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.