GEOMETRY OF 3-MANIFOLDS AND QUANTUM INVARIANTS

纽结的体积猜想指出，有色琼斯多项式的渐近行为，著名的琼斯多项式的推广，决定了纽结补的单纯体积。这个猜想是由R. Kashaev提出的双曲纽结，并由H. Murakami和J. Murakami的一般结。通过H. Murakami，J. Murakami，M. Okamoto，T.高田和作者。现在，许多几何学家和拓扑学家都对这个问题感兴趣。本研究的目的是研究三维流形的几何与量子不变量之间的关系，其动机是体积猜想，它暗示了纽结补的几何与有色琼斯多项式之间的关系。在Murakami的基础上，我们证明了，对于8字形纽结，有色琼斯多项式的某个极限不仅支配着体积， ...更多信息 不仅给出了补的空间的体积，而且给出了由Dehn手术得到的闭3-流形的体积，从而纠正了S. Gukov提出了体积猜想的一个新的推广，它也解释了有色Jones多项式的递归公式与纽结的A-多项式之间的意外关系.我们在2003年于爱丁堡和日内瓦举行的国际研讨会上报告了这一结果，同时还报告了关于量子6 j符号与双曲四面体体积之间关系的最新结果。作者进一步证实了8字结的论点也适用于所谓的扭结，并在2004年波茨坦举行的国际研讨会上报告了这一结果。另一方面，当作者于2004年访问日内瓦大学时，R. Kashaev和作者利用量子双对数函数证明了纽结的色Jones多项式可以用高维环面上的简单积分表示，其渐近性质是众所周知的。我们认为这一结果是解决体积猜想的一个重大进展，因为我们可以利用鞍点方法和莫尔斯理论证明来估计环面上这类积分的渐近性质.我们已经在2005年初在亚特兰大举行的美国数学学会会议上报告了这一结果。少

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