By theorems of Bennequin, Wrinkle and Orevkov-Shevchishin, transverse links in the unique tight contact structure on $R3$ may thought of as closed braids. For a word $h$ in the braid group on $n$ strands, let us denote by $T_h$ the corresponding transverse link. In this talk, I'll describe a relationship, for two braid words $h$ and $g$, between the transverse link invariants in Floer homology associated to $T_g$ and $T_h$ with the invariant associated to $T_{hg}$. And I'll describe how this relationship can be used to produce a plethora of new prime transversely non-simple link types.