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.
|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|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be