A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand. November 2014. / Knot theory is a branch of algebraic topology that is concerned with studying the interesting geometrical structures known as knots. The idea of a knot in the theory of knots is entirely different from everyday’s idea of knots, that is, a knot has free ends. In knot theory a knot is defined as a knotted loop of string which does not have free ends unless we cut it using a pair of scissors. The interesting aspect of knot theory is that it enables us to transfer techniques from graph theory, algebra, topology, group theory and combinatorics to study different classes of knots. In this dissertation we are only concerned with the relationship between knot theory and graph theory. It is widely known in knot theory literature that a knot has a corresponding signed planar graph and that a signed planar graph also has a corresponding knot which depends on the signs of the edges of its signed planar graph. This provides a foundation of a solid relationship between knot theory and graph theory, and it allows for some of the notions in graph theory to be transferred to knot theory. In this dissertation we study the path width and component number of links through their corresponding graphs.

Ho Wing Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 67-68). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Basics of Differential Topology and Knot Theory --- p.7 / Chapter 1.1 --- Elementary Differential Topology --- p.7 / Chapter 1.2 --- Elementary Knot Theory --- p.13 / Chapter 1.2.1 --- Basic Definition --- p.13 / Chapter 1.2.2 --- Regular Projection --- p.14 / Chapter 1.2.3 --- Classical Knot Invariants --- p.15 / Chapter 2 --- Thickness of Knots --- p.17 / Chapter 2.1 --- Thickness and its Curvature --- p.17 / Chapter 2.2 --- Consequences of the Main Theorem --- p.27 / Chapter 2.3 --- Polygonal Representation of Knot --- p.30 / Chapter 2.4 --- Self-distance and Main Theorem --- p.35 / Chapter 3 --- Energy --- p.37 / Chapter 3.1 --- Normal Energy and Symmetric Energy --- p.37 / Chapter 3.2 --- Mobius Energy --- p.48 / Chapter 3.3 --- Another Formulation of Mobius Energy --- p.57 / Chapter 3.4 --- A Non Self-Repulsive Energy Function --- p.62 / Bibliography --- p.67

Hui, Wing San. / Thesis submitted in: November 2003. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 58-60). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Outline of Thesis --- p.2 / Chapter 2 --- Basic Knowledge of Knot Theory --- p.3 / Chapter 2.1 --- Preparation --- p.3 / Chapter 2.1.1 --- "Knots, Knot Equivalence and Isotopic Knot" --- p.3 / Chapter 2.1.2 --- Tame and Wild Knots --- p.5 / Chapter 2.2 --- Some Invariants and Quantities about Knot --- p.7 / Chapter 2.2.1 --- Projection of Knot and Crossing Number --- p.7 / Chapter 2.2.2 --- Braids and Braid Index --- p.7 / Chapter 3 --- Minimal Stick Number --- p.11 / Chapter 3.1 --- History and Definition --- p.11 / Chapter 3.2 --- Minimal Stick Number on Some Simple Knots --- p.12 / Chapter 3.3 --- Some Theorems on the Minimal Stick Number --- p.14 / Chapter 4 --- Superbridge Index --- p.22 / Chapter 4.1 --- "Definitions of Bridge Index, Superbridge Index and Total Curvature" --- p.22 / Chapter 4.2 --- Superbridge Index and Braid Index --- p.25 / Chapter 4.3 --- "Relations between Bridge Index, Superbridge Index and Total Curvature" --- p.29 / Chapter 4.4 --- Superbridge Index and Minimal Stick Number --- p.36 / Chapter 5 --- The Geometric Knot Space --- p.37 / Chapter 5.1 --- Definition of the Geometric Knot Space --- p.37 / Chapter 5.2 --- "Geometric Equivalence and Topological Properties of the Geometric Knot Space, Geo(n)" --- p.39 / Chapter 5.3 --- "The Spaces Geo(3), Geo(4) and Geo(5)" --- p.40 / Chapter 5.4 --- Topology of the Space Geo(6) --- p.43 / Chapter 6 --- Concluding Remarks --- p.52 / Chapter 6.1 --- Other Results on the Minimal Stick Number --- p.52 / Chapter 6.2 --- Minimal Stick Number and Superbridge Index of the Torus Knot --- p.54 / Chapter 6.3 --- Explorations of the Geometric Knot Spaces --- p.56 / Bibliography --- p.58

Liu Chun-Lung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 98-100). / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Fundamentals of thickness --- p.7 / Chapter 2.1 --- Definition of thickness --- p.9 / Chapter 2.2 --- Basic theorem and corollaries --- p.10 / Chapter 2.3 --- Equivalent definitions --- p.17 / Chapter 3 --- Ropelength minimizer --- p.19 / Chapter 3.1 --- Existence of ropelength minimizer --- p.20 / Chapter 3.2 --- Some ropelength minimizers --- p.23 / Chapter 4 --- Thickness computation --- p.27 / Chapter 4.1 --- Definitions --- p.28 / Chapter 4.2 --- Octrope algorithm --- p.33 / Chapter 4.3 --- Minor improvements --- p.45 / Chapter 5 --- Arc presentation --- p.54 / Chapter 5.1 --- Definitions --- p.55 / Chapter 5.2 --- Basic Theorems --- p.57 / Chapter 5.3 --- Ropelength upper bound --- p.64 / Chapter 6 --- Hamiltonian knot projection --- p.74 / Chapter 6.1 --- Hamiltonian RPG --- p.75 / Chapter 6.2 --- Embedding of RPG --- p.78 / Chapter 6.3 --- Ropelength upper bound --- p.95 / Bibliography --- p.98

Thesis (Ph. D.)--Byrn Mawr College. / From the Transactions of the Royal Society of Edinburgh, v. 52 (1917).

Thesis (M.A.) -- Central Connecticut State University, 2008. / Thesis advisor: Nelson Castaneda. "... in partial fulfillment of the requirements for the degree of Master of Arts in Mathematical Sciences." Includes bibliographical references (leaves 98-99). Also available via the World Wide Web.

Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 83-84). Available also in a digital version from Dissertation Abstracts.

Thesis supervisor: Charles Frohman. Includes bibliographic references (p. 60).

Thesis supervisor: Oguz Durumeric. Includes bibliographic references (p. 58).