The Geometry, Topology and Asymptotics of the Jones Polynomial

琼斯多项式的几何、拓扑和渐近

• 批准号: 0805078
• 负责人: Stavros Garoufalidis
• 金额: $ 40.05万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805078&HistoricalAwards=false
• 关键词: Geometry; Topology; Asymptotics; Jones; Polynomial

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.

一个结是一个嵌入在三维空间中的柔性圆，它的开始和结束连接在一起，可以在三维空间中移动而不会交叉。结是基本的三维物体，丰富到足以描述任何可能的三维形状，也可以描述许多四维形状。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）上。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。本研究的目的是研究琼斯多项式的几何和拓扑，通过渐近展开式揭示，特别是它与双曲几何和基本群的表示的关系。拟议的研究将拓扑学、几何、代数、数论、分析、量子场论和组合学的思想联系在一起。这需要解决困难的分析和组合问题，并最终对3维和4维的规范理论有更深的几何理解。