Dissertation
Tabulating 2-string tangles with methods to count and generate them
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2024
DOI: 10.25820/etd.007313
Abstract
An $n$-tangle is a ball containing $n$ disjoint properly embedded strings with ends attached to the boundary of a ball, and a possibly non-empty set of interior loops. Tangles were originally defined by Conway and used as building blocks to construct unique knots and links. This allowed him to duplicate and extend the enumeration of knots done by C. N. Little, which originally took 6 years, in "just an afternoon.'' The ends of the strings in a tangle may be allowed to move freely along the boundary ball, or remain fixed in place. This dissertation will focus on 2-tangles and consider both the free boundary and fixed boundary case. 2-Tangles may be separated into two large groups; algebraic tangles, and non-algebraic tangles. There are several fully classified types of algebraic tangles, namely the rational, Montesinos, and generalized Montesinos tangles. The classification of these tangles include description of unique canonical diagrams and minimal crossing diagrams.
Using these diagrams, we develop a method for generating rational, Montesinos, and generalized Montesinos tangles with $N$ crossings. For each class of tangle, we consider how the total number of crossings are distributed throughout the diagram. We may then use results from number theory involving compositions and the Fibonacci sequence to encode crossing distributions. Combining our generating methods with the combinatorial properties of compositions, we then develop counting formulae for each of these classified tangles which depend only on the total number of crossings $N$.
We finally tabulate algebraic tangles with 6 and 7 crossings which are not rational, Montesinos, or generalized Montesinos. This process uses properties of compositions and knowledge of rational, Montesinos, and generalized Montesinos tangles to identify symmetries of these algebraic tangles to create unique minimal crossing diagrams.
Details
- Title: Subtitle
- Tabulating 2-string tangles with methods to count and generate them
- Creators
- Zachary C. Bryhtan
- Contributors
- Isabel Darcy (Advisor)Charles Frohman (Committee Member)Ryan Kinser (Committee Member)Colleen Mitchell (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2024
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.007313
- Number of pages
- xi, 175 pages
- Copyright
- Copyright 2024 Zachary C. Bryhtan
- Comment
- This thesis has been optimized for improved web viewing. If you require the original version, contact the University Archives at the University of Iowa: https://www.lib.uiowa.edu/sc/contact/
- Language
- English
- Date submitted
- 04/22/2024
- Description illustrations
- illustrations (some color)
- Description bibliographic
- Includes bibliographical references (page 173-175).
- Public Abstract (ETD)
Knot theory is a field in mathematics which focuses is on the properties of many shapes including knots, links, and tangles. A knot can be thought of as a single piece of string which has been tied up and intertwined with itself and gluing the two ends of the string together, forming a single “knotted” loop. Links are similar, but are made up of several knots “linked” together like a chain necklace. While knots and links are made from shapes that have been glued closed, a tangle is what is created if the ends of the string are instead glued to the boundary a ball. There are several types of tangles which have been fully classified. In this dissertation, we describe a method for generating classified tangles and establish counting formulae. We also identify some simple tangles which do not belong to a fully classified type.
- Academic Unit
- Mathematics
- Record Identifier
- 9984647255702771
Metrics
6 File views/ downloads
18 Record Views