Friezes, weak friezes, and T-paths

TL;DR

This paper introduces weak friezes and T-paths related to polygon dissections, establishing their properties, how they can be combined, and their connection to classical frieze patterns within algebraic and geometric contexts.

Contribution

It extends the concept of friezes to weak friezes with respect to polygon dissections and characterizes them via the T-path expansion formula.

Findings

Weak friezes satisfy the T-path expansion formula.

Weak friezes can be glued to form new weak friezes.

A weak frieze is a frieze if and only if each component is a frieze.

Abstract

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T-path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued.