Comultiplication in link Floer homology and transversely nonsimple links

Abstract

For a word $w$ in the braid group $B_n$, we denote by $T_w$ the corresponding transverse braid in $\left({ℝ}^3,{\xi }_{rot}\right)$. We exhibit, for any two $g,h\in B_n$, a “comultiplication” map on link Floer homology $\tilde{\Phi }:\widetilde{\mathit{HFL}}\left(m\left(T_{hg}\right)\right)\to \widetilde{\mathit{HFL}}\left(m\left(T_g\#T_h\right)\right)$ which sends $\tilde{\theta }\left(T_{hg}\right)$ to $\tilde{\theta }\left(T_g\#T_h\right)$. We use this comultiplication map to generate infinitely many new examples of prime topological link types which are not transversely simple.