Quiddity sequences for $\mathrm{SL}_3$-frieze patterns

TL;DR

This paper explores the relationship between (3,n)-frieze patterns and SL_3-frieze patterns, introducing superimposed triangulations to better understand quiddity sequences in this context.

Contribution

It establishes connections between classes of (3,n)-frieze patterns and SL_3-frieze patterns, and introduces superimposed triangulations as a new tool for analysis.

Findings

Established links between (3,n)-frieze and SL_3-frieze patterns.

Introduced superimposed triangulations for understanding quiddity sequences.

Clarified the role of superimposed triangulations in SL_3-frieze patterns.

Abstract

The notion of a $(k,n)$-frieze pattern was introduced by the author as a generalisation of the classical frieze patterns. In this article we describe connections between classes of $(3,n)$-frieze patterns and classes of $\mathrm{SL}_3$-frieze patterns. We introduce the idea of a superimposed triangulation and clarify how superimposed triangulations may be used to understand quiddity sequences for $\mathrm{SL}_3$-frieze patterns.