Friezes from surfaces and Farey triangulation

TL;DR

This paper classifies positive integral friezes on bordered surfaces, linking them to ideal triangulations with rescaling constants, and establishes finiteness and unitarity properties of these friezes.

Contribution

It provides a complete classification of positive integral friezes on bordered surfaces, connecting them to triangulations with rescaling constants and proving finiteness.

Findings

Positive integral friezes correspond to ideal triangulations with rescaling constants.

The set of constants for each triangulation is finite and determined by vertex valencies.

The number of non-equivalent friezes on bordered surfaces is finite; all on unpunctured surfaces are unitary.

Abstract

We provide a classification of positive integral friezes on marked bordered surfaces in the style of Conway and Coxeter. More precisely, we prove that positive integral friezes are in one-to-one correspondence with ideal triangulations supplied with a collection of rescaling constants assigned to punctures. For every triangulation the set of the collections of constants is finite and is completely determined by the valencies of vertices in the triangulation. In particular, it follows that the number of non-equivalent friezes on bordered surfaces is finite, and all friezes on unpunctured surfaces are unitary.