The Geometry, Topology and Asymptotics of the Jones Polynomial

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

A knot is an embedded flexible circle in 3-space, with beginning and ends joined together, which allows to move in 3-space without crossing itself. Knots are fundamental 3-dimensional objects, rich enough to describe any possible 3-dimensional shape, and many 4-dimensional shapes, too. Knots appear microscopically on the atomic level (possibly used in topological quantum computing), in the molecular level (in the structure of DNA), and in the astronomical level (in the structure of black holes). The Jones polynomial is a powerful knot invariant that keeps tightly locked key information about the geometry and topology of the knot complement. The purpose of the proposed research is to study the Geometry and Topology of the Jones polynomial that is revealed via asymptotic expansions, and especially its relation to hyperbolic geometry and representations of the fundamental group. The proposed research links together ideas from topology, geometry, algebra, number theory, analysis, quantum field theory and combinatorics. This requires the solution of hard analytic and combinatorial problems, and ultimately to a deeper geometric understanding of gauge theories in dimensions 3 and 4.