Date of Completion
Cluster algebras, Surface cluster algebras, Snake graphs
Field of Study
Doctor of Philosophy
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. Furthermore it is known that the product of two cluster variables can be described geometrically in the surface using skein relations. In this thesis work, we identify each cluster variable with its snake graph, and interpret relations among the cluster variables in terms of these graphs. Taking a more general viewpoint, we introduce the notion of abstract snake graphs and develop a graphical calculus for abstract snake graphs. Moreover, we give a new proof of skein relations of two cluster variables.
Canakci, Ilke, "Snake Graph Calculus and Cluster Algebras from Surfaces" (2013). Doctoral Dissertations. 165.