Transverse and Legendrian invariants of cables in combinatorial link Floer homology

We study the Ozsv'{a}th-Szab'{o}-Thurston transverse invariant in combinatorial link Floer homology for certain transverse cables ${\mathcal{L}}_{p,q}$ of transverse link $L$ in $S^3$. Transverse cables ${\mathcal{L}}_{p,q}$ are constructed from the grid diagram of $L$. The main result is $\widehat{\theta }({\mathcal{L}}_{p,q})=0$ if and only if $\widehat{\theta }(L)=0$ for $\frac{q}{p}$ sufficiently large. We also prove a similar result for invariants of Legendrian knots. Our proof uses an inclusion map $i$ of certain grid complexes associated to $L$ and $L_{p,q}$. We use these results to generate many infinite families of examples of Legendrian and transversely non-simple topological link types.