Cosmetic crossing changes of fibered knots

Kalfagianni Efstratia

doi:10.1515/crelle.2011.148

Abstract

We prove the nugatory crossing conjecture for fibered knots. We also show that if a knot K is n-adjacent to a fibered knot K′, for some n > 1, then either the genus of K is larger than that of K′ or K is isotopic to K′.

Highlights

An open question in classical knot theory is the question of when a crossing change on a knot changes the isotopy class of the knot
Using geometric properties of fibered knot complements, the problem reduces to the question of whether a power of a Dehn twist on the surface ∂N along he curve ∂D, can be written as a single commutator in the mapping class group of the surface
A result of Kotschick implies that a product of Dehn twists of the same sign, along a collection of disjoint, homotopically essential curves on an orientable surface cannot be written as a single commutator in the mapping class group of the surface

Summary

Introduction

An open question in classical knot theory is the question of when a crossing change on a knot changes the isotopy class of the knot. Using geometric properties of fibered knot complements, the problem reduces to the question of whether a power of a Dehn twist on the surface ∂N along he curve ∂D, can be written as a single commutator in the mapping class group of the surface. A result of Kotschick implies that a product of Dehn twists of the same sign, along a collection of disjoint, homotopically essential curves on an orientable surface cannot be written as a single commutator in the mapping class group of the surface Using this result, we show that the assumption that K is isotopic to K′ implies that ∂D bounds a disc in the complement of K. We organize the paper as follows: In Section 2 we summarize the mapping class group results that we need for the proof of Theorem 1.1 and in Section 3 we summarize known properties of fibered knot complements. Throughout the paper we work in the PL or the smooth category

Commutator length and Dehn twists

Uniqueness properties of knot fibrations

Splittings of fibered knot complements

Crossing changes and Dehn twists

Adjacency to fibered knots