Dissertation
Classification and tabulation of 2-string tangles: the astronomy of subtangle decompositions
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2021
DOI: 10.17077/etd.005978
Abstract
Tangles are a type of knot theoretic structure originally introduced by Conway to tabulate knots. Intuitively, tangles can be thought of as the building blocks of mathematical knots. The two principal goals of this dissertation are classification and tabulation of 2-string tangle diagrams, and we approach both endeavors through the careful study of diagram notations. Broadly speaking, tangles can be divided into two types: algebraic and non-algebraic. Many special cases of algebraic tangle are well understood and have been fully classified. We begin by reviewing the structure of these special types of tangles, each of which may be uniquely represented using a canonical diagram.
Unlike these special cases, generic algebraic and non-algebraic tangle diagrams are not fully classified. To better understand these types of tangles, we propose a graph-based notation for tangle diagrams, referred to as subtangle planar diagram (SPD) code. SPD code can be used to represent generic tangle diagrams using decompositions into subtangles. Based on this description, we show how algebraic and non-algebraic tangles can be distinguished using planar graphs. This description enables any algebraic diagram to be converted into an equivalent semi-canonical form. Furthermore, we show how certain types of planar graphs known as constellations arise in connection with subtangle decompositions for non-algebraic diagrams. All 2-string tangle diagrams with at most 9 crossings can be represented using a combination of tangle algebra and the first ten constellations.
In addition to this theoretical work, we also discuss a number of related computational results. In particular, we give an overview of an algorithm to compute several variations of the SPD code, which has been successfully implemented as a program in C/C++. This algorithm has been used in combination with tangle tabulation programs to classify tangle diagrams. We conclude by discussing the progress on the prototype of a web-accessible database of 2-string tangles currently being developed to catalog these results.
Details
- Title: Subtitle
- Classification and tabulation of 2-string tangles: the astronomy of subtangle decompositions
- Creators
- Nicholas Connolly
- Contributors
- Isabel K Darcy (Advisor)Keiko Kawamuro (Committee Member)Ryan Kinser (Committee Member)Benjamin Cooper (Committee Member)Charles Frohman (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Summer 2021
- DOI
- 10.17077/etd.005978
- Publisher
- University of Iowa
- Number of pages
- xx, 307 pages
- Copyright
- Copyright 2021 Nicholas Connolly
- Language
- English
- Description illustrations
- color illustrations
- Description bibliographic
- Includes bibliographical references (pages 304-307).
- Public Abstract (ETD)
Knots, the principal objects of study in the mathematical field of knot theory, can be visualized as an entwined piece of string with the two ends glued together. This dissertation is concerned with studying the parts of a knot obtained by chopping up this string into pieces while preserving the shape. These parts, known as tangles, can be thought of as the building blocks of mathematical knots; they can be pieced back together to build more complicated types of structures. There are many different types of tangles that can be identified based on how these pieces are put together.
Understanding, classifying, and tabulating these different types of tangles is the principal goal of this research. Some types of tangles are well understood in the literature, and we begin by reviewing these known results. Other types of tangles are not yet uniquely classified. In this dissertation, we propose a method for identifying these different types of tangles based on how they decompose into smaller pieces known as subtangles. In addition to this theoretical work, we also discuss progress on a web-accessible database of tangles currently under development.
- Academic Unit
- Mathematics
- Record Identifier
- 9984124571002771
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