Volume conjecture of knots and its applications

本研究的目的是证明3球中双曲结点的体积猜想，并将结果应用于寻找结点补的双曲结构的变形空间的理想点和最大环面上的最短测地线。本研究的结果是给出了从结图中计算双曲结补的chen - simons不变量的一种简单方法，证明了5交结的体积猜想（与日内瓦大学的R. Kashaev合作），证明了6交结的体积猜想（与京都大学的T. Ohtsuki合作），并给出了一种利用有色琼斯多项式积分表达式中出现的势函数的变形计算顶点形状的简单方法。

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.