Digitized Honors Theses (2002-2017)

Date of Award

5-2004

Document Type

Undergraduate Thesis

Degree Name

BS

Department

Mathematics

Faculty Mentor

Susan Williams

Advisor(s)

Scott Carter, Dan Silver

Abstract

By a knot, mathematicians mean a knotted closed loop in space. To every knot there is associated a polynomial with integer coefficients called the Alexander polynomial of the knot. A quantity called the Mahler measure M(f) can be calculated for every polynomial/. We study the Mahler measures of the Alexander polynomials of knots. One goal is to gain insight into a question proposed by D.H. Lehmer: For any positive number e, are there any polynomials with integer coefficients for which I < M(f) < I +e?

For one class of knots, the odd pretzel knots, we found a simple new expression for the Alexander polynomial. As a special case, we examined the odd fibered pretzel knots. These knots form a two-parameter family kcnj), where n represents the number of positive half twists and j represents the number of negative triple half twists in the pretzel diagram. It was found that for eachj, the Mahler measures of the corresponding Alexander polynomials approach 2j as n approaches infinity. When n > 4.68j, the Mahler measure is greater than 1.698.

Comments

©2004 Shaun Muhammad Zaki Bhatty ALL RIGHTS RESERVED

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