$S{L}_2$-tilings with translational symmetry

An $S{L}_2$-tiling is a bi-infinite matrix in which all adjacent $2\times 2$ minors are equal to $1$. Positive integral $S{L}_2$-tilings were introduced by Assem, Reutenauer and Smith as generalisations of classical Conway–Coxeter frieze patterns. We show that positive integral $S{L}_2$-tilings with translational symmetry are in bijection with triangulations of annuli. We use this correspondence to study the properties of periodic positive integral $S{L}_2$-tilings.