Cuntz, M.; Holm, T.; Pagano, C.: Frieze patterns over algebraic numbers. In: Bulletin of the London Mathematical Society (2024), online first. DOI: https://doi.org/10.1112/blms.13003
Abstract: | |
Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jorgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field Q(√d) where d < 0. We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non-zero frieze patterns for any given height. We then show that apart from the cases d is an element of d ϵ {-1,-2,-3,-7,-11} all non-zero frieze patterns over the rings of integers Ο(d) for d < 0 have only integral entries and hence are known as (twisted) Conway-Coxeter frieze patterns. | |
License of this version: | CC BY-NC-ND 4.0 Unported |
Document Type: | Article |
Publishing status: | publishedVersion |
Issue Date: | 2024 |
Appears in Collections: | Fakultät für Mathematik und Physik |
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