Date of Completion

5-3-2018

Embargo Period

5-3-2018

Keywords

Cluster Algebra, Markov numbers, Continued Fractions, Lattice path, Christoffel

Major Advisor

Ralf Schiffler

Associate Advisor

Tom Roby

Associate Advisor

Jerzy Weyman

Field of Study

Mathematics

Degree

Doctor of Philosophy

Open Access

Open Access

Abstract

In this work we present results from three different, albeit related, areas. First, we construct an explicit formula for the F-polynomial of a cluster variable in a surface type cluster algebra. Second, we define lattice paths and order them by the number of perfect matchings of their associated snake graphs. Lastly, we prove two conjec- tures from Martin Aigner’s book, Markov’s theorem and 100 years of the uniqueness conjecture that determine an ordering on subsets of the Markov numbers based on their corresponding rational.

The common thread throughout this work is the interplay between cluster alge- bras, lattice paths, snake graphs, Markov numbers and their connections to continued fractions. In the first section we give the necessary background on finite continued fractions and then in each of the following three sections, we introduce a topic and follow it with our related results.

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