  We study the Ozsv\'{a}th-Szab\'{o}-Thurston transverse invariant in
combinatorial link Floer homology for certain transverse cables
$\mathscr{L}_{p,q}$ of transverse link $L$ in $S^3$. Transverse cables
$\mathscr{L}_{p,q}$ are constructed from the grid diagram of $L$. The main
result is $\hat{\theta}(\mathscr{L}_{p,q})=0$ if and only if
$\hat{\theta}(L)=0$ 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.
