#### Volume 17, issue 4 (2017)

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A categorification of the Alexander polynomial in embedded contact homology

### Gilberto Spano

Algebraic & Geometric Topology 17 (2017) 2081–2124
##### 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 $\stackrel{̂}{ECK}\left(K,Y,\alpha \right)$ of embedded contact homology for $K$ and conjectured that it is isomorphic to the knot Floer homology $\stackrel{̂}{HFK}\left(K,Y\right)$.

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 $\stackrel{̂}{ECK}$ 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