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A Transverse Braiding Theorem with Rational Open Book Decomposition

Ivan So

TL;DR

The work proves that every transverse link in a compact $3$-manifold $M$ with a contact structure $\xi$ supported by a rational open book $(B,\pi)$ can be transversely isotoped to a braid relative to $(B,\pi)$, extending Pavelescu's result from integral open books. The method builds a family of contact structures $\{\xi_T\}$ via Gray's theorem, extends a compatible contact form over the binding, and uses a page-fixing flow together with a wrinkling technique to manage singularities; the final step uses a Bennequin-type isotopy in a standard neighborhood of the binding to realize the transverse braid. A corollary delivers the corresponding topological braid version, and the results broaden the applicability of braiding techniques in contact topology to rational open books, with potential implications for transverse invariants and rational surgeries.

Abstract

By generalizing the argument of Pavelescu \cite{Pav12}, we show that every transverse link $ K $ in a compact contact 3-manifold can be transversely isotoped to a braid with respect to a rational open book decomposition.

A Transverse Braiding Theorem with Rational Open Book Decomposition

TL;DR

The work proves that every transverse link in a compact -manifold with a contact structure supported by a rational open book can be transversely isotoped to a braid relative to , extending Pavelescu's result from integral open books. The method builds a family of contact structures via Gray's theorem, extends a compatible contact form over the binding, and uses a page-fixing flow together with a wrinkling technique to manage singularities; the final step uses a Bennequin-type isotopy in a standard neighborhood of the binding to realize the transverse braid. A corollary delivers the corresponding topological braid version, and the results broaden the applicability of braiding techniques in contact topology to rational open books, with potential implications for transverse invariants and rational surgeries.

Abstract

By generalizing the argument of Pavelescu \cite{Pav12}, we show that every transverse link in a compact contact 3-manifold can be transversely isotoped to a braid with respect to a rational open book decomposition.

Paper Structure

This paper contains 5 sections, 8 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Rational open book decomposition and contact geometry
  4. Braiding with respect to an open book decomposition
  5. Proof of the main theorem

Key Result

Theorem 1.1

Suppose $(B,\pi)$ is an open book decomposition for the 3-manifold $M$ and $\xi$ is the contact structure supported by $(B,\pi)$. Let $K\subset M$ be a transverse link. Then $K$ can be transversely isotoped to a braid with respect to $(B,\pi)$.

Figures (1)

  • Figure 1: A wrinkling on $L$ such that the a problematic point is avoided in negative segments.

Theorems & Definitions (11)

  • Theorem 1.1: Pav12, Theorem 3.2
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: Gra59
  • Corollary 2.2: Gra59
  • Definition 2.3
  • Theorem 2.4: BEV12, Theorem 1.7
  • Definition 2.5: Sko92
  • Definition 2.6: Sko92
  • Theorem 2.7: Ben83
  • ...and 1 more