Equivariant unknotting numbers of strongly invertible knots

TL;DR

This paper investigates symmetric crossing changes in strongly invertible knots, revealing that the equivariant unknotting number does not behave additively under connected sum, challenging previous conjectures.

Contribution

It introduces the concept of equivariant unknotting number for strongly invertible knots and demonstrates its non-additivity under connected sum, contrary to prior beliefs.

Findings

Equivariant unknotting number is not additive under connected sum.

Symmetric crossing change operations are central to the study.

Challenges the longstanding conjecture of unknotting number additivity.

Abstract

We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture that unknotting number is additive.