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 $L_{p,q}$ of transverse link $L$ in $S^3$. Transverse cables $L_{p,q}$ are constructed from the grid diagram of $L$. The main result is $\hat{\theta}(L_{p,q})=0$ if and only if $\hat{\theta}(L)=0$. We also prove a related 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 prove that $i$ induces inclusion on homology. We use these results to generate many infinite families of examples of Legendrian and transversely non-simple topological link types.