Conway’s approach to enumerating knots from the 1970s highlighted an interesting ambiguity wherein pairs of tangles assembled in different ways can give rise to different knots. The process relating the resulting knots has come to be known as knot mutation, and because this often leads to a subtle and difficult-to-detect change to a knot, has received considerable attention ever since. This talk will focus on the history of knot mutation in the context of the Jones polynomial and its categorification known as Khovanov homology. The former highlights how one might view mutation as pointing to hidden symmetries in the definition of a knot invariant; the proof that the Jones polynomial is unchanged under mutation is surprisingly simple from this perspective. By contrast, the latter is a much more subtle story, which ultimately makes a surprising appeal to the homological mirror symmetry of the 3-punctured sphere. This last step is part of a project with Artem Kotelskiy and Claudius Zibrowius.