Publication Date

8-2015

Date of Final Oral Examination (Defense)

5-21-2015

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Supervisory Committee Chair

Uwe Kaiser, Ph.D.

Supervisory Committee Member

Samuel Coskey, Ph.D.

Supervisory Committee Member

Zachariah Teitler, Ph.D.

Abstract

In the mid 1980s, it was realized that solutions to what is known as the Knizhnik- Zamolodchikov equation, or KZ equation, provided a pathway to representations of the braid group Bn on n strands, with early mathematical treatments of the topic by Kohno and Drinfel'd. Such representations are typically referred to as monodromy representations of the braid group along solutions of the KZ equation. These linear representations are of great interest within topology, integral to the construction of isotopy invariants of knots and links, such as the well known Jones polynomial. More current discussions of the KZ equation and the associated monodromy representations are available in [6] and [9]. The former provides extensive algebraic background, while assuming a broad knowledge of differential geometry and eschewing certain calculable details of an explicit monodromy representation. The latter is more elementary, while containing nontrivial gaps and irregularities in the presentation. The following is intended to be a complement to both. Chapter 3 provides details of the argument by which solutions of the KZ equation induce representations of the braid group Bn for arbitrary n. Chapter 4 solves the KZ equation in the cases of n = 2,3 and carries out explicit calculation of the monodromy representation on generators of the respective braid groups. From the work of Sections 3.1, 3.2, and 4.2.1, it is observed that the representation property of the KZ representations may be reduced to the uniqueness of solution to a particular initial value problem.

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