RUI: Applications of Symbolic and Algebraic Dynamics to Knot Theory

University Of South Alabama, Mobile AL

Investigators

Abstract

The investigators will combine their two areas of expertise, topology and dynamical systems, to investigate open questions in knot theory. In particular, they will investigate Lehmer's Question. Seventy years ago D.H. Lehmer began constructing large prime numbers using polynomials that are small in a precise sense: integral polynomials with Mahler measure close to but different from 1. Lehmer could do no better than 1.17628..., a value that he achieved with a remarkable polynomial of degree 10. He then asked if that value could improved. Lehmer's Question remains open despite the best efforts of many. Lehmer's polynomial continues to appear in surprisingly separated fields. The investigators will attack Lehmer's Question from the perspective of topology and geometry. They will investigate applications of Mahler measure to the study of knots and links. Their methods will include traditional ones from commutative algebra, group theory, geometry and topology, as well as new techniques, many from dynamical systems. Computer methods will be used to develop examples. The mathematical theory of knots arose from physical theories of the nineteenth century. Since then, the field has expanded greatly, attracting the interests of scientists in many fields, including biology, chemistry and physics. Some reasons for the attraction are not hard to see. DNA, solar plasma filaments and fluid flow, for example, all exhibit knotting or linking behaviour. The investigators will combine their two areas of expertise, topology and dynamical systems, to investigate open questions in knot theory. The proposed research will provide new understanding of knots and links by using techniques from symbolic dynamical systems, a mathematical branch of information theory. It will promote and strengthen interaction between researchers in different fields. Undergraduate and graduate students will participate in the project.