A categorification of the Alexander polynomial in embedded contact homology

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.

48 pages, 4 figures

References in corpus (3)

• Intersection theory of punctured pseudoholomorphic curves
• Sutures and contact homology I
• Instanton Floer homology and the Alexander polynomial