Snake graph calculus and cluster algebras from surfaces II: Self-crossing snake graphs

Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula which is parametrized by the perfect matchings of a snake graph. In this paper, we continue our study of snake graphs from a combinatorial point of view. We introduce the notions of abstract snake graphs and abstract band graphs, their crossings and self-crossings, as well as the resolutions of these crossings. We show that there is a bijection between the set of perfect matchings of (self-)crossing snake graphs and the set of perfect matchings of the resolution of the crossing. In the situation where the snake and band graphs are coming from arcs and loops in a surface without punctures, we obtain a new proof of skein relations in the corresponding cluster algebra.