The Jones polynomial is a powerful knot invariant that keeps tightly locked key information about the geometry and topology of the knot complement. The principal investigator and the principal co-investigator propose to study the Geometry and Topology of the Jones polynomial and its 3-dimensional counterpart, the Witten-Reshetikhin-Turaev invariant, that is revealed via asymptotic expansions, and especially its relation to hyperbolic geometry via representations of the fundamental group and to number theory via modular properties of the asymptotic expansions. A side project growing out of the proposal which concerns only classical invariants is the investigation of the growth rate of torsions of finite coverings of 3 manifolds and its relations to geometry/topology of the manifolds.
The study of knotted curves in 3-dimensional manifolds is intimately related to our physical 3-dimensional space. With a long history reaching back to the early stage of topology, it is fundamental to many areas of mathematics and physics, and also has applications in biology, chemistry, and quantum computation. In the last few decades, Jones and Witten's work and ideas from physics have led to the discovery of new types of invariants for knots which create interactions of low dimensional topology with geometry, algebra, number theory, analysis, quantum field theory and combinatorics. The aim of this project is to better understand the nature of these new invariants and their relations with classical invariants.