Mahler Measure and the Alexander Polynomials of Pretzel Links

Author

Shaun Bhatty, University of South AL

Date of Award

5-2004

Document Type

Undergraduate Thesis

Degree Name

BS

Department

Mathematics

Faculty Mentor

Susan Williams, Ph.D.

Advisor(s)

Scott Carter, Ph.D., Dan Silver, Ph.D.

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 f. 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 ɛ, are there any polynomials with integer coefficients for which 1 < M(f)< 1 + ɛ?

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 k_(nj), 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 each j, the Mahler measures of the corresponding Alexander polynomials approach 2^j as n approaches infinity. When n > 4.68j, the Mahler measure is greater than 1.698.

Recommended Citation

Bhatty, Shaun, "Mahler Measure and the Alexander Polynomials of Pretzel Links" (2004). Digitized Honors Theses (2002-2017). 70.
https://jagworks.southalabama.edu/honors_theses-boundprint/70