## Details

Original language | English |
---|---|

Pages (from-to) | 153-188 |

Number of pages | 36 |

Journal | Journal of Combinatorial Algebra |

Volume | 3 |

Issue number | 2 |

Publication status | Published - 27 Mar 2019 |

## Abstract

We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed), generalising the classic Conway Coxeter theory. This model is thus also a model forthe set of specializations of cluster algebras of Dynkin type A in which all cluster variables are integers. Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset R of the complex numbers. Under certain conditions on R, we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if R is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over R. Using this, we disprove a conjecture of Fontaine, by showing that for a complex d-th root of unity _d there are only finitely many non-zero frieze patterns for a given height over R D Z. if and only if d 2 f1; 2; 3; 4; 6g. Mathematics Subject Classification (2010). 05E15, 05E99, 13F60, 51M20.

## Keywords

- Cluster algebra, Frieze pattern, Polygon, Quiddity cycle, Triangulation.

## ASJC Scopus subject areas

- Mathematics(all)
**Algebra and Number Theory**- Mathematics(all)
**Discrete Mathematics and Combinatorics**

## Cite this

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**Frieze patterns over integers and other subsets of the complex numbers.**/ Cuntz, Michael; Holm, Thorsten.

In: Journal of Combinatorial Algebra, Vol. 3, No. 2, 27.03.2019, p. 153-188.

Research output: Contribution to journal › Article › Research › peer review

*Journal of Combinatorial Algebra*, vol. 3, no. 2, pp. 153-188. https://doi.org/10.48550/arXiv.1711.03724, https://doi.org/10.4171/JCA/29

*Journal of Combinatorial Algebra*,

*3*(2), 153-188. https://doi.org/10.48550/arXiv.1711.03724, https://doi.org/10.4171/JCA/29

}

TY - JOUR

T1 - Frieze patterns over integers and other subsets of the complex numbers

AU - Cuntz, Michael

AU - Holm, Thorsten

PY - 2019/3/27

Y1 - 2019/3/27

N2 - We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed), generalising the classic Conway Coxeter theory. This model is thus also a model forthe set of specializations of cluster algebras of Dynkin type A in which all cluster variables are integers. Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset R of the complex numbers. Under certain conditions on R, we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if R is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over R. Using this, we disprove a conjecture of Fontaine, by showing that for a complex d-th root of unity _d there are only finitely many non-zero frieze patterns for a given height over R D Z. if and only if d 2 f1; 2; 3; 4; 6g. Mathematics Subject Classification (2010). 05E15, 05E99, 13F60, 51M20.

AB - We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed), generalising the classic Conway Coxeter theory. This model is thus also a model forthe set of specializations of cluster algebras of Dynkin type A in which all cluster variables are integers. Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset R of the complex numbers. Under certain conditions on R, we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if R is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over R. Using this, we disprove a conjecture of Fontaine, by showing that for a complex d-th root of unity _d there are only finitely many non-zero frieze patterns for a given height over R D Z. if and only if d 2 f1; 2; 3; 4; 6g. Mathematics Subject Classification (2010). 05E15, 05E99, 13F60, 51M20.

KW - Cluster algebra

KW - Frieze pattern

KW - Polygon

KW - Quiddity cycle

KW - Triangulation.

UR - http://www.scopus.com/inward/record.url?scp=85074716753&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1711.03724

DO - 10.48550/arXiv.1711.03724

M3 - Article

AN - SCOPUS:85074716753

VL - 3

SP - 153

EP - 188

JO - Journal of Combinatorial Algebra

JF - Journal of Combinatorial Algebra

SN - 2415-6302

IS - 2

ER -