Snake graph calculus and cluster algebras from surfaces

TL;DR

This paper explores the relationship between snake graphs and cluster algebras from surfaces, providing new insights into cluster variable formulas and skein relations through graph interpretations.

Contribution

It introduces a novel interpretation of cluster variables as snake graphs and offers a new proof of skein relations using this graphical approach.

Findings

Cluster variables correspond to snake graphs.

Relations among cluster variables are interpreted via graph operations.

A new proof of skein relations is provided.

Abstract

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 identify each cluster variable with its snake graph, and interpret relations among the cluster variables in terms of these graphs. In particular, we give a new proof of skein relations of two cluster variables.