Double Dimer Covers on Snake Graphs from Super Cluster Expansions

TL;DR

This paper introduces a new combinatorial approach using double dimer covers on snake graphs to compute super $$-lengths, extending previous dimer formulas and providing an alternative to existing methods.

Contribution

It presents an alternative combinatorial formula for super -lengths using double dimer covers, generalizing prior dimer-based formulas for cluster expansions.

Findings

Provides a new combinatorial expression for super -lengths.

Generalizes existing dimer formulas to super cluster expansions.

Connects super cluster variables with double dimer covers on snake graphs.

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.