Collaborative Research: FRG: Hyperbolic Geometry and Jones Polynomials

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The works of Thurston and Jones revolutionized low dimensionaltopology. Thurston established the ubiquity of hyperbolic structure inlow dimensions. Jones' work, via such physical notions as quantumgroups and path integrals, led to vast families of topologicalinvariants associated with diagrammatic descriptions of topologicalobjects. The investigators have striking experimental evidence for adirect link between these disparate approaches. Establishing such alink is of fundamental importance to low dimensional topology. Thisproposal aims to establish this link, with particular focus on thegeneralized volume conjecture, which relates the most importantgeometric and quantum invariants: the hyperbolic volume andChern-Simons invariant, and the colored Jones polynomials of aknot. The theory of L2-invariants provides a combinatorial frameworkto study hyperbolic volume. Deforming this construction along thecurve of representations given by the A-polynomial involves thetwisted Alexander polynomial and the colored Jones polynomials. Treeentropy of graphs provides the bridge between L2-torsion and coloredJones polynomials.The volume conjecture relates classical geometric invariants ofthree-dimensional spaces with topological invariants motivated byideas from quantum physics. This conjecture originated in the theoryof quantum gravity, which cannot yet be verified experimentally. Themathematically rigorous verification sought by this project of thisand related conjectures will support the internal consistency ofquantum gravity. Unifying quantum and geometric invariants is also ofintrinsic mathematical importance, which will yield important newinsights in other fields. Computer programs to study geometricinvariants and tabulation of knots and their invariants areessential tools for this research. Undergraduate and graduate studentsinvolved in this project will be exposed to sophisticated mathematicsand computer tools.