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 $\gamma$-symmetric perfect matchings and $\gamma$-compatible pairs of $\gamma$-symmetric perfect matchings, respectively. We simplify and unify these approaches by considering good matchings of loop graphs.