In a 2011 breakthrough, Greene uses the "Tait graph" construction for knots, and a lattice-valued invariant of graphs, to prove that the Heegaard-Floer homology of the double branched cover is a complete mutation invariant of alternating knots. I will describe our recent work generalising this construction to knots on surfaces, and showing - by counterexample - that the analogous invariant is not complete up to mutation. I will give brief summary of the computational methods used - which are interesting in their own right - and end with a list of open questions.
This talk is based on joint work with Hans Boden, Damian Lin and Tilda Wilkinson-Finch.