Cosmetic operations and Khovanov multicurves

TL;DR

This paper proves an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots using Khovanov multicurve invariants, and applies these techniques to related conjectures and detection problems.

Contribution

It introduces an equivariant approach to the Cosmetic Surgery Conjecture leveraging Khovanov multicurve invariants, extending previous results and detection capabilities.

Findings

Proves an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots.

Reproves a result of Wang on the Cosmetic Crossing Conjecture and split links.

Shows that Khovanov and Bar-Natan invariants detect split Conway tangles.

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 $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$. 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 $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$ detect if a Conway tangle is split.