Fractional Dehn twists in knot theory and contact topology

TL;DR

This paper explores fractional Dehn twists as invariants in knot theory and contact topology, providing methods for their estimation and computation, and linking them to stabilization and overtwistedness in contact structures.

Contribution

It introduces techniques to estimate and compute fractional Dehn twist invariants and relates these to stabilization problems and overtwisted contact structures.

Findings

Methods for estimating fractional Dehn twists

Characterization of overtwisted contact structures

Connections to stabilization in knot theory

Abstract

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in classical knot theory, general open book decompositions, and contact topology. We include an elementary characterization of overtwistedness for contact structures described by open book decompositions.