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

By Ilke Çanakçi, Ralf Schiffler. 2014

55 citations recorded by the source corpus.

Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula whose terms are 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 advance the study of abstract snake graphs by introducing the notions of abstract band graphs, self-crossings of abstract snake graphs as well as their resolutions. We show that there is a bijection between the set of perfect matchings of crossing or 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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f1944c330b995b71f917f251537e9d499bf3322b

Citations in the corpus

• Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings (2012, unknown): “What are the snake graphs corresponding to the skein relations? In this thesis work [2], we introduce the notion of an abstract snake graph, which is not necessarily related to an arc in a surface.”