A categorification of the Alexander polynomial in embedded contact homology

Abstract

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

We define here a full version $ECK\left(K,Y,\alpha \right)$ and generalize the definitions to the case of links. We prove then that if $Y=S^3$, then $ECK$ and $\widehat{ECK}$ 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.