A family of transverse link homologies

Wu, Hao

doi:10.2140/agt.2016.16.41

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A family of transverse link homologies

Hao Wu

Algebraic & Geometric Topology 16 (2016) 41–127

DOI: 10.2140/agt.2016.16.41

Abstract

We define a homology $ℋ_{N}$ for closed braids by applying Khovanov and Rozansky’s matrix factorization construction with potential $a x^{N + 1}$ . Up to a grading shift, $ℋ_{0}$ is the HOMFLYPT homology defined by Khovanov and Rozansky. We demonstrate that for $N \geq 1$ , $ℋ_{N}$ is a $ℤ_{2} \oplus ℤ^{\oplus 3}$ –graded $ℚ [a]$ –module that is invariant under transverse Markov moves, but not under negative stabilization/destabilization. Thus, for $N \geq 1$ , this homology is an invariant for transverse links in the standard contact $S^{3}$ , but not for smooth links. We also discuss the decategorification of $ℋ_{N}$ and the relation between $ℋ_{N}$ and the $s l (N)$ Khovanov–Rozansky homology.

Keywords

transverse link, Khovanov–Rozansky homology, HOMFLYPT polynomial

Mathematical Subject Classification 2010

Primary: 57M25, 57R17

References

Publication

Received: 8 April 2014

Revised: 12 February 2015

Accepted: 15 April 2015

Published: 23 February 2016

Authors

Hao Wu
Department of Mathematics The George Washington University Monroe Hall, Room 240 2115 G Street, NW Washington DC 20052 USA

Volume 16, issue 1 (2016)