Knots without cosmetic crossings

TL;DR

This paper proves that certain satellite knots and Whitehead doubles of prime, non-cable knots do not admit cosmetic crossing changes, advancing understanding of knot invariants and the nugatory crossing conjecture.

Contribution

It establishes that satellite knots with specific properties and Whitehead doubles of prime, non-cable knots have no cosmetic crossing changes, confirming the nugatory crossing conjecture in these cases.

Findings

Satellite knots with winding number zero and pattern K' admit no cosmetic crossing changes.

Whitehead doubles of prime, non-cable knots satisfy the nugatory crossing conjecture.

The results extend the class of knots known to have no cosmetic crossing changes.

Abstract

Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-cable knots.