Graduation Year

2010

Document Type

Dissertation

Degree

Ph.D.

Degree Granting Department

Mathematics

Major Professor

Masahiko Saito, Ph.D.

Committee Member

Brian Curtin, Ph.D.

Committee Member

Xiang-Dong Hou, Ph.D.

Committee Member

Nataˇsa Jonoska, Ph.D.

Keywords

Random links, Knot determinant, Fox colorings, Cohomology, Yang-Baxter equation

Abstract

In this dissertation, we study two areas of interest in knot theory: Random knots in the unit cube, and the Yang-Baxter solutions constructed from Frobenius algebras.

The study of random knots can be thought of as a model of DNA strings situated in confinement. A random knot with n vertices is a polygonal loop formed by selecting n distinct points in the unit cube, for a positive integer n, and connecting these points by straight line segments successively, such that the last point selected is joined with the first one. We present a step by step description of our algorithm and Maple codes for generating random knots in the unit cube, with a given vertex number n. To detect non-trivial knots, we use a knot invariant called the determinant. We present an algorithm and its Maple code for computing the determinant for random knots. For each vertex number n, we generate large number of random knots and form data sets of values of the determinant. Then we analyze our data sets in various ways. For instance, for each vertex number n, we form data sets of the number of p-colorable random knots by finding the set of prime divisors of each determinant output. We define the stick number for p-colorability to be the minimum number of line segments required to form a p-colorable knot. We use our data sets to find upper bounds for stick numbers for p-colorability, for primes p _ 191. We also find distributions of p-colorable knots and small determinant values.

The second topic on random knots is the linking number of random links. A random link is a collection of disjoint random knots produced simultaneously. We present descriptions of our algorithm and its Maple code for constructing random links of two components, and calculating their linking numbers in detail. By running the code for 1000 times, for the vertex number n less than or equal to 30, we obtain data sets of linking numbers for two-component random links such that each component is a random knot with n vertices. Then we find the distribution of linking numbers and calculate upper bounds for the stick number for the linking numbers ` _ 15.

The second area we investigate is applications of Fobenius algebras to knot theory. Chain complexes and Yang-Baxter solutions (R-matrices) are constructed by the skein theoretic approach using Frobenius algebras, and deformed R-matrices are constructed by using 2-cocyles. We compute cohomology groups, Yang-Baxter solutions and their cocycle deformations for group algebras, polynomial algebras and complex numbers. We construct knot and link invariants using these R-matrices from Frobenius algebras via Turaev’s criteria. Then a series of skein relations of the invariant are introduced for oriented knot or link diagrams. We also present calculations of the Frobenius skein invariant for various knots and links.

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