JournalsjcaVol. 3, No. 2pp. 153–188

Frieze patterns over integers and other subsets of the complex numbers

  • Michael Cuntz

    Leibniz Universität Hannover, Germany
  • Thorsten Holm

    Leibniz Universität Hannover, Germany
Frieze patterns over integers and other subsets of the complex numbers cover

A subscription is required to access this article.

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 for the set of specializations of cluster algebras of Dynkin type AA 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 RR of the complex numbers. Under certain conditions on RR, we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if RR is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over RR. Using this, we disprove a conjecture of Fontaine, by showing that for a complex dd-th root of unity ζd\zeta_d there are only finitely many non-zero frieze patterns for a given height over R=Z[ζd]R=\mathbb{Z}[\zeta_d] if and only if d{1,2,3,4,6}d\in \{1,2,3,4,6\}.

Cite this article

Michael Cuntz, Thorsten Holm, Frieze patterns over integers and other subsets of the complex numbers. J. Comb. Algebra 3 (2019), no. 2, pp. 153–188

DOI 10.4171/JCA/29