Cosmetic surgery in L-spaces and nugatory crossings

TL;DR

This paper proves the cosmetic crossing conjecture for certain knots in L-spaces, including many with up to nine crossings, using Dehn surgery techniques and homological conditions.

Contribution

It introduces a novel approach using Dehn surgery characterization to verify the conjecture for a broad class of knots in integer homology spheres.

Findings

Proves the conjecture for all knots with up to nine crossings.

Establishes the conjecture for knots in L-spaces satisfying specific homological conditions.

Provides new examples of knots where the conjecture holds, including pretzel and symmetric union knots.

Abstract

The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology spheres whose branched double covers are L-spaces satisfying a homological condition. This includes as a special case all alternating and quasi-alternating knots with square-free determinant. As an application, we prove the cosmetic crossing conjecture holds for all knots with at most nine crossings and provide new examples of knots, including pretzel knots, non-arborescent knots and symmetric unions for which the conjecture holds.