\(\mathrm{SL}_2(\mathbb{Z})\)-tilings of the torus, Coxeter-Conway friezes and Farey triangulations (Q5963020)

Summary: The notion of \(\mathrm{SL}_2\)-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic \(\mathrm{SL}_2\)-tilings that contain a rectangular domain of positive integers. Every such \(\mathrm{SL}_2\)-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.