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

Guanheng Chen

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.

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

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.
Paper Structure (7 sections, 7 theorems, 40 equations)

This paper contains 7 sections, 7 theorems, 40 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Main results
  4. Quantitative Heegaard Floer homology
  5. Continuous morphisms
  6. Filtration
  7. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

Suppose that the link $\underline{L}$ is $\eta$-admissible (see Definition def1) with $k$ components. Then there exists a positive constant $\varepsilon_{\underline{L}}$ depending on $\underline{L}$ such that the following property holds. Given $\varphi \in Ham_{\underline{L}}(\mathbb{D}, \omega)$,

Theorems & Definitions (23)

  • Theorem 1
  • Corollary 1.1
  • proof
  • Definition 2.1
  • Remark 1
  • Remark 2
  • Definition 2.2
  • Remark 3
  • Remark 4
  • Lemma 2.3
  • ...and 13 more