Dissertation/Thesis Abstract

An Algorithmic Definition of Gabai Width
by Lee, Ricky, M.S., California State University, Long Beach, 2020, 55; 27838206
Abstract (Summary)

Intuitively, knot theory is the study of loops in three dimensional space. Applications of knot theory have arisen in many diverse fields of study such as DNA replication and protein folding. As a result, many different perspectives have been developed on how the complexity of a knot should be measured. One of the most successful measures of complexity is Gabai width. Gabai width has been the central tool in the resolution of many important problems in low-dimensional topology over the past few decades. Some examples are Gabai's proof of the property R conjecture and Gordan and Luecke's resolution of the knot complement problem. While Gabai width is a central tool in the study of knot theory, computing the Gabai width of a specific knot is a notoriously challenging task.

In this thesis, we define the Wirtinger width, a diagrammatic reformulation of Gabai width that is more computationally accessible. The calculation of Gabai width involves taking a knot which is embedded in a well behaved manner with respect to a certain height function, and analyzing the ordering of its maxima and minima by height. However, complexity from the perspective of Wirtinger width is calculated only on the level of knot diagrams without reference to any specific height function. This is what makes Wirtinger width amenable to algorithmic calculations.

We first show Wirtinger width is equal to Gabai width. Then, we describe an algorithm we wrote in Python which calculated for the first time the Gabai width of over 50000 tabulated knots.

Indexing (document details)
Advisor: Blair, Ryan
Commitee: Brevik, John, Sack, Joshua
School: California State University, Long Beach
Department: Mathematics and Statistics
School Location: United States -- California
Source: MAI 82/6(E), Masters Abstracts International
Subjects: Mathematics, Genetics, Bioinformatics
Keywords: Algorithmic definition, Knot theory, Protein folding, DNA replication, Python software
Publication Number: 27838206
ISBN: 9798698587699
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