Link Floer Homology and a Hofer Pseudometric on Braids

TL;DR

This paper introduces Hofer-type pseudonorms on braid groups using Link Floer Homology, establishing a norm for two-strand braids and providing lower bounds for more strands, linking symplectic geometry and braid theory.

Contribution

It defines a new family of Hofer-type pseudonorms on braid groups based on Link Floer Homology, connecting Hamiltonian dynamics with braid invariants.

Findings

For two-strand braids, the pseudonorm is a true norm.

Lower bounds are established for braids with more strands.

Constructs quasimorphisms sensitive to linking numbers.

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.