Generalized frieze pattern determinants and higher angulations of polygons

TL;DR

This paper extends the concept of frieze patterns from triangulations to d-angulations of polygons, analyzing associated matrices, their determinants, and properties of the resulting patterns, revealing invariance and combinatorial criteria.

Contribution

It generalizes the computation of determinants and Smith normal forms for matrices from polygon d-angulations, showing invariance and establishing conditions for frieze pattern properties.

Findings

Determinant is a power of d-1, independent of the d-angulation.

Elementary divisors are only d-1 and 1.

Adjacent 2x2-determinants are 0 or 1, with criteria for when they are 1.

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.