Recent joint work with Artem Kotelskiy and Claudius Zibrowius gives a proof that the reduced version of Khovanov homology is insensitive to Conway mutation of knots over any coefficient field. In order to get our hands on this fact, we studied the relative version of Khovanov's invariant for Conway tangles, as developed by Bar-Natan, which can be expressed in terms of immersed curves. Crucially, taking this approach one is forced to rule out certain immersed curves as arising as tangle invariants. This is the part of the story I will focus on, as it makes a surprising appeal to the Homological Mirror Symmetry of the 3-punctured sphere.