Volume conjecture of knots and its applications

证明了三维球面上双曲纽结的体积猜想，并将其应用于求纽结补的双曲结构的变形空间的理想点和求最大环面上的最短测地线，给出了由纽结图计算双曲纽结补的Chern-Simons不变量的简单方法，证明了5-交纽结的体积猜想（与R. Kashaev在日内瓦大学），证明了6-交叉结点的体积猜想（与T. Ohtsuki在京都大学），并给出了一个简单的方法来计算尖点形状从变形的势函数出现在积分表达式的有色琼斯多项式。

The purpose of this research is to prove the volume conjecture for hyperbolic knots in 3-sphere, and to apply the results to find the ideal points of the deformation space of the hyperbolic structures of the knot complement and to find the shortest geodesic on the maximal torus.The results of this research are to give a simple method to compute the Chern-Simons invariant of hyperbolic knot complements from knot diagrams, the proof of the volume conjecture for 5-crossing knots(joint work with R. Kashaev at University of Geneva), the proof of the volume conjecture for 6-crossing knots(joint work with T. Ohtsuki at Kyoto University), and to give a simple method to compute the cusp shape from the deformation of the potential function appearing in the integral expression of the colored Jones polynomial.