Cosmetic operations and Khovanov multicurves.

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 Kh~ and 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 Kh~ and BN~ detect if a Conway tangle is split.