6.7. Let
with indices constrained by 0 s b and 0 l k b s. Proposition 6.8 ([HRW21, Proposition 3.12]). We have
where v , h , c are pairwise anti-commuting di↵erentials given as follows: • the vertical di↵erential v : P k,l,s ! P k,l 1,s is the direct sum of Koszul di↵erentials, up to the sign ( 1) k
; its component
and the connecting di↵erential c are uniquely characterized by h + c = H from Proposition 6.6, together with
In other words, h is the part of H which preserves s-degree and c is the part of H which increases s-degree by 1. Since the di↵erentials v and h preserve s-degree, we may reorganize the direct sum (101) as follows:
The skein relation essentially amounts to a topological interpretation of the s th summand above.
Proposition 6.9 ([HRW21, Theorem 3.4]). The complexes We now aim to promote Proposition 6.9 to the curved setting.
To begin, we immediately note that the complex
provides a curved lift of KMCS a,b . We will write VKMCS a,b explicitly as a one-sided twisted complex
◆ where we slightly abuse notation in writing H for VK( H ). Note that each of the objects VK(W k ) is itself a curved complex, whose di↵erentials we draw as arrows pointing downward and upward in illustrations such as (11) and Example 6.14 below.
Our goal is to write down the appropriate curved version of Proposition 6.8. First, consider the
2 )v i . Lemma 6.10. In terms of the ⇣-basis we have
Proof. The formula for the uncurved di↵erential is given above in Proposition 6.6; the formula for is immediate from (100). ⇤ Proposition 6.11. We have
where v , h , c are as in Proposition 6.8, and
Moreover, the endomorphisms v , v , h , c , c satisfy the following relations:
Proof. The first statement holds by construction, and everything else follows by taking components in ( + )
Since v preserves s, we may define curved lifts of MCCS s a,b as follows.
Definition 6.12.
This is a well-defined 1-morphism in V a,b by Proposition 6.11.
The following holds by construction.
Proposition 6.13. For all integers a, b 0 we have
in which the Maurer-Cartan element c + c increases the index s by one. ⇤ Example 6.14. We illustrate the complex VKMCS 2,2 , as well as the subquotients P •,•,s = q s t s MCCS s 2,2 for 0 s 2. We use the symbol • instead of ⌦ to declutter the diagram.
Black and blue horizontal arrows correspond to components of h . All other black and blue arrows indicate non-zero components of v . The curved twist v is indicated by red arrows. Finally the connecting di↵erential c and its curved correction c are marked using grey horizontal arrows and grey dashed arrows, respectively. We begin by defining these curved lifts. Since MCCS s a,b is not invertible for s 6 = 0, b, we cannot simply invoke Lemma 4.15 to define the curved lift. Instead, we will bootstrap from the s = 0 case.
More precisely, I (s) : C a,`! C a,`+s is defined as the composition of first applying ( ) ⇥ 1 s and then horizontal pre-and post-composing with 1 a ⇥ (`,s) S (`+s) and 1 a ⇥ (`+s) M (`,s) , respectively. Here, (`,s) S (`+s) and (`+s) M (`,s) denote the split and merge bimodules from (25), and |L| = `= |L 0 |. Convention 6.17. When considering endomorphisms of complexes of the form I (s) (X), we will often use the alphabet naming convention indicated below:
In other words, the above diagram indicates how Sym(X 1 |X 0 1 |X 2 |X 0 2 |L|L 0 |B) acts on the functor I (s) by natural transformations.
Set b = `+ s. We now extend I (s) to categories of curved complexes I (s) : V a,`! V a,b . Note that since (`,s) S (`+s) and (`+s) M (`,s) are not 1-morphisms in Y(SSBim), we cannot simply invoke the 2-categorical operations from §4.3. Definition 6.18. Let I (s) : V a,`! V a,b be the functor defined on objects by
and on morphisms by the map
induced from the map
and the assignment (67) (with L = X [a+1,...,a+`] and L 0 = X 0 [a+1,...,a+`] ).
Note that I (s) preserves the curvature elements by (68), together with the observation that
). Thus, this functor is indeed well-defined.
Let k , mimicking our notation for the v-variables. First, note that
) is a direct sum (with shifts) of terms of the form I (s) (W
k ]. [HRW21, Lemma 3.28] establishes an isomorphism
Crucially for our present considerations, this isomorphism is Sym(M (k) )-linear.
To prove (110), we only need to verify the commutativity of squares of the form
k ])
k ])
b ]
since the commutativity of analogous squares for v and h has already been established in the proof of [HRW21, Proposition 3.27]. Equation (107) shows that the action of v on W
( 1)
(Here, and in the following, we slightly abuse notation in the ordering of our tensor factors; since both h i (M (k) ) and all v-variables have even cohomological degree, this does not cause any hidden sign issues.) Now, we apply I (s) to this, obtaining (111)
On the other hand,
and it su ces to show that (112) equals (111). By comparing coe cients of
The latter is an application of Lemma 2.9 with X = M (k) , Y = L M (k) , r = m b + s, and c = b s j. ⇤ Corollary 6.20 (Curved skein relation). We have a homotopy equivalence in V a,b of the form
where the twist on the left-hand side strictly increases the parameter s and ( 0 ) c is V-irrelevant.
Proof. The right-hand side is a shift of VK(MCS) ' VKMCS a,b , as observed in (104). Proposition 6.13 shows this is homotopy equivalent to tw c + c L b s=0 q s(b 1) t s VMCCS s a,b where the twist c + c raises s-degree by one. In Proposition 6.19 we have seen that VMCCS s a,b ' I (s) (VFT a,b s ). Homological perturbation (i.e. Proposition 4.1) now extends the direct sum of these equivalences to a homotopy equivalence:
for some twist ( 0 ) c + ( 0 ) c that necessarily increases s-degree by the formula for the induced twist in Proposition 4.1. ⇤
Recall from Convention 6.15 that FT a,b = MCCS 0 a,b , which is a simplified model for the Rickard complex FT a,b = MCCS 0 a,b of the (a, b)-colored full twist braid. Let VFT a,b := VMCCS 0 a,b be its curved lift from Definition 6.12. In this section, we study a canonical closed morphism
) that is the colored analogue of the "link splitting" map from [GH]. We will use the notation from §4.8 throughout. 7.1. Characterization of the splitting map. To begin, we show that, up to homotopy and scalar multiple, there is a unique closed degree-zero map VFT a,b ! 1 a,b . This morphism admits an abstract description as a perturbation of the following.
Proof. Note that FT a,b equals the finite one-sided twisted complex tw v L b l=0 R l , where R l := L b k=l P k,l,0 , h and v decreases the l-degree by one. It follows that
! is likewise a one-sided twisted complex (( v ) ⇤ now increases l-degree by one). Considering the inclusion
)-linear isomorphism. The first result now follows from Definition 7.1. Next, we consider the V-deformation. Let
with wt(↵) = q 0 t 1 . Here, we use the shorthand
) for some twist ↵ 0 , which must be zero since 1 a,b is supported in a single cohomological degree. Thus, as desired. ⇤ Remark 7.3. Note that the morphism ⌃ a,b "constructed" in the proof of Proposition 7.2 is not explicitly specified, since we have not specified the constituent morphisms (this will be done in Definition 7.8 below). However, any two closed, degree-zero lifts of S a,b are necessarily homotopic, so ⌃ a,b is uniquely determined up to homotopy. We call this map the (deformed) splitting map. Indeed, the di↵erence between any two such curved lifts of S a,b is a closed, degree-zero element in the
) and the only such element in the latter is zero (which is nullhomotopic).
Example 7.4. For a b = 1, and using the notation introduced in §4.8, the curvature element in V a,1 is (x a+1 x 0 a+1 )v 1 . A curved lift ⌃ a,1 : VFT a,1 ! 1 a,1 of the splitting map is then given by:
(116)
Note that ⌃ a,1 = + 0 mod v1 . Remark 7.5. Since VFT a,b ' VFT a,b , we also obtain a degree-zero closed splitting map ⌃ a,b : VFT a,b ! 1 a,b , that is unique up to homotopy. Using this, we can construct a deformed splitting map for the colored positive full twist braid on any number of strands. Indeed, let FT b denote the Rickard complex associated to the full twist braid with strands colored b 1 , . . . , b m . The full twist braid can be written as a composition of (pure) braids of the form:
(here, in the interest of space, we break our conventions and write the braid vertically). Thus, the 2-strand splitting maps 7.2. Explicit description of the full twist splitting map. In this section, we explicitly describe the splitting map ⌃ a,b : VFT a,b ! 1 a,b as a morphism in V a,b . We will adopt the convention for alphabets labeling the web W k as in (98) and the convention for deformation parameters in V a,b as in §4.8. In particular, we consider deformation parameters (k) and V are therefore related by the substitution rule (67), with M (k) = X [a+1,a+k] , and we
we have the elements
which, as elements of Q[X, X 0 , V], do not depend on k so long as 1 k b.
Remark 7.6. The algebra Q[X, X 0 , V] Sa⇥S k ⇥S b k acts on the web W k , thought of as an object of
i and appropriately partially symmetric expressions in the y i may be regarded as an endomorphism of W k .
We begin with a reformulation of Lemma 4.33.
Lemma 7.7. For 1 r b, we have
Here the operators @ i are considered Q[V (k) ]-linear.
Proof. Example 3.17 shows that e k i (X [a+1,a+k 1] ) k i=1 and ( 1) i 1 x i 1 a+k k i=1 are dual bases with respect to the Sylvester operator
We thus compute
We now give an explicit model for the splitting map from Proposition 7.2.
) have non-zero components as indicated by the following diagram:
(118)
The map ⌃ k a,b is given as ( 1) ( b 2 ) times the composition
Here, the first map (denoted ⇡) is a "slanted identity"
which has weight q k(a b+k) t 0 .
Remark 7.9. The morphism X + : W k ! 1 a,b is given in extended graphical calculus as
In the perpendicular graphical calculus from §3.4, we have i=1 that are adapted to the "columns" VK(W k ). Recall from (100) that the change of variables between these sets of variables is lower-triangular, so we have identifications
using Lemma 2.8. In the latter case, note that this is zero unless l > k. Thus, (67) shows that in the variables v(k) i the map v takes the simplified form
Next, we compute h | P k,k 1,0 in the variables {⇠ i } k i=1 . This is simply given by the composition
where H is induced by 0 . In particular, H is determined in Koszul degree 1 by sending
i=1 implies that h still acts by 0 on terms indexed by (products of) ⇠ 1 , . . . , ⇠ k 1 . However, for ⇠ k , we compute
The value of h (⇠ k ) is obtained by projecting onto the span of {⇣
( 1) k i+1 e k i (M (k 1) )⇠ i .
k ], it therefore remains to verify that the diagram
anti-commutes for 1 i k (here we have omitted the q-degree shifts on the bimodules, and the overall factor of ( 1) ( b 2 ) on the components of ⌃ a,b ). To do so, we use the perpendicular graphical calculus. Recall from (29) and Remark 7.9 that
We then compute
. Now, we may simplify this latter diagram since the middle portion is locally the composition
i.e. it is multiplication by the element
and the result follows. (Here, we used that (
We record an immediate consequence, which will be used below.
Corollary 7.11. Let a,b : 1 a,b ! VFT a,b be the morphism of weight q 2b t 2b given by the composition
Proof. The first identity follows directly from the explicit description of ⌃ a,b in Theorem 7.10. Next, VFT a,b is invertible since it is (homotopy equivalent to) a curved Rickard complex, so we have where I (s) is the functor from Definition 6.18. We use this isomorphism to identify these complexes, which in turn allows us to identify the left-hand side of equation ( 109) with the complex:
In light of the topological interpretation of I (s) (VFT a,b s ) ⇠ = VMCCS s a,b a↵orded by Proposition 6.19, we will refer to this as the curved complex of threaded digons. The curved splitting map suggests the consideration of the analogous (curved) complex wherein each instance of VFT a,b s is replaced by the corresponding identity bimodule 1 a,b s .
Definition 7.12. The complex of (unthreaded) digons is the complex for the digon webs appearing in 1 a ⇥ VTD b (0). In this notation, the components of the di↵erential d take the form d s = id 1a ⇥ d 0 s , where d 0 s admits the following descriptions in perpendicular and extended graphical calculus:
. We now observe that the complex 1 a ⇥ VTD b (0) is homotopically trivial.
Lemma 7.13. The complex 1 a ⇥ VTD b (0) is contractible, with null homotopy given by k
Moreover, the components satisfy k s 1 k s = 0.
Proof. We illustrate the chain complex and the homotopy as: The main goal of this section is to assemble these into a chain map VTD b (a) ! 1 a ⇥VTD b (0) as follows.
Theorem 7.14 (Skein relation splitter). There is a degree-zero closed morphism
whose component r,s : q s(b 1) t s VMCCS s a,b ! q r(b 1) t r Dig r a,b is homotopic to the splitting map I (s) (⌃ a,b s ) if s = r, and zero otherwise.
The proof of this theorem will occupy the remainder of this section. We begin with a number of preparatory lemmata. First, the following implies that Theorem 7.14 holds in the undeformed setting. Proof. Since S a,b s is supported on the minimal t-degree summand, the same is true for
⇠ = q a(b s) W b s (see e.g. Definition 6.7 and equation ( 108)). By definition (see Proposition 6.8) the connecting di↵erential restricted to such summands is given by All of the squares are readily seen to commute. The Frobenius extension structure discussed in §3.3 implies that the triangle commutes, and commutativity of the hexagon can be verified using an explicit computation (or is a consequence of [QR16, Equation (3.8)] and foam isotopy). ⇤
Adjoining the squares (122) for 0 s b gives a chain map
which is the uncurved analogue of Theorem 7.14. It remains to deform this map, and our argument will proceed in two steps. First, we will show that the a deformation of the square (122) commutes up to homotopy. Second, we will "straighten" these maps to give the desired closed morphism.
To aid in the former, we next establish a particular instance of an adjunction involving the functors
b ] Here, we follow Convention 6.17 and denote the alphabet on the s-labeled edge in Dig s a,b by B and the alphabet on the (b s)-labeled edge by L. We view E a,(b s,s) as a module over
Lemma 7.16. Let X be a curved complex in V a,b s , then there is an isomorphism
of dg E a,(b s,s) -modules that is natural in X.
Proof. First, we consider the undeformed setting. Let X be a 1-morphism in C a,b , then we have that
Using Proposition 3.19 and Corollary A.2, we compute (123)
In our present notation, we have that End (b s,s) . Using the isomorphism in (123), we obtain
this implies our result. Indeed, it remains to see that the isomorphism is an isomorphism of complexes, and this follows from Definition 6.18, which implies that the di↵erential on both complexes is induced from that on X. ⇤ Proof. The preceding lemma gives the commutative diagram
Proposition 7.2 gives that the left vertical map is a homotopy equivalence, thus the right vertical map is as well. ⇤
We now have a curved analogue of Lemma 7.15.
Lemma 7.18. The square As in Remark 7.3, any degree-zero element in the latter that is V-irrelevant is zero, hence nullhomotopic. Thus s+1) (1 a,b s 1 ) , as desired. ⇤
Using this, we can now give the following.
Proof of Theorem 7.14. We will prove that there exist (strictly) commutative squares of the form
as desired. ⇤
As conjectured in [GNR21] and shown in [GH], the relation between Soergel bimodules and Hilbert schemes is mediated by the (positive) full twist braid. In this section, we speculate on the extension to the colored setting, culminating in Conjecture 8.15. In the following §9, we establish this conjecture in the 2-strand case. ), and denote the corresponding alphabets of cardinality b i acting on the left and right by X i and X 0 i , respectively (for 1 i m). Following the notation in (51), we denote the corresponding dg category of curved complexes by V b . Recall that the deformation parameters therein are denoted by V i = {v i,1 , . . . , v i,bi } and the curvature element is
Let One consequence of this conjecture would be that the splitting map ⌃ b is unique up to homotopy and nonzero scalar. One may fix the scalar by prescribing the restriction of ⌃ b to the term in cohomological degree zero. For the remainder of §8, we will assume Conjecture 8.1 and refer to ⌃ b as the splitting map.
We now introduce the main objects of interest.
Definition 8.2. Extending (115), we let It is an important problem to compute H(M b ) and J b explicitly. Indeed, in the case that b = 1 m := (1, . . . , 1), they were shown to be isomorphic, and the latter ideal was described explicitly by E. Gorsky and the first-named author [GH]. In turn, this computation was used to provide an explicit connection between triply-graded Khovanov-Rozansky homology and the geometry of the Hilbert scheme of points in C 2 . As such, we anticipate that an explicit presentation of J b for general b will provide an analogous relation between colored Khovanov-Rozansky homology and the geometry of the Hilbert scheme.
The explicit description of H(M 1 m ) ⇠ = J 1 m in [GH] relies on work of Elias and the first-named author [EH19]. Therein, they show that the uncolored (m, m)-torus link T (m, m) is parity (Definition 5.51). Based on this, and Conjecture 1.24, we propose the following. As Theorem 5.52 shows, the following is an immediate consequence of parity.
Proposition 8.6. If Conjecture 8.5 holds, then there is an isomorphism of bigraded Q[V]-modules:
We next record an important consequence of the expected "flatness" statement in Proposition 8.6. x k 1 j vk we have that y a+1 • • • y a+b is the sum of the monomial (v 1 ) b and terms involving the alphabet X 2 that are lower order with respect to the alphabet V (if we impose the monomial order v1 >
L , V], this implies that multiplication by y a+1 • • • y a+b is injective. Hence, YH KR (⌃ a,b ) is injective as well.
The general case follows analogously, using the description of the full twist braid as the horizontal composition of two-strand full twists from Remark 7.5. See also Theorem 10.10 below for a more general result. ⇤
Assuming the validity of Conjecture 8.5, the computation of the deformed colored homology of the colored torus link T (m, m; b) in lowest Hochschild-degree amounts to finding a presentation of the ideal J b E b . We now aim to formulate a precise conjectural description of J b . To help motivate this, we now recall the description in the uncolored case from [GH]. We begin by establishing some conventions.
Convention 8.8. As above, we write 1 N for the sequence (1, . . . , 1) of length N . If b = 1 N then each alphabet X i consists of a single variable x i and we have a single deformation parameter for each i that we denote by y i := v i,1 . The latter thus satisfy wt(y i ) = q 2 t 2 . In this case, define the total alphabets by
Convention 8.9. Below, we will consider various algebras E and ideals I C E that are generated by collections of elements S ⇢ E. Since we will consider di↵erent algebras that contain the same subsets S, we will denote such ideals by I = E • S in order to make clear the algebra in which each ideal lives.
The following combines the torus link computations of [EH19] (see also [Mel17,HM19]) with [GH, Theorem 6.16 and Corollary 6.17]. (We take the liberty of renaming J N from [GH] as J 1 N , in order to avoid clashes with our notation for the colored case.)
Theorem 8.10. The parity conjecture (Conjecture 8.5) holds when b = 1 N . Further, the full twist ideal J 1 N is equal to the ideal I 1 N Q[X, Y] generated by polynomials in Q[X, Y] that are antisymmetric with respect to the diagonal S N -action.
Inspired by this theorem, and also our results in §9 below, we now formulate a conjecture concerning the ideals J b for general b = (b 1 , . . . , b m ) that extends Theorem 8.10.
We write X = {x 1 , . . . , x N } where N := b 1 + • • • + b m and thus identify the alphabets X i (of cardinality b i ) with the subsets
Definition 8.12. Introduce alphabets
that is analogous to (55).
Let S N act on Q[X, Y] by simultaneously permuting the variables x i , y i .
denote the subalgebra inclusion given by the assignment (127).
given by the assignment (127) is S b1 ⇥ • • • ⇥ S bm -equivariant, provided we let this group act trivially on the alphabet V. Assume f 2 Q[X, Y] is antisymmetric with respect to S N . Then f is antisymmetric with respect to S b1 ⇥ • • • ⇥ S bm , and the same is true of ⇢(f ). Thus, each V-coe cient of ⇢(f ) is an antisymmetric polynomial in the alphabets X i , hence is divisible by each (X i ). The (X i ) are relatively prime, as they live in distinct tensor factors of
Definition 8.14. Define the ideal (128) One may study the colored homology of the full twist by comparison with the 2-strand case, as was done in the uncolored setting in [GH]. To make this precise, let A i,j be the generator of the pure braid group described in Remark 7.5, which we will regard as a colored braid via b. Let VA i,j denote the curved Rickard complex for A i,j . Consider the ideal in E b generated by J bi,bj ⇢ Sym(X i |X j )[V i , V j ] ⇢ E b , which we will denote E b J bi,bj . It is straightforward to check that E b J bi,bj is the ideal associated to A i,j ; in other words E b J bi,bj is the image of the map
induced by the splitting map VA i,j ! 1 b . Canonicity of splitting maps implies that ⌃ b : VFT b ! 1 b factors as the composition
Conjecture 8.16. We have J b = T i<j E b J bi,bj . In the uncolored case b = (1, . . . , 1) this was proven in [GH,Theorem 6.16]. Combining Conjecture 8.15 and Conjecture 8.16 gives the following, purely algebraic, conjecture.
Conjecture 8.17. We have an equality of ideals in E b :
where S bi+bj acts by simultaneously permuting variables within the alphabets X i [ X j and Y i [ Y j .
In the uncolored case this was proven by Haiman; see Lemma 8.18 below. 8.2. Full twists and Hilbert schemes. Pioneering work of Haiman [Hai01] shows that the ideal
appearing in Theorem 8.10 plays a crucial role in the study of the Hilbert scheme Hilb N (C 2 ) of N points in C 2 . Recall that the isospectral Hilbert scheme X N is defined to be the reduced fiber product
and that in [Hai01, Proposition 3.4.2] it is shown that X N is isomorphic to the blowup of (C 2 ) N at the ideal
Haiman goes on to to show that X N is normal, Cohen-Macaulay, and Gorenstein. The following seemingly (but not) straightforward result plays a role in these considerations, and will be used below.
Lemma 8.18 ([Hai01, Corollary 3.8.3]).
Inspired by the connections between uncolored, triply-graded link homology and the isospectral Hilbert scheme X N , we propose the following as a "colored" analogue of the latter. We expect that the relation between Soergel bimodules and the (isospectral) Hilbert scheme in [GNR21, OR, GH] has an analogue for singular Soergel bimodules, with X b playing the role of "colored" isospectral Hilbert scheme. In particular, let B b be the colored "full-twist dg algebra",
Post-composing with the splitting maps VFT We save investigations along these lines for future work.
In this section we prove Conjecture 8.5 and Conjecture 8.15 for the (positive) Hopf link, using the explicit description of the splitting map from §7.2, as well as the curved skein relation from §6. After the general setup from §8 we now return to the notational conventions for the 2-strand case that were established in §4.8.
We start by showing that the colored Hopf link is parity, which implies that the link splitting map to the corresponding colored unlink homology is injective. In §9.2 we compute generators for the Hopf link homology in lowest a-degree and in §9.3 we show that their images under the splitting map are (reduced) Haiman determinants. This implies the inclusion J a,b ⇢ I a,b between the ideals from Definitions 8.3 and 8.14, in the 2-strand case. We will refer to the ideal J a,b as the Hopf link ideal, since it is isomorphic to the (lowest Hochschild summand of the) deformed colored homology of the (a, b)colored Hopf link. In Sections 9.5 and 9.6, we establish the opposite inclusion, thus proving Conjecture
The following is now an immediate consequence of Lemma 9.1 together with Proposition 4.1.
Proposition 9.4. The (a, b)-colored Hopf link is parity.
Proof. Up to a global shift of magnitude a ab q 2ab t ab (see Definition 5.16), the complex C KR ( \ FT a,b ) associated to the closure of the (a, b)-colored full twist is given by HH
for some twist ↵ which strictly increases the index l. Here, the right-hand side is a finite one-sided twisted complex (see Remark 4.2), hence Proposition 4.1 (the homological perturbation lemma) allows us to apply the homotopy equivalences from Lemma 9.1 term-wise. We thus obtain
for some Maurer-Cartan element . Since the Maurer-Cartan element has t-degree one, it must be zero since it is acting on a complex which is supported in even cohomological degrees. ⇤
For the reader's convenience, we include also the Hochschild cohomology version of the above (in particular note that the shift a l disappears): Proof. This is an immediate consequence of Corollary 8.7. ⇤
Below, we will compute the image of the map H(⌃ a,b ) :
and that the homology of the latter is isomorphic to the lowest Hochschild degree summand of YH KR (T (2, 2; a, b) (up to shift).
where (as before) the map denoted ⇡ is a slanted identity of W b l .
We can regard CM l as a degree-zero map
Lemma 9.7. If we regard Hom SSBim (1 a,b , W b l ) as a complex with zero di↵erential, then the morphism CM l from Definition 9.6 is closed. Furthermore, the induced map of complexes
is surjective in homology.
Proof. The di↵erential of CM l sends ' 2 Hom SSBim (1 a,b , W b l ) to ( h + v + v ) '; here we identify P b l,b l,0 with W b l (up to a shift). Each of the di↵erentials v and h restrict to zero at the corners, thus to see that CM l is closed it su ces to show that v ' = 0 for all ' 2 Hom(1 a,b , W b l ). This follows since v is the Koszul di↵erential associated to the action of h i (X 2 X 0 2 ) on W b l for 1 i b l, and the central elements h i (X 2 X 0
2 ) act by zero on Hom SSBim (1 a,b , W b l ). This proves the first statement. For the second statement, Corollary 8.7 implies that post-composing with the splitting map Here, we have written ' a,b,l ( ) using the perpendicular graphical calculus from §3.4. In particular, taking = ; gives us the canonical map ' a,b,l (;) : 1 a,b ! W b l of weight q (a l)(b l) given by cr (digon creation on the b-labeled strand) followed by zip.
We now show that these maps span Hom SSBim (1 a,b , W b l ). Recall the alphabet labeling conventions from Convention 6.17, and observe that
This gives an action of Sym(X 1 |L|B) on W b l , and hence on Hom SSBim (1 a,b , W b l ) by post-composition. In our current situation, |B| = l and |L| = b l.
For use here and below, we record the following. 9.3. Haiman determinants and the Hopf link. We now compute the generators ⌃ a,b CM(' a,b,l ( )) 2 J a,b explicitly. We will see that these elements are special cases of the Haiman determinants from §2.4.
Definition 9.13. For a b l 0 and a partition 2 P (l, b l), the associated key shape Key l ( ) is defined to be the set of monic monomials in Q[x, y]: Key l ( ) := {x a+b l 1 , . . . , x, 1} [ {x 1 +l 1 y, . . . , x l y} ordered as indicated.
Convention 9.14. We will identify finite sets S of monic monomials in Q[x, y] with finite subsets of Z 0 ⇥ Z 0 , illustrated by collections of boxes living in the 4 th quadrant. In general, we will call such collections of boxes shapes.
For example, for a = 5, b = 3 and l = 2 there are three key shapes:
which are associated with the partitions ;, (1) and (1, 1) inside the 2 ⇥ 1 box, respectively. In a similar way, every partition specifies a finite set of monic monomials via the coordinates of the boxes in the Young diagram for . We caution the reader that this set of monomials naïvely associated to a partition is unrelated to both the key shape Key l ( ) and the set of monomials M N ( ) from Definition 2.13.
Example 9.15. The Haiman determinant associated to a key shape takes the form (130)
x l a+b y a+b Lemma 9.16 (Key lemma). For a b l 0 and a partition 2 P (l, b l), we have the following identity in E a,b :
Here, X = X 1 [X 2 = {x 1 , . . . , x a+b }, Y = {y 1 , . . . , y a+b }, and ⇡ is the reduction functor from Definition 4.40 (as discussed before Lemma 9.12 above).
Proof. We will use Definition 2.13 to help abbreviate parts of the proof. Specifically, for each c 1 and each weakly decreasing sequence = ( 1 • • • c 0) of length c, let M c ( ) = {x 1 +c 1 , . . . , x c 1 1 , x c } be the associated list of monomials. We note the following properties of M c ( ):
(1) M c (;) = {x c 1 , . . . , x, 1}.
(2) hdet M c ( ) = s (X) (X), where |X| = c.
(3) If 2 P (c, d), then the dual complementary partition b 2 P (d, c) satisfies
Now, consider the determinant Key l ( ) , depicted in (130), regarded as an element of Q[X, V] via (127). If we apply the algebra endomorphism ⇡, the result is:
a+b This determinant will be computed by (multiple column) Laplace expansion in the first a columns. Let M denote the matrix appearing on the right-hand side of (132), and let C = {1, . . . , a} be the indexing set for the selected columns. The result of the expansion is of the form: Thus, Laplace expansion yields the following identity:
where the sign is obtained from shu✏ing {x a+b l 1 , . . . , x, 1} into M a ( ) [ M b l ( b ).
We now compute the left-hand side of (131) using perpendicular graphical calculus. Up to the ± sign coming from (119), this equals
Here, M l ( )y := {x 1 +l 1 y, . . . , x l y}, and we use the equality
In the middle step, we can slide Q j y j through the top vertex (which acts via the Demazure operator associated with the longest element from Example 3.16) since it is S l -symmetric. Now, we can express the right-hand side of (133) as:
.
and the result follows. ⇤ Example 9.17. Let a b and consider the determinant associated to the "maximal" key shape Key b (;). We compute
x a 1 a x a 1 a+1 . . . x a 1 a+b . . . . . . . . . . . . 1 . . . 1 1 . . . 1 0 . . . 0 x b 1 a+1 y a+1 . . . x b 1 a+b y a+b . . . . . . . . . . . . 0 . . . 0 y a+1 . . . y a+b = y a+1 . . . y a+b (X 1 ) (X 2 ) by expanding the determinant in the first a columns. The product y a+1 . . . y a+b is familiar from Corollary 7.11 (which, in fact, had already shown that y a+1 . . . y a+b 2 J a,b
). 9.4. Reduced vs. unreduced. Recall that Conjecture 8.15 proposes an explicit algebraic description of the full twist ideal. In the present (two-strand) case, it posits that the Hopf link ideal J a,b is equal to the ideal (135)
Here, X = X 1 [ X 2 = {x 1 , . . . , x a+b } and Y = {y 1 , . . . , y a+b }. In order to establish a relation between the full twist ideal J a,b and the ideal I a,b , we now relate (certain) reduced and unreduced Haiman determinants.
To begin, we recall the method of computing determinants via the Schur complement. Proof. Gaussian elimination. ⇤
This immediately implies the following.
Corollary 9.19. Let Key l ( ) be a key shape and let
Note that the ± sign occurs here since our ordering of monomials di↵ers from the conventions for Haiman determinants (established in Definition 2.16) that is used in Definition 9.13. We will continue to use this ordering for the remainder of this section, since, for our current considerations, Proposition 9.18 is best-adapted to this ordering. Corollary 9.19 motivates the study of the matrix D l ( ) C l ( )A 1 B. Our next result computes the entries of this matrix. Lemma 9.20. Let z be an indeterminate and let r, s 0. If A is the (Vandermonde) matrix from (136), then
for all x i 2 X 1 (thus is the unique such polynomial).
Proof. Note that E a,0 = Sym(X 1 )[V L ]. By equation (127) and Corollary 2.10, there exists a polynomial m(z) 2 E a,0 [z] of degree a 1 that satisfies m(x i ) = x r i y s i for all x i 2 X 1 . Now, by definition, m r,s X1,V L (z) is a polynomial with coe cients in the field of fractions of Q[X 1 , Y 1 ] of degree a 1 satisfying (137). Since there is a unique such polynomial and E a,0 is a subring of this field of fractions, we have m r,s
We now relate the entries of D l ( ) C l ( )A 1 B in the unreduced and reduced setting.
Lemma 9.21. Let x j 2 X 2 and let r 0, then
L,k .
(By convention, the summation on the right-hand side is zero when r = 0.) Proof. To begin, note that Corollary 2.10 implies that
( 1) a t s (c a|a t) (X 1 )z t 1 when c a. Next, suppose that x i 2 X 1 , then
L,k .
On the other hand, recall from (62) that we have
L,b+i .
There are now two cases. First, suppose that b a r, then
L,k as desired. When b a r, the computation is similar:
L,k . ⇤ Lemmata 9.20 and 9.21 immediately relate the (unreduced) Haiman determinants Key l ( ) (X, Y) to certain reduced Haiman determinants ⇡( S (X, Y)). Namely, recall from Convention 9.14 that we identify sets of monic monomials S with finite subsets of Z 0 ⇥ Z 0 which are illustrated as collections of boxes that we call shapes. Definition 9.22. for 0 l b, let S l be the collection of all subsets of monomials S of the form
In other words, S l consists of shapes that only have boxes in the first two rows, with exactly l boxes in the second row confined to the first b positions, and with a + b l boxes in the first row confined to the first a + b positions and necessarily occupying the first a boxes in that row. In particular, note that Key l ( ) 2 S l .
Proposition 9.23. Let S 2 S l , then
with A and B as in (136). Lemma 9.20 then implies that (140)
The result now follows by applying Lemma 9.21 to the last l rows of (140). Indeed, this expresses each row of M S as the sum of the corresponding row in ⇡(M S ) and a Q[V L ]-linear combination of rows corresponding to monomials x s for a s a + b 1. Thus, we have that (1 a,b s ) for threaded digons. For the duration of this section, we will use Convention 6.17 for notation relevant to the functor I (s) . In particular, we will consider |B| = s and |L| = `(often we have `= b s). are alphabets as in (58). Further, we identify E a,(`,0) = E a,`( from Definition 8.2, with L in place of X 2 ) and regard this as a subalgebra of E a,(`,s) via the identification:
that is analogous to (67).
The relevance of this algebra stems from the following result.
Lemma 9.27. Let X be a curved complex in V a,`, then there is an isomorphism Hom V a,`+s (1 a,`+s , I (s) (X)) ⇠ = q `s Hom V a,`( 1 a,`, X) ⌦ E a,(`,0) E a,(`,s)
of dg E a,(`,s) -modules that is natural in X.
Proof. The endomorphism algebra End V a,b (I (s) (X)) is a Sym(X 1 |X 0
L , V (`+s) R ]-module. (As a reminder, we denote alphabets as in Convention 6.17.) The induced action on Hom V a,b (1 a,b , I (s) (X)) factors through the quotient in which we identify X 1 = X 0 1 and X 2 = X 0 2 , and thus also L = L 0 . Hence, we see that Hom V a,`+s (1 a,`+s , I (s) (X)) is indeed a dg E a,(`,s) -module. (The di↵erential is induced from the di↵erential on X, which commutes with this action.)
We now establish the isomorphism. Lemma 9.9 gives that Hom C a,`+s (1 a,`+s , I (s) (X)) ⇠ = q `s Hom C a,`( 1 a,`, X) ⌦ Sym(B) .
and in the deformed setting, this gives Hom V a,`+s (1 a,`+s , I (s) (X)) = Hom C a,`+s (1 a,`+s , I (s) (X))[V ] ⇠ = q `s Hom V a,`( 1 a,`, X) ⌦ E a,(`,0) E a,(`,s) .
The result follows since Definitions 6.18 and 9.26 show that this is an isomorphism of complexes. ⇤ In the next section, we will achieve our goal of showing that J a,b = I a,b by using an inductive argument involving the collection {J a,(b s,s) } b s=0 . To this end, we now observe that J a,(`,0) = J a,c ompletely determines J a,(`,s) . Lemma 9.30. For `+ s a, we have J a,(`,s) = E a,(`,s) • J a,(`,0) . Proof. Lemma 9.27 gives that M a,(`,s) ⇠ = M a,(`,0) ⌦ E a,(`,0) E a,(`,s) .
Taking homology and applying Corollary 9.28 gives J a,(`,s) ⇠ = J a,(`,0) ⌦ E a,(`,0) E a,(`,s) which is a restatement of the desired result. ⇤
Motivated by this, we introduce the following family of ideals that generalize the ideal I a,b C E a,b .
Definition 9.31. Let X (a+`) = X 1 [ L = {x 1 , . . . , x a+`} and Y (a+`) = {y 1 , . . . , y a+`} and set I a,(`,s) := E a,(`,s) `) , Y (a+`) ] is antisymmetric for S a+` .
Analogous to Lemma 9.30, the following holds (essentially by definition):
Lemma 9.32. For `+ s a, we have I a,(`,s) = E a,(`,s) • I a,(`,0) .
Proof. The ideals I a,(`,0) = I a,`a nd I a,(`,s) both have generators of the form f (X (a+`)
,Y (a+`) ) (X1) (L) , which give elements of E a,`a nd E a,(`,s) by expanding {y i } a+ì =1 in the alphabets V Lemma 9.35. I a,1 ⇢ J a,1 (and thus J a,1 = I a,1 ).
Proof. Recall that generators for I a,1 take the form f (X,Y) (X1) where f (X, Y) is anti-symmetric for the diagonal action of S a+1 on Q[X, Y]. Here, X = {x 1 , . . . , x a+1 }, X 1 = X r {x a+1 }, and Y = {y 1 , . . . , y a+1 }. Fix such a generator and note that f (X, Y) 2 I a+1 (see §8.2). Lemma 8.18 gives have that Hence,
and
• {e a (X 1 {x a+1 }), y a+1 } Thus, we have that
• {e a (X 1 {x a+1 }), y a+1 } and so f (X, Y) (X 1 ) 2 Q[X, V] • {e a (X 1 {x a+1 }), y a+1 } .
Further, since f (X,Y) (X1) 2 Q[X, V] is invariant under the S a -action, this implies that in fact f (X, Y) (X 1 ) 2 Sym(X 1 |{x a+1 })[V] • {e a (X 1 {x a+1 }), y a+1 } = ⇡(J a,1 ) ⇢ J a,1 and thus I a,1 ⇢ J a,1 , as desired. ⇤
Our proof of Theorem 9.33 for b 2 relies on Theorem 7.14, which has the following consequence.
Lemma 9.36. There is a (contractible) complex of the form We will occasionally abbreviate the notation of these complexes to J • ⇢ E • when there is no confusion as to the values of a and b.
Proof. The complex E • is obtained by applying the functor Hom ◆ from Definition 7.12 and using Lemma 9.27, which gives that Hom V a,b (1 a,b , q s(b 1) t s Dig s a,b ) ⇠ = q s(s 1) E a,(b s,s) . The component of the di↵erential q s(s 1) t s E a,(b s,s) ds ! q (s+1)s t s E a,(b s 1,s+1) is explicitly given by: q s(s 1) E a,(b s,s) =q s(s 1) Sym(X 1 |L|B)[V R ] = q (s+1)s E a,(b s 1,s+1) where here @ 1,s is (the tensor product of identity morphisms with) the Sylvester operator @ 1,s : Sym({x a+b s }|B) ! Sym({x a+b s } [ B) from Example 3.17, which has degree 2|B| = 2s. (The complex E • is contractible by Lemma 7.13.) Finally, the di↵erential on E a,(b •,•) restricts to a map J a,(b s,s) ! J a,(b s 1,s+1) by Theorem 7.14. ⇤
We next investigate the homology of the subcomplex J a,(b •,•) , since we are interested in identifying the quotient complex J a,(b,0) of the latter. We now work towards the proof of the following: ⌘ as s-filtered. Theorem 7.14 then implies that is a filtered chain map. Further, by Corollary 9.28 and Definition 9.29, the chain map (142) induces an isomorphism in homology for the associated graded complexes. Since the s-filtration is bounded, a straightforward argument using the long exact sequence associated to a short exact sequence of chain complexes and the five lemma implies 14 that induces an isomorphism We are therefore interested in the complex:
(144) Hom is supported on the far right. ⇤
We now establish Theorem 9.33 and Corollary 9.34.
14 Alternatively, this follows from the standard fact that if a morphism of spectral sequences is an isomorphism on a certain page, then it is an isomorphism on all subsequent pages.
Proof of Theorem 9.33. We show that I a,`= J a,`f or 1 `< a implies that I a,`+1 = J a,`+1 . The result then follows inductively from the base case 15 I a,1 = J a,1 that was established above in Lemma 9.35.
To begin, we first claim that the inclusion E a,(b,0) ,! E a,(b 1,1) restricts to an inclusion I a,(b,0) ,! I a,(b 1,1) , i.e. it sends elements in the former to the latter. Indeed, recall that the ideal I a,(b,0) is generated over E a,(b,0) by expressions f (X, Y) (X 1 ) (X 2 ) where f 2 Q[X, Y] is anti-symmetric for the diagonal action of S a+b . Similarly, Definition 9.31 implies that I a,(b 1,1) is generated over E a,(b 1,1) by expressions g(X r {x a+b }, Y r {y a+b }) (X 1 ) (X 2 r {x a+b }) ,
where g 2 Q[X, Y] is anti-symmetric for the diagonal action of S a+b 1 ⇥ S 1 . Thus, for S a+b -antisymmetric f 2 Q[X, Y], it su ces to show that (146)
f (X, Y) 2 E a,(b 1,1) -span
Note that (X 2 ) (X 2 r {x a+b }) 1 = Q a+b 1 j=a+1 (x j x a+b ). By the E a,(b,0) -linearity of the inclusion E a,(b,0) ,! E a,(b 1,1) , it su ces to check (146) in the case when f is an antisymmetrized monomial, i.e. when f (X, Y) = S (X, Y) = hdet(S) for some collection S = {m i (x, y)} a+b i=1 ⇢ Q[x, y] of monic monomials. In this case, we now see that (146) follows from Laplace expansion in the last column (j = a + b), after performing certain column operations on f (X, Y) = |m i (x j , y j )| a+b i,j=1 . To describe these column operations we use the identity a+b X j=a+1
( 1) a+b j (X 2 r {x j }) (X 2 )
• m i (x j , y j ) = @ a+1 • • • @ a+b 1 (m i (x a+b , y a+b ( 1) a+b j (X 2 r {x j }) (X 2 r {x a+b }) • m i (x j , y j ) = (X 2 ) (X 2 r {x a+b }) @ a+1 • • • @ a+b 1 (m i (x a+b , y a+b )) .
Laplace expansion in the last column then gives |m i (x j , y j )| a+b i,j=1 = a+b X k=1
( 1) a+b+k @ a+1 • • • @ a+b 1 (m k (x a+b , y a+b )) (X 2 ) (X 2 r {x a+b }) |m i (x j , y j )| i6 =k,j6 =a+b , which verifies (146). (Note that we have to slightly extend scalars by inverting the expression (X 2 r {x a+b }) to perform (147); the value of the determinant remains una↵ected regardless.)
15 Our use of Proposition 9.37 later in the proof will make clear that we cannot simply induct up from the obvious equality I a,0 = J a,0 .
Suppose now that 2 ` b and that we have shown that I a,m = J a,m for 1 m < `. In particular, Lemmata 9.30 and 9.32 imply that I a,(` s,s) = J a,(` s,s) for 1 s ` 2. We thus consider the following commutative diagram: I a,(`,0) I a,(` 1,1) I a,(` 2,2) 0 J a,(`,0) J a,(` 1,1) J a,(` 2,2)
where the vertical inclusion is given by Proposition 9.25. Since d 1 d 0 = 0 (by Lemma 9.36), we have that I a,(`,0) ⇢ ker I a,(` 1,1) d1 ! I a,(` 2,2) . However, ker I a,(` 1,1) d1 ! I a,(` 2,2) = ker J a,(` 1,1) d1 ! J a,(` 2,2)
and Proposition 9.37 implies that the latter equals J a,(`,0) . Thus I a,`= I a,(`,0) = J a,(`,0) = J a,`, which establishes the result by induction. ⇤
We record a consequence of Theorem 9.33, which gives a generating set for the ideal I a,b C E a,b of cardinality 2 b . Note that its statement (unlike its proof) appears to have nothing to do with link homology! for some 0 , 00 2 Br m (Z 1 ). Let b := min( 00 (b) i , 00 (b) i+1 ) (i.e. the minimal color involved in the crossing in the Artin generator ± i ) and let ! 2 S m . Then, there exists a pair of degree zero morphisms + : YC( + b,! ) ! YC( b,! ) , : YC( b,! ) ! q 2b t 2b YC( + b,! ) such that + ⇠ id YC( 0 ) ? (D ( 00 !)(i),( 00 !)(i+1) • id YC( 1 i ) ) ? id YC( 00 b,! ) and + ⇠ id YC( 0 ) ? (D ( 00 !)(i),( 00 !)(i+1) • id YC( i ) ) ? id YC( 00 b,! ) .
Proof. This is an immediate consequence of Lemma 10.4 using horizontal composition in Y(SSBim). ⇤
Note that if the crossing in Proposition 10.5 occurs between strands that correspond to the same component of the closure of a balanced, colored braid, then the relevant transparifer acts null-homotopically on the complex YC KR ( b ) from Definition 5.27 (since the alphabets X i and X j become identified, as do Y i and Y j ). Thus, in the setting of link homology, Proposition 10.5 is most interesting when the crossing is between strands in distinct components of the closure. Now, given a link L presented as the closure b of a braid , recall that it is possible to unlink a component of L from the remaining components via a sequence of crossing changes. More precisely, if L has components {L i } ì=1 , then there is a sequence of crossing changes that take the braid to a braid 0 where b
In the latter display, the square cup t denotes the split union, which is given by placing two links in disjoint 3-balls in S 3 . Passing to the complex YC KR ( b ) from Definition 5.27 (which is the relevant complex when considering braid closures), now immediately gives the following.
Proposition 10.6. Let L and L 0 be colored links presented as the closures of colored braids b and 0 b , and suppose that there is a sequence of crossing changes taking b to 0 b . Let p i,j and n i,j denote the number of positive-to-negative and negative-to-positive crossing changes between the components L i and L j , then there exist closed morphisms (Here, by slight abuse of notation, D i,j denotes the transparifer evaluated at the relevant quadrupel of alphabets associated with the i th and j th components of L.)
Intuitively speaking, "turning on" strand-wise curvature of the form h a 0 +1 (X X 0 )v a 0 +1 with v a 0 +1 invertible makes s behave as if its color is a 0 . We might say that such a strand now has e↵ective thickness a 0 .
Let us illustrate this with a concrete example. Extending our notation from §7.3, let TD b (a) denote the complex of (uncurved) threaded digons appearing on the left-hand side in the undeformed colored skein relation (95). Omitting all explicit grading shifts, this is
(150) TD b (a) := 0 @ u v a a b 0 b } ~! u v a a b 1 b } ~! • • • ! u v a a b b b } ~1 A .
By [HRW21, Proposition 2.31 and Theorem 3.4], TD b (a) ' 0 when a < b, so (150) detects the "thickness" of the a-labeled strand. If instead a b, this complex is not contractible, but becomes so after deforming to give the a-labeled strand e↵ective thickness smaller than b. Thus, this deformation forces the a-labeled strand to "act" as if its label was smaller than b.
To see this, it is convenient to replace TD b (a) by the (homotopy equivalent) complex on the righthand side of the colored skein relation ( 95 (Again, we are suppressing grading shifts.) Recall from Definition 6.1 that = P 1ib h i (X 2 X 0 2 )⌦⇠ ⇤ i . Fix an integer a 0 0 and "turn on" a curved Maurer-Cartan element with curvature h a 0 +1 (X 1 X 0 1 )v a 0 +1 with v a 0 +1 invertible. If a 0 < b, we claim that this deformed complex is contractible. Indeed, since X 1 + X 2 = X 0 1 + X 0 2 on KMCS a,b , equation (20) gives that h r (X 1 X 0 1 ) = h r (X 0 2 X 2 ) = r X j=1 h r j (X 0 2 X 2 )h j (X 2 X 0 2 ) so we may take j=1 h a 0 +1 j (X 0 2 X 2 )⇠ j does not break contractibility.
Since the present paper is already quite long, we save further investigations along these lines for future work. However, to come full circle, we do comment that our discussion here informs Conjecture 1.25 from the introduction. Recall that part of this conjecture asserts that a generalization of (150) should be interpreted as the b th "elementary symmetric function" of certain braids associated with the strands threading the digons. Some motivation is thus: as with (150), when the (e↵ective) size of the inputs to the elementary symmetric function e b ( ) is smaller than b, it vanishes.
Appendix A. Some Hom-space computations We first recall the following.
The indicated signs are required to give a well-defined 2-functor from the categorified quantum group in[KLMS12] to the 2-category of singular Soergel bimodules. We will always depict the signs in green for signed diagrams that are sent to the "naïve" (unsigned) movie of webs in the image.
Since b is balanced, ! 1 (b) = b, but we wish to emphasize the formal similarity to the complex YC( b,! 1 ).
) ⌦ ^[⇠ 1 , . . . , ⇠ b ]