$S{L}_2(\mathbb{Z})$-tilings of the torus, Coxeter-Conway friezes and Farey triangulations

The notion of $S{L}_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $S{L}_2$-tilings that contain a rectangular domain of positive integers. Every such $S{L}_2$-tiling corresponds to a pair of frieze patterns and a unimodular $2\times 2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph.