The Geometry and Topology of the Jones Polynomial

1985年，琼斯给一个结分配了一个整数系数的多项式，1989年，威滕提出了这个多项式的路径积分解释。在取平行线时，我们可以考虑一个多项式序列。体积猜想将这个序列的渐近行为（在单位的复根处评估）与结补的黎曼（主要是双曲）几何联系起来。pi建议研究这一点以及与量子拓扑和几何化有关的相关猜想。三维空间中的绳结是一种灵活的绳子，它可以在不交叉的情况下变形。绳子的宽度和长度不相关，它的起点和终点重合。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）是由结构成的；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，弦理论提供了一个类似的世界观，即基本粒子在宇宙结构中沿着打结的圆圈振动。弦理论能否解释世界是一个关键问题。同时，众所周知，弦理论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.