#### Volume 16, issue 1 (2016)

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A family of transverse link homologies

### Hao Wu

Algebraic & Geometric Topology 16 (2016) 41–127
##### Abstract

We define a homology ${\mathsc{ℋ}}_{N}$ for closed braids by applying Khovanov and Rozansky’s matrix factorization construction with potential $a{x}^{N+1}\phantom{\rule{0.3em}{0ex}}$. Up to a grading shift, ${\mathsc{ℋ}}_{0}$ is the HOMFLYPT homology defined by Khovanov and Rozansky. We demonstrate that for $N\ge 1$, ${\mathsc{ℋ}}_{N}$ is a ${ℤ}_{2}\oplus {ℤ}^{\oplus 3}\phantom{\rule{0.3em}{0ex}}$–graded $ℚ\left[a\right]$–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 ${\mathsc{ℋ}}_{N}$ and the relation between ${\mathsc{ℋ}}_{N}$ and the $\mathfrak{s}\mathfrak{l}\left(N\right)$ Khovanov–Rozansky homology.

##### Keywords
transverse link, Khovanov–Rozansky homology, HOMFLYPT polynomial
##### Mathematical Subject Classification 2010
Primary: 57M25, 57R17