  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.
