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.

Received:
Revised:
Accepted:
Published online:
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
@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 = {https://alco.centre-mersenne.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  - https://alco.centre-mersenne.org/articles/10.5802/alco.180/
DO  - 10.5802/alco.180
LA  - en
ID  - ALCO_2021__4_4_741_0
ER  - 
%0 Journal Article
%A Cuntz, Michael
%A Holm, Thorsten
%T Subpolygons in Conway–Coxeter frieze patterns
%J Algebraic Combinatorics
%D 2021
%P 741-755
%V 4
%N 4
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.180/
%R 10.5802/alco.180
%G en
%F ALCO_2021__4_4_741_0
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/

[1] Conway, John H.; Coxeter, Harold S. M. Triangulated polygons and frieze patterns, Math. Gaz., Volume 57 (1973), p. 87-94 and 175–183 | DOI | MR | Zbl

[2] Coxeter, Harold S. M. Frieze patterns, Acta Arith., Volume 18 (1971), pp. 297-310 | DOI | MR | Zbl

[3] Coxeter, Harold S. M. Regular complex polytopes, Cambridge University Press, Cambridge, 1991, xiv+210 pages | Zbl

[4] Cuntz, Michael On Wild Frieze Patterns, Experimental Mathematics, Volume 26 (2017) no. 3, pp. 342-348 | DOI | MR | Zbl

[5] Cuntz, Michael; Holm, Thorsten; Jørgensen, Peter Frieze patterns with coefficients, Forum Math. Sigma, Volume 8 (2020), Paper no. e17, 36 pages | DOI | MR | Zbl

[6] Grobe, J. Frieze patterns with coefficients, Masters thesis, Leibniz Universität Hannover (2020)

[7] Morier-Genoud, Sophie 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] Oda, Tadao 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] Perron, Oskar Die Lehre von den Kettenbrüchen, Teubner Verlag, Leipzig, 1929 | Zbl

[10] Propp, James The combinatorics of frieze patterns and Markoff numbers, Integers, Volume 20 (2020), Paper no. A12, 38 pages | MR | Zbl

Cited by Sources: