The Geometry, Topology and Asymptotics of the Jones Polynomial

结是三维空间中嵌入的柔性圆，起点和终点连接在一起，允许在三维空间中移动而不交叉。结是基本的三维对象，丰富到足以描述任何可能的三维形状，以及许多四维形状。纽结在微观上出现在原子水平（可能用于拓扑量子计算）、分子水平（DNA结构）和天文水平（黑洞结构）。琼斯多项式是一个强大的纽结不变量，它紧紧锁定了关于纽结补的几何和拓扑的关键信息。拟议的研究的目的是研究琼斯多项式的几何和拓扑，通过渐近展开，特别是它与双曲几何和基本群的表示的关系。 拟议的研究链接在一起的想法，从拓扑学，几何学，代数，数论，分析，量子场论和组合。这需要解决困难的分析和组合问题，并最终对3维和4维规范理论有更深的几何理解。

A knot is an embedded flexible circle in 3-space, with beginning and ends joined together, which allows to move in 3-space without crossing itself. Knots are fundamental 3-dimensional objects, rich enough to describe any possible 3-dimensional shape, and many 4-dimensional shapes, too. Knots appear microscopically on the atomic level (possibly used in topological quantum computing), in the molecular level (in the structure of DNA), and in the astronomical level (in the structure of black holes).The Jones polynomial is a powerful knot invariant that keeps tightly locked key information about the geometry and topology of the knot complement. The purpose of the proposed research is to study the Geometry and Topology of the Jones polynomial that is revealed via asymptotic expansions, and especially its relation to hyperbolic geometry and representations of the fundamental group. The proposed research links together ideas from topology, geometry, algebra, number theory, analysis, quantum field theory and combinatorics. This requires the solution of hard analytic and combinatorial problems, and ultimately to a deeper geometric understanding of gauge theories in dimensions 3 and 4.