Comultiplication in link Floer homology and transversely non-simple links

For a word w in the braid group on n-strands, we denote by T_w the corresponding transverse braid in the rotational symmetric tight contact structure on S^3. We exhibit a map on link Floer homology which sends the transverse invariant associated to T_{ws_i} to that associated to T_w, where s_i is one of the standard generators of B_n. This gives rise to a “comultiplication” map on link Floer homology. We use this to generate infinitely many new examples of prime topological link types which are not transversely simple.