A categorification of the Alexander polynomial in embedded contact homology

TL;DR

This paper introduces a full version of embedded contact homology for knots and links in contact 3-manifolds, proving it categorifies the multivariable Alexander polynomial, extending prior work on the hat version.

Contribution

It defines a full embedded contact homology for knots and links and proves it categorifies the multivariable Alexander polynomial, generalizing previous hat version results.

Findings

ECK categorifies the multivariable Alexander polynomial in S^3.

The full ECK extends the hat version to links.

Results provide new algebraic invariants for knots and links in contact manifolds.

Abstract

Given a transverse knot $K$ in a three dimensional contact manifold $(Y,\alpha)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,\alpha)$, and conjecture that it is isomorphic to the knot Floer homology $\widehat{HFK}(K,Y)$. We define here a full version $ECK(K,Y,\alpha)$ and generalise the definitions to the case of links. We prove then that, if $Y = S^3$, $ECK$ and $\widehat{ECK}$ categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogue to that for knot and link Floer homologies in the plus and, respectively, hat versions.