A Family of Transverse Link Homologies

TL;DR

This paper introduces a new family of homologies for closed braids, extending Khovanov-Rozansky constructions, which serve as invariants for transverse links in contact 3-spheres but not for smooth links.

Contribution

It defines the homology $\\mathcal{H}_N$ for closed braids using matrix factorizations, generalizing HOMFLYPT homology and establishing its invariance properties for transverse links.

Findings

or $N q 0$, $\\mathcal{H}_N$ is a graded module with specific invariance properties.

or $N q 0$, $\\mathcal{H}_N$ is an invariant of transverse links in $S^3$.

iscusses the relation between $\\mathcal{H}_N$ and $\\mathfrak{sl}(N)$ Khovanov-Rozansky homology.

Abstract

We define a homology $\mathcal{H}_N$ for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential $ax^{N+1}$. Up to a grading shift, $\mathcal{H}_0$ is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for $N \geq 1$, $\mathcal{H}_N$ 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 $N\geq 1$, 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 $\mathcal{H}_N$ and the relation between $\mathcal{H}_N$ and the $\mathfrak{sl}(N)$ Khovanov-Rozansky homology defined in arXiv:math/0401268.