RUI: Algebraic Dynamics of Knot Theory

University Of South Alabama, Mobile AL

Investigators

Abstract

The PIs will continue their program to understand the algebra arising from knots and links by using new techniques of algebraic dynamics. Modules over Laurent polynomial rings in d variables are replaced by their Pontryagin duals, which are compact abelian groups. Actions by the variables become commuting homeomorphisms, and corresponding dynamical invariants (e.g., numbers of periodic points, topological entropy, etc.) provide new topologicial invariants. The focus will be on Mahler measure, surface dynamics and twisted homology. The project will strengthen newly emerging connections between knot theory, dynamical systems and number theory. Although knots and links arise in a host of scientific phenomena such as DNA, solar flares and fluid flows, their mathematical study is fairly new. Fueled by ideas from many areas of mathematics, progress in knot theory has exploded during the past two decades, and the subject now attracts wide interest from physicists, biologists and engineers as well as mathematicians. This project will continue the investigators' program of applying ideas from another mathematical field, dynamical systems, in order to gain a new perspective. Computer methods will be used to develop hypotheses. The PIs will also write a monograph, the first survey of this interdisciplinary study, designed for graduate students and researchers.