Periodic Infinite Frieze Patterns of Type $\Lambda_{p_1,\ldots,p_n}$ and Dissections on Annuli

TL;DR

This paper explores the relationship between infinite frieze patterns with algebraic entries and geometric dissections of annuli and punctured discs, providing algorithms and new combinatorial interpretations.

Contribution

It extends prior work by connecting infinite frieze patterns to dissected surfaces and introduces quotient dissections and generalized combinatorial interpretations.

Findings

Developed an algorithm to identify frieze patterns from dissected surfaces.

Introduced quotient dissections for unrealizable frieze patterns.

Provided two new combinatorial interpretations for frieze pattern entries.

Abstract

Finite frieze patterns with entries in $\mathbb{Z}[\lambda_{p_1},\ldots,\lambda_{p_s}]$ where $\{p_1,\ldots,p_s\} \subseteq \mathbb{Z}_{\geq 3}$ and $\lambda_p = 2 \cos(\pi/p)$ were shown to have a connection to dissected polygons by Holm and Jorgensen. We extend their work by studying the connection between infinite frieze patterns with such entries and dissections of annuli and once-punctured discs. We give an algorithm to determine whether a frieze pattern with entries in $\mathbb{Z}[\lambda_{p_1},\ldots,\lambda_{p_s}]$, finite or infinite, comes from a dissected surface. We introduce quotient dissections as a realization for some frieze patterns unrealizable by an ordinary dissection. We also introduce two combinatorial interpretations for entries of frieze patterns from dissected surfaces. These interpretations are a generalization of matchings introduced by Broline, Crowe, and Isaacs for finite frieze patterns over $\mathbb{Z}$.