Double Dimers and Super Ptolemy Relations

Ptolemy’s theorem relates the lengths of the diagonals and sides of a quadrilateral inscribed in a circle, and this is the inspiration for the mutation relation in a cluster algebra associated to a triangulated surface. A super-symmetric version of the Ptolemy relation was introduced recently by Penner and Zeitlin, involving anticommuting variables. Previous work of the first author and Schiffler gave a formula for cluster variables in terms of perfect matchings of some planar graph. Motivated by this, we investigate certain algebraic expressions, obtained via iterating the super Ptolemy relation, that may be given as a sum over double dimer covers of this graph.