Khovanov homology and the Fukaya category of the three-punctured sphere

About 20 years ago, Dror Bar-Natan described an elegant generalisation
of Khovanov homology to tangles with any number of endpoints, by
considering certain quotients of two-dimensional relative cobordism
categories.  I claim that these categories are in general not
well-understood (not by me in any case).  However, if we restrict to
tangles with four endpoints, things simplify and Bar-Natan's category
turns out to be closely related to the wrapped Fukaya category of the
four-punctured sphere.  This relationship gives rise to a symplectic
interpretation of Khovanov homology that is useful both for doing
calculations and for proving theorems.  I will discuss joint work in
progress with Artem Kotelskiy and Liam Watson where we investigate what
happens when we fill in one of the punctures.