Collaborative Research: FRG: Hyperbolic Geometry and Jones Polynomials

瑟斯顿和琼斯的工作彻底改变了低维拓扑学。瑟斯顿建立了低维双曲结构的普遍性。琼斯的工作，通过量子群和路径积分等物理概念，导致了与拓扑对象的图解描述相关的大量拓扑不变量族。研究人员有惊人的实验证据证明这些不同的方法之间存在直接联系。建立这种链路对低维拓扑结构具有重要意义。本提案旨在建立这种联系，特别关注广义体积猜想，它涉及最重要的几何和量子不变量：双曲体积和chen - simons不变量，以及aknot的彩色琼斯多项式。l2不变量理论为研究双曲体积提供了一个组合框架。沿着a -多项式给出的表示曲线变形这个结构涉及到扭曲的亚历山大多项式和有色的琼斯多项式。图的树熵提供了l2 -扭转和彩色琼斯多项式之间的桥梁。体积猜想将三维空间的经典几何不变量与量子物理思想激发的拓扑不变量联系起来。这个猜想起源于量子引力理论，但还不能通过实验加以验证。该项目对这一猜想和相关猜想所寻求的数学上严格的验证将支持量子引力的内部一致性。统一量子不变量和几何不变量也具有内在的数学重要性，这将在其他领域产生重要的新见解。研究几何不变量和结及其不变量表的计算机程序是本研究的重要工具。参与该项目的本科生和研究生将接触到复杂的数学和计算机工具。

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.