Combinatorics of Cluster Algebras from Surfaces

Abstract

We construct a periodic infinite frieze using a class of peripheral elements of a cluster algebra of type D or affine A􏰕. We discover new symmetries and formulas relating the entries of this frieze and bracelet elements. We also present a correspondence between Broline, Crowe and Isaacs’s classical matching tuples and various recent interpretations of elements of cluster algebras from surfaces. We extend a T-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D. We further generalize our work and present T-path formulas for tagged arcs with one or two notchings on a marked surface with punctures.