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.
Revised:
Accepted:
Published online:
Keywords: Frieze pattern, tame frieze pattern, quiddity cycle, cluster algebra, polygon, triangulation.
@article{ALCO_2021__4_4_741_0, author = {Cuntz, Michael and Holm, Thorsten}, title = {Subpolygons in {Conway{\textendash}Coxeter} frieze patterns}, journal = {Algebraic Combinatorics}, pages = {741--755}, publisher = {MathOA foundation}, volume = {4}, number = {4}, year = {2021}, doi = {10.5802/alco.180}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.180/} }
TY - JOUR AU - Cuntz, Michael AU - Holm, Thorsten TI - Subpolygons in Conway–Coxeter frieze patterns JO - Algebraic Combinatorics PY - 2021 SP - 741 EP - 755 VL - 4 IS - 4 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.180/ DO - 10.5802/alco.180 LA - en ID - ALCO_2021__4_4_741_0 ER -
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. http://www.numdam.org/articles/10.5802/alco.180/
[1] Triangulated polygons and frieze patterns, Math. Gaz., Volume 57 (1973), p. 87-94 and 175–183 | DOI | MR | Zbl
[2] Frieze patterns, Acta Arith., Volume 18 (1971), pp. 297-310 | DOI | MR | Zbl
[3] Regular complex polytopes, Cambridge University Press, Cambridge, 1991, xiv+210 pages | Zbl
[4] On Wild Frieze Patterns, Experimental Mathematics, Volume 26 (2017) no. 3, pp. 342-348 | DOI | MR | Zbl
[5] Frieze patterns with coefficients, Forum Math. Sigma, Volume 8 (2020), e17, 36 pages | DOI | MR | Zbl
[6] Frieze patterns with coefficients, Masters thesis, Leibniz Universität Hannover (2020)
[7] Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 895-938 | DOI | MR | Zbl
[8] Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 15, Springer-Verlag, Berlin, 1988, viii+212 pages (An introduction to the theory of toric varieties, Translated from the Japanese) | MR | Zbl
[9] Die Lehre von den Kettenbrüchen, Teubner Verlag, Leipzig, 1929 | Zbl
[10] The combinatorics of frieze patterns and Markoff numbers, Integers, Volume 20 (2020), A12, 38 pages | MR | Zbl
Cited by Sources: