An Alexander Polynomial Obstruction to Cosmetic Crossing Changes

TL;DR

This paper introduces an Alexander polynomial-based criterion that prevents cosmetic crossing changes in certain knots, including all alternating knots, and proves the conjecture for a specific family of pretzel knots.

Contribution

It generalizes previous work by providing an Alexander polynomial obstruction for cosmetic crossing changes in knots with L-space branched covers, including all alternating knots.

Findings

Proves the cosmetic crossing conjecture for a five-parameter family of pretzel knots.

Provides an Alexander polynomial condition that obstructs cosmetic crossing changes.

Discusses the status of the conjecture for alternating knots with eleven crossings.

Abstract

The cosmetic crossing conjecture posits that switching a non-trivial crossing in a knot diagram always changes the knot type. Generalizing work of Balm, Friedl, Kalfagianni and Powell, and of Lidman and Moore, we give an Alexander polynomial condition that obstructs cosmetic crossing changes for knots with $L$-space branched double covers, a family that includes all alternating knots. As an application, we prove the cosmetic crossing conjecture for a five-parameter infinite family of pretzel knots. We also discuss the state of the conjecture for alternating knots with eleven crossings.