Equivariant Unknotting Number and Involutive Khovanov Homology

Abstract

We demonstrate that the equivariant unknotting number $\tilde{u}(K)$ 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 $u(K)<\tilde{u}(K)$ holds.

30 pages, 23 figures. Comments are welcome!