Cluster algebraic interpretation of infinite friezes

TL;DR

This paper explores the connection between infinite friezes, cluster algebras, and surface combinatorics, revealing new symmetries and formulas that deepen understanding of their algebraic and geometric structures.

Contribution

It introduces new symmetries and formulas for infinite friezes with Laurent polynomial entries and links classical matching tuples to cluster algebra elements from surfaces.

Findings

Discovered new symmetries in infinite friezes with Laurent polynomial entries

Established formulas relating frieze entries to each other

Linked classical matching tuples to cluster algebra interpretations

Abstract

Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to matching numbers in the once-punctured disk and annulus. In this paper, we study such infinite friezes with an eye towards cluster algebras of type D and affine A, respectively. By examining infinite friezes with Laurent polynomial entries, we discover new symmetries and formulas relating the entries of this frieze to one another. Lastly, we also present a correspondence between Broline, Crowe and Isaacs's classical matching tuples and combinatorial interpretations of elements of cluster algebras from surfaces.