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

Abstract

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.

On cluster algebras from unpunctured surfaces with one marked point

“…For precise definitions, we refer to [MSW2]. For the snake graph calculus used in this section, see [CS2]. Let L be the Laurent polynomial associated to the loop around the boundary.…”

“…This is shown by exhibiting certain identities in the cluster algebra, that allow to write the Laurent polynomials in question as polynomials in cluster variables. The main ingredients for the proof of these identities are the snake graph calculus developed in [CS,CS2,CS3], and the skein relations proved in [MW].…”

F-polynomial formula from continued fractions

“…Notice that in figure 4 we show these subgraphs with their completed minimal perfect matchings, P i − . In [CS,CS2,CS3] identities in the cluster algebra have been expressed in terms of snake graphs. Equation (2.1) below follows from the grafting with a single edge formula from Theorem 7.3 of [CS2], where n ≥ 2 and the grafting takes place at tile G n−1 .…”

Book Review: Lecture notes on cluster algebras

“…Thus the curves in the surface completely determine the algebraic and combinatorial structure of the cluster algebra. In [11][12][13] this structure was reinterpreted purely in terms of the snake graphs, bypassing the arcs in the surface and providing a more efficient method for computation.…”