Friezes for a pair of pants

TL;DR

This paper investigates frieze patterns linked to a pair of pants surface, showing all positive integral friezes are derived from triangulations with unit lambda-lengths, connecting cluster algebra theory with hyperbolic geometry.

Contribution

It extends the theory of frieze patterns to bordered surfaces with hyperbolic metrics, proving all positive integral friezes are unitary for a pair of pants.

Findings

All positive integral friezes are unitary.

Frieze entries correspond to lambda-lengths of arcs.

Connection established between frieze patterns and hyperbolic geometry.

Abstract

Frieze patterns are numerical arrangements that satisfy a local arithmetic rule. These arrangements are actively studied in connection to the theory of cluster algebras. In the setting of cluster algebras, the notion of a frieze pattern can be generalized, in particular to a frieze associated with a bordered marked surface endowed with a decorated hyperbolic metric. We study friezes associated with a pair of pants, interpreting entries of the frieze as lambda-lengths of arcs connecting the marked points. We prove that all positive integral friezes over such surfaces are unitary, i.e. they arise from triangulations with all edges having unit lambda-lengths.