## Mathematical Sciences Theses

#### Title

Alexander Polynomials of 3-Braid Knots

4-2020

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematical Sciences

Dr. Nafaa Chbili

Ho Hon Leung

#### Abstract

A knot is an embedding of a circle S1 into the three-dimensional sphere S3. A component link is an embedding of n disjoint circles Ⅱᵢ=1 S1 into S3. The main objective of knot theory is to classify knots and links up to natural deformations called isotopies. While there is no simple algorithm that helps decide whether two given knots (or links) are equivalent, various topological invariants have been developed to help distinguish between non-equivalent knots and links. The Alexander polynomial ∆L(t) is one of the oldest such tools. It was originally defined from the Seifert surface of the knot. A simpler recursive definition of this invariant has been introduced later using Conway skein relations on the link diagram.

The aim of this study is to compute the Alexander polynomial and obtain an explicit formula for some families of alternating knots of braid index 3. This formula is used to prove that ∆L(t) satisfies Fox’s trapezoidal conjecture. This conjecture states that the coefficients of the Alexander polynomial of an alternating knot are trapezoidal. In other words, these coefficients increase, stabilize then decrease in a symmetrical way. The main tool in this study is the Burau representation of the braid group.

COinS
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