Frieze patterns and Farey complexes

TL;DR

This paper introduces a combinatorial model for frieze patterns over rings using Farey complexes, extending previous models and providing new classifications and enumerations for patterns over various rings.

Contribution

It develops a general combinatorial framework for frieze patterns over arbitrary rings via Farey complexes, expanding the scope beyond integers.

Findings

Provides a combinatorial model for frieze patterns over any rings using Farey complexes.

Classifies frieze patterns modulo n that lift to integer frieze patterns based on Farey complex topology.

Recovers and generalizes existing results on Farey graph diameters and frieze enumeration over finite fields.

Abstract

Frieze patterns have attracted significant attention recently, motivated by their relationship with cluster algebras. A longstanding open problem has been to provide a combinatorial model for frieze patterns over the ring of integers modulo $n$ akin to Conway and Coxeter's celebrated model for positive integer frieze patterns. Here we solve this problem using the Farey complex of the ring of integers modulo $n$; in fact, using more general Farey complexes we provide combinatorial models for frieze patterns over any rings whatsoever. Our strategy generalises that of the first author and of Morier-Genoud et al. for integers and that of Felikson et al. for Eisenstein integers. We also generalise results of Singerman and Strudwick on diameters of Farey graphs, we recover a theorem of Morier-Genoud on enumerating friezes over finite fields, and we classify those frieze patterns modulo $n$ that lift to frieze patterns over the integers in terms of the topology of the corresponding Farey complexes.