Cluster algebras were introduced by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)], motivated by combinatorial aspects of canonical bases in Lie theory. They are commutative algebras, whose generators and relations are constructed by a recursive process. This distinguished set of generators, called the cluster variables, are collected into groups of \(n\) elements called clusters connected by local transition rules. By construction, cluster variables are Laurent polynomials with integer coefficients. Moreover, these coefficients are conjectured to be positive.
This positivity conjecture was proved for a large class of cluster algebras that are constructed from triangulations of surfaces, by Musiker, Shiffler and Williams [G. Musiker et al., Adv. Math. 227, No. 6, 2241–2308 (2011; Zbl 1331.13017)]. The proof uses a direct combinatorial formula for the cluster variables. This formula is parametrized by perfect matchings of so-called snake graphs, which are originally constructed from arcs on the surface. A snake graph is a connected graph consisting of a finite sequence of square tiles each sharing exactly one edge whose configuration depends on the arc. Hence, it makes sense to say that two snake graphs are crossing when the corresponding arcs are crossing. Also, one can define the resolution of two crossing snake graphs, as the pair of snake graphs associated with the resolution of the two crossing arcs.
The aim of this article is to develop the notion of abstract snake graphs which are not necessarily related to arcs on a surface. The authors give a combinatorial definition of snake graphs and define combinatorially what it means for two abstract snake graph to cross, and how to construct the resolution of this crossing as a new pair of snake graphs. When the abstract snake graphs are usual snake graphs and come from arcs on a surface, they prove that the notions of crossing and resolution coincide.