Frieze patterns and Farey complexes

Ian Short, Matty Van Son, Andrei Zabolotskii

Published 2023-12-20Version 1

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. In so doing we generalise the strategy of the first author 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 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.