  In arXiv:1308.3152, the author proved that the Khovanov-Rozansky homology
$\mathcal{H}_N$ with potential $ax^{N+1}$ is an invariant for transverse links
in the standard contact $3$-sphere. In the current paper, we study the
$\mathbb{Z}_2 \oplus \mathbb{Z}^{\oplus 3}$-graded $\mathbb{Q}[a]$-module
structure of $\mathcal{H}_N$, which leads to better understanding of the effect
of stabilization on $\mathcal{H}_N$. As an application, we compute
$\mathcal{H}_N$ for all transverse unknots.
