Comultiplication in link Floer homology and transversely non-simple links

TL;DR

This paper introduces a comultiplication map in link Floer homology related to transverse braids, enabling the construction of numerous prime links that are not transversely simple, thus advancing understanding of link invariants.

Contribution

It presents a novel comultiplication map in link Floer homology that links transverse invariants of braids, leading to new examples of non-transversely simple links.

Findings

Defined a map on link Floer homology connecting transverse invariants.

Generated infinitely many prime links that are not transversely simple.

Provided new tools for studying transverse link invariants.

Abstract

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.