Bases for Cluster Algebras from surfaces without punctures

In previous work with Ralf Schiffler and Lauren Williams, we have provided combinatorial formulas, involving snake graphs, for Laurent expansions for cluster variables coming from a bordered surface. In this talk, I will discuss extensions of this work involving band graph formulas for closed loops, and how these formulas allow us to construct vector space bases for such algebras. This is work in progress with Schiffler and Williams.