Double Dimer Covers on Snake Graphs from Super Cluster Expansions

Abstract

In a recent paper, the authors gave combinatorial formulas for the Laurent expansions of super $\lambda$-lengths in a marked disk, generalizing Schiffler's $T$-path formula. In the present paper, we give an alternate combinatorial expression for these super $\lambda$-lengths in terms of double dimer covers on snake graphs. This generalizes the dimer formulas of Musiker, Schiffler, and Williams.

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