Equivariant Unknotting Number and Involutive Khovanov Homology

TL;DR

This paper establishes a lower bound for the equivariant unknotting number of strongly invertible knots using involutive Bar-Natan homology, and identifies specific knots where this bound is strict.

Contribution

It introduces a new lower bound for equivariant unknotting number based on involutive homology and applies it to specific prime knots.

Findings

Lower bound for equivariant unknotting number via involutive homology

Identification of five prime knots with strict inequality between unknotting numbers

Extension of previous bounds to an equivariant setting

Abstract

We demonstrate that the equivariant unknotting number $\widetilde{u}(K)$ of a strongly invertible knot $K$ is bounded below by the $H$-torsion order $\widetilde{\mathrm{ord}}(K)$ of the involutive Bar-Natan homology $\widetilde{\mathrm{BN}}(K)$. This result serves as an equivariant analogue to the bound established by Alishahi. As an application, we identify five strongly invertible prime knots with crossing numbers at most $9$ for which the strict inequality $u(K) < \widetilde{u}(K)$ holds.