SL_2-tilings and triangulations of the strip

Abstract

SL_2-tilings were introduced by Assem, Reutenauer, and Smith in connection with frieses and their applications to cluster algebras. An SL_2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2 x 2-submatrix has determinant 1. We construct a large class of new SL_2-tilings which contains the previously known ones. More precisely, we show that there is a bijection between our class of SL_2-tilings and certain combinatorial objects, namely triangulations of the strip.

25 pages

References in corpus (1)

• Linearly recursive sequences and Dynkin diagrams

Cited by in corpus (8)

• Coxeter's frieze patterns at the crossroads of algebra, geometry and combinatorics
• Generalized frieze pattern determinants and higher angulations of polygons
• Arithmetic infinite friezes from punctured discs
• Decomposition of Pointwise Finite-Dimensional S^1 Persistence Modules
• Cluster categories of type ${\mathbb{A}}_{\infty }^{\infty }$ and triangulations of the infinite strip
• Cotorsion pairs in cluster categories of type $A_{\infty }^{\infty }$
• Infinite friezes and triangulations of the strip
• Classifying SL${}_2$-tilings