Frieze patterns and Farey complexes
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916726710468608 |
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| author | Short, Ian Van Son, Matty Zabolotskii, Andrei |
| author_facet | Short, Ian Van Son, Matty Zabolotskii, Andrei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12953 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Frieze patterns and Farey complexes Short, Ian Van Son, Matty Zabolotskii, Andrei Combinatorics Number Theory 05E16 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. |
| title | Frieze patterns and Farey complexes |
| topic | Combinatorics Number Theory 05E16 |
| url | https://arxiv.org/abs/2312.12953 |