Subpolygons in Conway–Coxeter frieze patterns
Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 741-755.

Friezes with coefficients are maps assigning numbers to the edges and diagonals of a regular polygon such that all Ptolemy relations for crossing diagonals are satisfied. Among these, the classic Conway–Coxeter friezes are the ones where all values are positive integers and all edges have value 1. Every subpolygon of a Conway–Coxeter frieze yields a frieze with coefficients over the positive integers. In this paper we give a complete arithmetic criterion for which friezes with coefficients appear as subpolygons of Conway–Coxeter friezes. This generalizes a result of our earlier paper with Peter Jørgensen from triangles to subpolygons of arbitrary size.

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DOI: 10.5802/alco.180
Classification: 05E15,  05E99,  13F60,  51M20
Keywords: Frieze pattern, tame frieze pattern, quiddity cycle, cluster algebra, polygon, triangulation.
Cuntz, Michael 1; Holm, Thorsten 1

1 Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Welfengarten 1, D-30167 Hannover, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Cuntz, Michael; Holm, Thorsten. Subpolygons in Conway–Coxeter frieze patterns. Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 741-755. doi : 10.5802/alco.180. https://alco.centre-mersenne.org/articles/10.5802/alco.180/

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