Frieze patterns over integers and other subsets of the complex numbers

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Original languageEnglish
Pages (from-to)153-188
Number of pages36
JournalJournal of Combinatorial Algebra
Volume3
Issue number2
Publication statusPublished - 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.

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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 journalArticleResearchpeer review

Cuntz M, Holm T. Frieze patterns over integers and other subsets of the complex numbers. Journal of Combinatorial Algebra. 2019 Mar 27;3(2):153-188. doi: 10.48550/arXiv.1711.03724, 10.4171/JCA/29
Cuntz, Michael ; Holm, Thorsten. / Frieze patterns over integers and other subsets of the complex numbers. In: Journal of Combinatorial Algebra. 2019 ; Vol. 3, No. 2. pp. 153-188.
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