The Geometry and Topology of the Jones Polynomial

琼斯多项式的几何和拓扑

• 批准号: 0505445
• 负责人: Stavros Garoufalidis
• 金额: --
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505445&HistoricalAwards=false
• 关键词: Geometry; Topology; Jones; Polynomial

In 1985 Jones assigned to a knot a polynomial with integer coefficients,and in 1989 Witten came up with a path-integral interpretation of thispolynomial. Upon taking parallels, one can consider a sequence of polynomials. The volume conjecture relates an asymptotic behavior of thissequence (evaluated at complex roots of unity) with the Riemannian (mostlyhyperbolic) geometry of the knot complement. The PIs propose to study thisand related conjectures that relate quantum topology and geometrization.A knot in 3-space is a flexible rope that is allowed to deform withoutcrossing itself. The width and length of the rope are irrevelant, and itsbeginning and end coincide. For example, a rubber band is an example of a knot. In the middle of 19th century, Kelvin postulated that chemical elements (such as gold, aluminum) are made out of knots; thus differentknot types explain the variation of observed chemical elements. More than a century later, string-theory offers a similar view of the world, whereelementary particles are vibrating in the fabric of cosmos, along knottedcircles. Whether string-theory explains the world is a key question. Meanwhile, it is known that string-theory has deeply influenced mathematics.

1985年，琼斯给一个结点分配了一个整数系数的多项式，1989年，维滕提出了这个多项式的路径积分解释。在取平行线时，可以考虑多项式序列。体积猜想涉及到这个序列的渐近行为（在复数的单位根上计算）与结补的黎曼几何（大部分是双曲的）。物理学家们建议研究这一点以及与量子拓扑学和几何化相关的相关理论。三维空间中的纽结是一条柔性的绳子，它可以变形而不交叉。绳子的宽度和长度是无关的，它的起点和终点是一致的。例如，橡皮筋就是结的一个例子。世纪中期，开尔文假设化学元素（如金、铝）是由结组成的，因此不同的结类型解释了观测到的化学元素的变化。世纪之后，弦理论提供了一个类似的世界观，基本粒子在宇宙结构中沿着沿着的圆圈振动。弦理论能否解释世界是一个关键问题。同时，众所周知，弦论对数学产生了深远的影响。