The works of Thurston and Jones revolutionized low dimensional topology. Thurston established the ubiquity of hyperbolic structure in low dimensions. Jones' work, via such physical notions as quantum groups and path integrals, led to vast families of topological invariants associated with diagrammatic descriptions of topological objects. The investigators have striking experimental evidence for a direct link between these disparate approaches. Establishing such a link is of fundamental importance to low dimensional topology. This proposal aims to establish this link, with particular focus on the generalized volume conjecture, which relates the most important geometric and quantum invariants: the hyperbolic volume and Chern-Simons invariant, and the colored Jones polynomials of a knot. The theory of L2-invariants provides a combinatorial framework to study hyperbolic volume. Deforming this construction along the curve of representations given by the A-polynomial involves the twisted Alexander polynomial and the colored Jones polynomials. Tree entropy of graphs provides the bridge between L2-torsion and colored Jones polynomials.
The volume conjecture relates classical geometric invariants of three-dimensional spaces with topological invariants motivated by ideas from quantum physics. This conjecture originated in the theory of quantum gravity, which cannot yet be verified experimentally. The mathematically rigorous verification sought by this project of this and related conjectures will support the internal consistency of quantum gravity. Unifying quantum and geometric invariants is also of intrinsic mathematical importance, which will yield important new insights in other fields. Computer programs to study geometric invariants and tabulation of knots and their invariants are essential tools for this research. Undergraduate and graduate students involved in this project will be exposed to sophisticated mathematics and computer tools.
