GEOMETRY OF 3-MANIFOLDS AND QUANTUM INVARIANTS

三流形的几何结构和量子不变量

基本信息

批准号：
15540089
负责人：
YOKOTA Yoshiyuki
金额：
$ 1.6万
依托单位：
TOKYO METROPOLITAN UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540089/
关键词：
knot Jones polynomial volume conjecture A-多項式

项目摘要

The volume conjecture of knots states that the asymptotic behavior of the colored Jones polynomial, a genaralization of the famous Jones polynomial, determines the simplicial volume of the knot complement. This conjecture was first proposed by R. Kashaev for hyperbolic knots, and generalized by H. Murakami and J. Murakami for general knots. This conjecture is further generalized to involve the Chern-Simons invariants through a computer experiment made by H. Murakami, J. Murakami, M. Okamoto, T. Takata and the author. Now, many geometers and topologists are interested in this problem. The purpose of this research is to investigate the relationship between the geometry of 3-manifolds and the quantum invariants, motivated by the volume conjecture which suggests a relationship between the geometry of knot complements and the colored Jones polynomial.In 2003, with H. Murakami, we proved that, for the figure eight knot, certain limit of the colored Jones polynomial dominates not only the vol … More ume of the complement but also the volumes of the closed 3-manifolds obtained by Dehn surgeries, which corrects the conjecture proposed by S. Gukov, and we proposed a new genaralization of the volume conjecture, which also explains the unexpected relationship between the recursive formula of the colored Jones polynomial and the A-polynomial of knots. We reported this result in the international workshops held at Edinburgh and Geneva in 2003 together with a newest result concerning the relationship between quantum 6j-symbols and volumes of hyperbolic tetrahedra. The author further confirmed that the argument for the figure eight knot is also available for so-called twist knots, and reported this result in the international workshop held at Potsdam in 2004.On the other hand, when the author visited the University of Geneva in 2004, R. Kashaev and the author proved that the colored Jones polynomial of knots can be expressed by simple integrals over higher dimensional tori by using quantum dilogarithm functions whose asymptotic behaviors are well-known. We consider this result is a big progress toward the solution of the volume conjecture, because we may estimate the asymptotic behavior of such integrals over tori, a well-known compact manifold, by using the saddle point method together with the Morse theoretic argument. We have already reported this result in the meeting of American Mathematical Society held at Atlanta in early 2005. Less

结的体积猜想指出，有色琼斯多项式（著名的琼斯多项式的推广）的渐近行为决定了结补的单纯体积。该猜想首先由 R. Kashaev 针对双曲结提出，并由 H 推广。 Murakami 和 J. Murakami 通过计算机实验将这个猜想进一步推广到涉及 Chern-Simons 不变量。 H. Murakami、J. Murakami、M. Okamoto、T. Takata 和作者们现在对这个问题感兴趣，这项研究的目的是研究 3-流形的几何形状与几何之间的关系。量子不变量，受体积猜想的启发，体积猜想表明了结补的几何形状与彩色琼斯多项式之间的关系。2003 年，我们与 H. Murakami 一起证明了，对于数字 8结，有色琼斯多项式的一定极限不仅支配补体的体积，而且支配Dehn手术得到的闭3-流形的体积，这纠正了S. Gukov提出的猜想，我们提出了一个新的猜想体积猜想的推广，这也解释了彩色琼斯多项式的递归公式与结的 A 多项式之间的意外关系，我们在举行的国际研讨会上报告了这一结果。 2003年爱丁堡和日内瓦提出了关于量子6j符号与双曲四面体体积关系的最新结果，作者进一步证实了8字结的论证也适用于所谓的扭结，并报道了这一结果。 2004年在波茨坦举行的国际研讨会上。另一方面，2004年作者访问日内瓦大学时，R. Kashaev和作者证明了纽结的彩色琼斯多项式可以通过使用渐近行为众所周知的量子双对数函数来表示为高维圆环上的简单积分，我们认为这个结果是解决体积猜想的一大进步，因为我们可以估计此类积分的渐近行为。 tori，一个著名的紧流形，利用鞍点法结合莫尔斯理论论证，我们早在亚特兰大举行的美国数学会会议上就已经报告了这个结果。 2005. 减