Stability of the braid types defined by the symplecticmorphisms preserving a link

TL;DR

This paper investigates the stability of braid types associated with Hamiltonian symplectic morphisms preserving a link, demonstrating that small Hofer distances imply identical braid types, thus establishing the nondegeneracy of a related pseudometric.

Contribution

It proves that Hamiltonian symplectic morphisms with small Hofer distance define the same braid type, confirming the nondegeneracy of Morabito's pseudometric.

Findings

Small Hofer distance implies identical braid types

Morabito's pseudometric is nondegenerate

Braid types are stable under small Hamiltonian perturbations

Abstract

Fix a suitable link on the disk. Recently, F. Morabito associates each Hamiltonian symplecticmorphism preserving the link to a braid type. Based on this construction, Morabito defines a family of pseudometrics on the braid groups by using the Hofer metric. In this paper, we show that two Hamiltonian symplecticmorphisms define the same braid type provided that their Hofer distance is sufficiently small. As a corollary, the pseudometric defined by Morabito is nondegenerate.