An official website of the United States government
Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3‐sphere in terms of their presentation as open books and as Seifert fiber spaces. We provide Morse–Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg–Witten Floer theory developed by Hutchings and Taubes, and use them to compute the knot filtered embedded contact homology, for odd and positive.
more »
« less
On the computation of torus link homology
We introduce a new method for computing triply graded link homology, which is particularly well adapted to torus links. Our main application is to the $(n,n)$ -torus links, for which we give an exact answer for all $$n$$ . In several cases, our computations verify conjectures of Gorsky et al. relating homology of torus links with Hilbert schemes.
more »
« less
- Award ID(s):
- 1702274
- PAR ID:
- 10228858
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 155
- Issue:
- 1
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 164 to 205
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Auroux, Denis; Biran, Paul; Donaldson, Simon; Mrowka, Tomasz (Ed.)We describe a weighted A-infinity algebra associated to the torus. We give a combinatorial construction of this algebra, and an abstract characterization. The abstract characterization also gives a relationship between our algebra and the wrapped Fukaya category of the torus. These algebras underpin the (unspecialized) bordered Heegaard Floer homology for three-manifolds with torus boundary, which will be constructed in forthcoming work.more » « less
-
We give a classification of Legendrian torus links. Along the way, we give the first classification of infinite families of Legendrian links where some smooth symmetries of the link cannot be realized by Legendrian isotopies. We also give the first family of links that are non-destabilizable but do not have maximal \tb invariant and observe a curious distribution of Legendrian torus knots that can be realized as the components of a Legendrian torus link. This classification of Legendrian torus links leads to a classification of transversal torus links. We also give a classification of Legendrian and transversal cable links of knot types that are uniformly thick and Legendrian simple. Here we see some similarities with the classification of Legendrian torus links, but also some differences. In particular, we show that there are Legendrian representatives of cable links of any uniformly thick knot type for which no symmetries of the components can be realized by a Legendrian isotopy, others where only cyclic permutations of the components can be realized, and yet others where all smooth symmetries are realizable.more » « less
-
We study families of metrics on the cobordisms that underlie the differential maps in Bloom’s monopole Floer spectral sequence, a spectral sequence for links in [Formula: see text] whose [Formula: see text] page is the Khovanov homology of the link, and which abuts to the monopole Floer homology of the double branched cover of the link. The higher differentials in the spectral sequence count parametrized moduli spaces of solutions to Seiberg–Witten equations, parametrized over a family of metrics with asymptotic behavior corresponding to a configuration of unlinks with 1-handle attachments. For a class of configurations, we construct families of metrics with the prescribed behavior, such that each metric therein has positive scalar curvature. The positive scalar curvature implies that there are no irreducible solutions to the Seiberg–Witten equations and thus, when the spectral sequences are computed with these families of metrics, only reducible solutions must be counted. The class of configurations for which we construct these families of metrics includes all configurations that go into the spectral sequence for [Formula: see text] torus knots, and all configurations that involve exactly two 1-handle attachments.more » « less
-
We prove that Khovanov homology with coefficients in\Z/2\Zdetects the(2,5)torus knot. Our proof makes use of a wide range of deep tools in Floer homology, Khovanov homology, and Khovanov homotopy. We combine these tools with classical results on the dynamics of surface homeomorphisms to reduce the detection question to a problem about mutually braided unknots, which we then solve with computer assistance.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/s0010437x18007571
