Number of triangulations of a Möbius strip

Bazier-Matte, Véronique; Huang, Ruiyan; Luo, Hanyi

doi:10.2140/involve.2023.16.547

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Number of triangulations of a Möbius strip

Véronique Bazier-Matte, Ruiyan Huang and Hanyi Luo

Vol. 16 (2023), No. 4, 547–562

DOI: 10.2140/involve.2023.16.547

Abstract

Consider a Möbius strip with $n$ chosen points on its boundary. A triangulation is a maximal collection of arcs among these points and cuts the strip into triangles and quasi-triangles. We prove that the number of all triangulations that one can obtain from a Möbius strip with $n$ chosen points on its boundary is given by $4^{n - 1} + (\binom{2 n - 2}{n - 1})$ . We then connect our finding with the number of clusters in the quasi-cluster algebra arising from the Möbius strip.

Keywords

quasi-cluster algebras

Mathematical Subject Classification

Primary: 13F60

Milestones

Received: 12 September 2020

Revised: 29 July 2022

Accepted: 5 August 2022

Published: 31 October 2023

Communicated by Kenneth S. Berenhaut

Authors

Véronique Bazier-Matte
Département de Mathématiques et de Statistiques Université Laval Québec, QC Canada

Ruiyan Huang
Yale University New Haven, CT United States

Hanyi Luo
Harvey Mudd College Claremont, CA United States

Vol. 16, No. 4, 2023