$SL_2$-tilings with translational symmetry
Veronique Bazier-Matte, Marie-Anne Bourgie, Anna Felikson, Pavel Tumarkin
TL;DR
The paper investigates positive integral $SL_2$-tilings with translational symmetry. It proves a bijection between these tilings and triangulations of annuli, translating algebraic questions about tilings into questions about annulus triangulations. This correspondence enables the study of periodic $SL_2$-tilings and yields structural results about their patterns. The work links generalized Conway–Coxeter friezes to geometric triangulations, providing a combinatorial framework for understanding translationally symmetric $SL_2$-tilings.
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.