The Geometry, Topology, Asymptotics and Number Theory of the Jones Polynomial

琼斯多项式是一个强大的结不变量，它可以紧紧锁定结补的几何和拓扑的关键信息。首席研究员和首席联合研究员提议研究琼斯多项式的几何和拓扑，以及它的三维对应物，Witten-Reshetikhin-Turaev不变量，这是通过渐近展开式揭示的，特别是它与双曲几何的关系，通过基本群的表示和通过渐近展开式的模性质。从这个只涉及经典不变量的提议中产生的一个副项目是研究3个流形的有限覆盖的扭转的增长率及其与流形的几何/拓扑的关系。三维流形中打结曲线的研究与我们的物理三维空间密切相关。拓扑学有着悠久的历史，可以追溯到拓扑学的早期阶段，它是数学和物理许多领域的基础，在生物学、化学和量子计算中也有应用。在过去的几十年里，Jones和Witten的工作和来自物理学的想法导致了结的新型不变量的发现，这些不变量创造了低维拓扑与几何、代数、数论、分析、量子场论和组合学的相互作用。这个项目的目的是更好地理解这些新的不变量的性质以及它们与经典不变量的关系。

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.