Algebraic & Geometric Topology

A categorification of the Alexander polynomial in embedded contact homology

Gilberto Spano

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Abstract

Given a transverse knot K in a three-dimensional contact manifold (Y,α), Colin, Ghiggini, Honda and Hutchings defined a hat version ECK̂(K,Y,α) of embedded contact homology for K and conjectured that it is isomorphic to the knot Floer homology HFK̂(K,Y ).

We define here a full version ECK(K,Y,α) and generalize the definitions to the case of links. We prove then that if Y = S3, then ECK and ECK̂ 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.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2081-2124.

Dates
Received: 9 February 2016
Revised: 5 December 2016
Accepted: 26 December 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841438

Digital Object Identifier
doi:10.2140/agt.2017.17.2081

Mathematical Reviews number (MathSciNet)
MR3685603

Zentralblatt MATH identifier
1373.57033

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57R17: Symplectic and contact topology 57R58: Floer homology

Keywords
embedded contact homology Alexander polynomial categorification

Citation

Spano, Gilberto. A categorification of the Alexander polynomial in embedded contact homology. Algebr. Geom. Topol. 17 (2017), no. 4, 2081--2124. doi:10.2140/agt.2017.17.2081. https://projecteuclid.org/euclid.agt/1510841438


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