Classifying integer tilings and hypertilings

Abstract

There are two objectives to this work: to classify all tame integer tilings and to classify all tame integer hypertilings. Motivation for the first objective comes from Conway and Coxeter's modelling of positive integer friezes using triangulated polygons, which has received significant attention since the discovery of cluster algebras by Fomin and Zelevinsky in 2002. Assem, Reutenauer, and Smith introduced $\text{SL}_2$-tilings as generalisations of friezes, and Bessenrodt, Holm, and J{\o}rgensen classified positive integer $\text{SL}_2$-tilings using infinite triangulated polygons. Here we consider $N$-tilings, of which $\text{SL}_2$-tilings are the case $N=1$. We provide a geometric model for all tame integer $N$-tilings using a generalisation of the Farey graph in the hyperbolic plane. Highlights of this model include classifications of all positive integer $N$-tilings and of all positive rational friezes, with entries encoded by lambda lengths or weight data of triangulated polygons. The second objective is motivated by Bhargava's celebrated study of binary quadratic forms using integer cubes and by an observation of Demonet et al. that there is essentially only one three-dimensional positive integer tiling with $\text{SL}_2$ cross sections. We consider a richer class of three-dimensional tilings, which we call hypertilings, using the Cayley hyperdeterminant. We classify all tame integer hypertilings using generalised Farey graphs; remarkably, those with Cayley hyperdeterminant 1 prove to have a simple description in terms of triple Hadamard products of integer pairs.