Link Floer Homology and a Hofer Pseudometric on Braids

Abstract

Following an idea of Fr\'ed\'eric le Roux, we define in this paper a family of Hofer-type pseudonorms on braid groups, computing the minimal energy of a Hamiltonian diffeomorphism which fixes a Lagrangian configuration of circles on the unit disc and realises that braid type. We prove that in the case of braids with two strands we have in fact a norm, and we give lower estimates for braids with more strands. The main tool is Link Floer Homology, recently defined by D. Cristofaro-Gardiner, V. Humili\`ere, C.-Y. Mak, S. Seyfaddini and I. Smith, which we use to construct a family of quasimorphisms on the group of compactly supported Hamiltonian diffeomorphisms which is sensitive to the linking number of diffeomorphisms fixing Lagrangian links.