The Geometry and Topology of the Jones Polynomial

1985年，琼斯给一个纽结分配了一个具有整系数的多项式，1989年，Witten提出了这个多项式的路径积分解释。取平行线时，可以考虑多项式序列。体积猜想将这个序列的渐近行为(在复数单位根处求值)与纽结补的黎曼几何(大多数是双曲的)联系起来。PI们建议研究这一点以及与量子拓扑学和几何化相关的猜想。3-空间中的结是一种灵活的绳索，可以在不脱离自身的情况下变形。绳子的宽度和长度是无关的，它的起点和终点是重合的。例如，橡皮筋就是打结的一个例子。19世纪中叶，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察到的化学元素的变化。一个多世纪后，弦理论提供了类似的世界观，基本粒子在宇宙的结构中沿着打结的圆圈振动。弦理论是否能解释世界是一个关键问题。同时，弦理论也对数学产生了深远的影响。

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.