Symmetric unions without cosmetic crossing changes

TL;DR

This paper investigates symmetric unions in knot theory to find an infinite family of hyperbolic three-bridge knots that challenge the cosmetic crossing conjecture, which states only trivial crossing changes preserve knot type.

Contribution

It introduces new examples of hyperbolic non-fibered three-bridge knots with constant determinant that satisfy the cosmetic crossing conjecture, expanding understanding of knot invariants.

Findings

Identified an infinite family of such knots.

Confirmed these knots satisfy the cosmetic crossing conjecture.

Provided new insights into symmetric unions and knot invariants.

Abstract

A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the well-known cosmetic crossing conjecture. This conjecture asserts that the only crossing changes which preserve the isotopy type of a knot are nugatory.