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
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