A Family of Transverse Link Homologies

Abstract

We define a homology ${\mathcal{H}}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $a{x}^{N+1}$. Up to a grading shift, ${\mathcal{H}}_0$ is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for $N\ge 1$, ${\mathcal{H}}_N$ is a ${\mathbb{Z}}_2\oplus {\mathbb{Z}}^{\oplus 3}$-graded $\mathbb{Q}[a]$-module that is invariant under transverse Markov moves, but not under negative stabilization/de-stabilization. Thus, for $N\ge 1$, 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 ${\mathcal{H}}_N$ and the relation between ${\mathcal{H}}_N$ and the $\mathfrak{sl}(N)$ Khovanov-Rozansky homology defined in arXiv:math/0401268.

56 pages, 24 figures. Second version: corrected typos and minor mistakes

References in corpus (1)

• A topological introduction to knot contact homology