A categorification of the Alexander polynomial in embedded contact homology

Spano, Gilberto

doi:10.2140/agt.2017.17.2081

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A categorification of the Alexander polynomial in embedded contact homology

Gilberto Spano

Algebraic & Geometric Topology 17 (2017) 2081–2124

DOI: 10.2140/agt.2017.17.2081

Abstract

Given a transverse knot $K$ in a three-dimensional contact manifold $(Y, α)$ , Colin, Ghiggini, Honda and Hutchings defined a hat version $\hat{ECK} (K, Y, α)$ of embedded contact homology for $K$ and conjectured that it is isomorphic to the knot Floer homology $\hat{HFK} (K, Y)$ .

We define here a full version $ECK (K, Y, α)$ and generalize the definitions to the case of links. We prove then that if $Y = S^{3}$ , then $ECK$ and $\hat{ECK}$ categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogous to that for knot and link Floer homologies in the minus and, respectively, hat versions.

Keywords

embedded contact homology, Alexander polynomial, categorification

Mathematical Subject Classification 2010

Primary: 57M27, 57R17, 57R58

References

Publication

Received: 9 February 2016

Revised: 5 December 2016

Accepted: 26 December 2016

Published: 3 August 2017

Authors

Gilberto Spano
Institut Fourier Université Grenoble Alpes 38000 Grenoble France

Volume 17, issue 4 (2017)