Cosmetic operations and Khovanov multicurves

Abstract

We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants $\widetilde{\text{Kh}}$ and $\widetilde{\text{BN}}$. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that $\widetilde{\text{Kh}}$ and $\widetilde{\text{BN}}$ detect if a Conway tangle is split.