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

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1999. / Includes bibliographical references (p. [85]-87).

Thesis (PH. D.)--University of Chicago, 1933. / Vita. Lithoprinted. "Private edition, distributed by the University of Chicago libraries, Chicago, Illinois." Bibliography: p. 28.

A knot invariant called the Jones polynomial will be defined in two ways, as the Kauffman Bracket polynomial and the Tutte polynomial. Three properties of the Jones polynomial are discussed. We also see how mutant knots share the same Jones polynomial.

This thesis concerns flat knots and their properties. We study various invariants of flat knots, such as the crossing number, the u-polynomial, the flat arrow polynomial, the flat Jones-Krushkal polynomial, the based matrices, and the φ-invariant. We also examine the behavior of these invariants under connected sum and cabling. We give a matrix-based algorithm to calculate the flat Jones-Krushkal polynomial. We take a special interest in certain subclasses of flat knots, such as almost classical flat knots, checkerboard colorable flat knots, and slice flat knots. We explore how the invariants can be used to obstruct a flat knot from being almost classical, checkerboard colorable, or slice. We show that any minimal crossing diagram of a composite flat knot is a con- nected sum, and we introduce a skein formula for the constant term of the flat arrow polynomial. A companion project to this thesis is the interactive website, FlatKnotInfo. It provides a curated dataset of examples and invariants of flat knots. It also features a tool for searching flat knots and another tool that crossreferences flat knots with virtual knots. FlatKnotInfo was used to develop many of the results in this thesis, and we hope others find it useful for their research on flat knots. The Python code for calculating based matrices and flat Jones-Krushkal polynomials is included in an appendix. / Dissertation / Doctor of Philosophy (PhD)

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No description available.

This paper begins with a basic overview of the key concepts of classical and virtual knot theory. After introductions to concepts such as knot diagrams, Reidemeister moves, and virtual links, the paper discusses the bikei algebraic structure and the fundamental bikei. The paper describes an algorithm that converts fundamental bikei presentations to matrix representations, and then completes the resulting matrices. These completed matrices can return the value of two link invariants.

Proteins play a vital role in all organic life. The structure of a protein is directly related to its function. Hence, how they fold and what they fold into is of great interest. Given the spontaneous manner in which many proteins fold, one would not expect complicated structures like knots to occur in native states. Nevertheless, current research has shown that proteins do indeed contain local knots; some with as many as 6 crossings. In general, the role of knots in proteins and how they are formed is not completely understood. This thesis develops models of protein knotting by using knot theory and tangles. Mathematically, a knot is just a topological embedding of a circle in Euclidean 3-space, R3, or the unit 3-sphere, S3. A tangle is defined as a pair, (B, T), where B is a 3-dimensional ball and T is a set of disjoint arcs properly embedded in B. We begin with 2-string tangles and use the tangle calculus developed by Ernst and Sumners to set up tangle equations. In this model the strings of the 2-tangles represent the protein chain. Solutions to these 2-string tangle equations are then found. Motivated by the hypothesized folding pathway of the knotted protein DehI, a more complicated 3-string tangle model is developed. It is hypothesized that a terminal end of the protein is threaded through two loops. In the proposed model, the threading of a terminal end of the protein through two loops is translated into a Γ;-move on 3-string tangles. A Γ;-move is a special type of 3-string tangle replacement. The 3-braids are utilized as a subset of 3-string tangles to find solutions in a limited case. Additionally, tangle models give insight into how to make specific knot types in proteins. We finish with a general result by proving that any knot of unknotting number 2 can be unknotted by the Γ;-move. With these models we determine which knots are the most biologically possible to occur in proteins.

In the 1960s French mathematician George de Rham found a relationship between two invariants of knots. He found that there exist representations of the fundamental group of a knot into a group G of upper right triangular matrices in C with determinant one that is described exactly by the roots of the Alexander polynomial. I extended this result to find that the representations of the fundamental group of a link into G are described by the multivariable Alexander polynomial of the link.