Khovanov homology and mutation

Mutation invariance of the Jones polynomial follows more or less immediately from the perspective of skein modules for tangles, as introduced by Kauffman. Mutation invariance in the context of Khovanov homology, the Jones polynomial's categorification, is a little more subtle. I will talk about Bar-Natan’s important contributions to the foundations of the theory, how this interacts with the Fukaya categories of punctured spheres, and how this ultimately gives rise to a proof that reduced Khovanov homology is insensitive to mutation on knots over any choice of coefficient field. Part of this story is joint work with Artem Kotelskiy and Claudius Zibrowius.