A family of transverse link homologies

Abstract

We define a homology ${\mathcal{\mathscr{H}}}_N$ for closed braids by applying Khovanov and Rozansky’s matrix factorization construction with potential $a{x}^{N+1}$. Up to a grading shift, ${\mathcal{\mathscr{H}}}_0$ is the HOMFLYPT homology defined by Khovanov and Rozansky. We demonstrate that for $N\ge 1$, ${\mathcal{\mathscr{H}}}_N$ is a ${\mathbb{Z}}_2\oplus {\mathbb{Z}}^{\oplus 3}$–graded $\mathbb{Q}\left[a\right]$–module that is invariant under transverse Markov moves, but not under negative stabilization/destabilization. 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 ${\mathcal{\mathscr{H}}}_N$ and the relation between ${\mathcal{\mathscr{H}}}_N$ and the $\mathfrak{s}\mathfrak{l}\left(N\right)$ Khovanov–Rozansky homology.