Publication date:
31 July 2017
Source:Advances in Mathematics, Volume 315
Author(s): Christine Bessenrodt, Thorsten Holm, Peter Jørgensen
An ${\mathrm{SL}}_2$-tiling is a bi-infinite matrix of positive integers such that each adjacent $2\times 2$-submatrix has determinant 1. Such tilings are infinite analogues of Conway–Coxeter friezes, and they have strong links to cluster algebras, combinatorics, mathematical physics, and representation theory. We show that, by means of so-called Conway–Coxeter counting, every ${\mathrm{SL}}_2$-tiling arises from a triangulation of the disc with two, three or four accumulation points. This improves earlier results which only discovered ${\mathrm{SL}}_2$-tilings with infinitely many entries equal to 1. Indeed, our methods show that there are large classes of tilings with only finitely many entries equal to 1, including a class of tilings with no 1's at all. In the latter case, we show that the minimal entry of a tiling is unique.