SL2SL2-tilings and triangulations of the strip

SL2SL2-tilings were introduced by Assem, Reutenauer, and Smith in connection with frieses and their applications to cluster algebras. An SL2SL2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2×22×2-submatrix has determinant 1.In this paper we define the class of SL2SL2-tilings with enough ones. It contains the previously known tilings as well as some which are new, and we show that it is in bijection with a certain class of combinatorial objects, namely “good” triangulations of the strip.