Generalized frieze pattern determinants and higher angulations of polygons

Abstract

Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper we consider d-angulations of polygons and generalize the combinatorial algorithm for computing the entries in the associated symmetric matrices; we compute their determinants and the Smith normal forms. It turns out that both are independent of the particular d-angulation, the determinant is a power of d-1, and the elementary divisors only take values d-1 and 1. We also show that in the generalized frieze patterns obtained in our setting every adjacent 2x2-determinant is 0 or 1, and we give a combinatorial criterion for when they are 1, which in the case d=3 gives back the Conway-Coxeter condition on frieze patterns.

12 pages, final accepted version

References in corpus (2)

• SL_2-tilings and triangulations of the strip
• Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object

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• Partitions of unity in $\mathrm{SL}(2,\mathbb{Z})$, negative continued fractions, and dissections of polygons
• From frieze patterns to cluster categories
• A Caldero-Chapoton map depending on a torsion class