Surface cluster algebra expansion formulae via loop graphs

Abstract

In 2011 Musiker, Schiffler and Williams obtained expansion formulae for cluster algebras from orientable surfaces. For singly and doubly notched arcs these formulae required the notion of ${\hat{I}}^3$-symmetric perfect matchings and ${\hat{I}}^3$-compatible pairs of ${\hat{I}}^3$-symmetric perfect matchings, respectively. We simplify and unify these approaches by considering good matchings of loop graphs.

43 pages, 30 figures. Comments welcome. We generalise the definition of loop graph to cover full generality; this main edit was first presented in a CSitN 2023 talk (https://jon-m-wilson.github.io/CSitN-2023/schedule). Many figures added to improve readability. References added to reflect developments in the literature since original preprint. Corollary regarding poset expansion formulae added