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1. Stable homotopy refinement of quantum annular homology

   https://doi.org/10.1112/S0010437X20007721  

   Akhmechet, Rostislav; Krushkal, Vyacheslav; Willis, Michael (April 2021, Compositio Mathematica)

   null (Ed.)

   We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $$r\geq ~2$$ we associate to an annular link $$L$$ a naive $$\mathbb {Z}/r\mathbb {Z}$$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $$L$$ as modules over $$\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology. 

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2. Extremal Khovanov homology and the girth of a knot

   https://doi.org/10.1142/S0218216522500833  

   Sazdanović, Radmila; Scofield, Daniel (November 2022, Journal of Knot Theory and Its Ramifications)

   We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to [Formula: see text], then the girth of the link is equal to [Formula: see text] 

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3. A Kirby color for Khovanov homology

   https://doi.org/10.4171/JEMS/1589  

   Hogancamp, Matthew; Rose, David_E V; Wedrich, Paul (February 2025, Journal of the European Mathematical Society)

   We construct a Kirby color in the setting of Khovanov homology: an ind-object of the annular Bar-Natan category that is equipped with a natural handle slide isomorphism. Using functoriality and cabling properties of Khovanov homology, we define a Kirby-colored Khovanov homology that is invariant under the handle slide Kirby move, up to isomorphism. Via the Manolescu–Neithalath 2-handle formula, Kirby-colored Khovanov homology agrees with the\mathfrak{gl}_{2}skein lasagna module, hence is an invariant of 4-dimensional 2-handlebodies. 

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4. K-theoretic Gromov–Witten invariants of line degrees on flag varieties

   https://doi.org/10.1142/S0217751X24460138  

   Buch, Anders S; Chen, Linda; Xu, Weihong (November 2024, International Journal of Modern Physics A)

   A homology class [Formula: see text] of a complex flag variety [Formula: see text] is called a line degree if the moduli space [Formula: see text] of 0-pointed stable maps to X of degree d is also a flag variety [Formula: see text]. We prove a quantum equals classical formula stating that any n-pointed (equivariant, [Formula: see text]-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety [Formula: see text]. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags [Formula: see text]. Our formulas make it straightforward to compute the big quantum [Formula: see text]-theory ring [Formula: see text] modulo the ideal [Formula: see text] generated by degrees d larger than line degrees. 

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5. Classifying module categories for generalized Temperley–Lieb–Jones ∗-2-categories

   https://doi.org/10.1142/S0129167X20500275  

   Ferrer, Giovanni; Hernández Palomares, Roberto (April 2020, International Journal of Mathematics)

   Generalized Temperley–Lieb–Jones (TLJ) 2-categories associated to weighted bidirected graphs were introduced in unpublished work of Morrison and Walker. We introduce unitary modules for these generalized TLJ 2-categories as strong ∗-pseudofunctors into the ∗-2-category of row-finite separable bigraded Hilbert spaces. We classify these modules up to ∗-equivalence in terms of weighted bi-directed fair and balanced graphs in the spirit of Yamagami’s classification of fiber functors on TLJ categories and DeCommer and Yamashita’s classification of unitary modules for [Formula: see text]. 

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