Date of Completion
7-19-2013
Embargo Period
7-19-2013
Keywords
Cluster algebras, Surface cluster algebras, Snake graphs
Major Advisor
Ralf Schiffler
Associate Advisor
Kyu-Hwan Lee
Associate Advisor
Milena Hering
Field of Study
Mathematics
Degree
Doctor of Philosophy
Open Access
Open Access
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. 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.
Recommended Citation
Canakci, Ilke, "Snake Graph Calculus and Cluster Algebras from Surfaces" (2013). Doctoral Dissertations. 165.
https://digitalcommons.lib.uconn.edu/dissertations/165