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.

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