$SL_2$-tilings with translational symmetry

TL;DR

This paper establishes a bijection between positive integral $SL_2$-tilings with translational symmetry and triangulations of annuli, providing insights into their periodic properties.

Contribution

It introduces a novel correspondence between symmetric $SL_2$-tilings and annulus triangulations, advancing understanding of their structure and periodicity.

Findings

Positive integral $SL_2$-tilings with symmetry correspond to annulus triangulations.

The study reveals properties of periodic $SL_2$-tilings.

Connections to classical frieze patterns are extended.

Abstract

An $SL_2$-tiling is a bi-infinite matrix in which all adjacent $2 \times 2$ minors are equal to $1$. Positive integral $SL_2$-tilings were introduced by Assem, Reutenauer and Smith as generalisations of classical Conway--Coxeter frieze patterns. We show that positive integral $SL_2$-tilings with translational symmetry are in bijection with triangulations of annuli. We use this correspondence to study the properties of periodic positive integral $SL_2$-tilings.