Topological field theory and some problems on 3-manifolds

本研究的目的是利用拓扑量子场论中的量子不变量来解决纽结和三维流形上的各种问题，得到了纽结和三维流形的几何结构与量子不变量的渐近行为之间的一种神秘的关系，即众所周知的纽结的体积猜想。然后，对于三维球面上的双曲纽结，我们发现了该函数的驻相方程与由补的理想三角剖分得到的双曲性方程之间惊人的对应关系。此外，我们观察到函数的临界值给出了补数的体积和Chern-Simons不变量。在这些结果的基础上，在2000年，我们通过在补的双曲结构的形变空间中加入一些来自三角剖分的项，得到了一族控制补的双曲结构变形空间的势函数。也就是说，势函数的定相方程与非完全双曲结构的双曲性方程重合，并且势函数的临界值不仅给出了闭合的双曲三维流形的体积，而且还给出了由适当的Dehn手术得到的闭合双曲三维流形的Chern-Simons不变量。最近，一些实验表明，双曲四面体的体积与量子6J符号有关，用于定义闭合三维流形的量子不变量。结合这些结果和我们的结果，我们现在有强烈的动机将体积猜想推广到闭的3-流形上。

The purpose of this research is to attack various problems on knots and 3-manifolds by using quantum invariants deribed from topological quantum field theory, and we obtain a mysterious relationship, which is well-known as the volume conjecture of knots, between the geometric structure of knots and 3-manifolds and the asymptotic behavior of quantum invariants as described below.In 1999, for a knot in 3-sphere, we first extracted a characteristic complex function from the integral approximation of the Jones invariant. Then, for a hyperbolic knot in 3-sphere, we found a surprising correspondence between the stationary phase equations for the function and the hyperbolicity equations deduced from an ideal triangulation of the complement. Furthermore, we observed that a critical value of the function gives the volume and the Chern-Simons invariant of the complement. Following these results, in 2000, we obtain a family of potential functions, which dominates the deformation space of hyperbolic structures of the complement, by adding some term coming from the triangulation to the function above. That is, the stationary phase equations for a potential function coincide with the hiperbolicity equations for a non-complete hyperbolic structure, and a critical value of the potential function gives not only the volume but also the Chern-Simons invariant of the closed, hyperbolic 3-manifold obtained from the complement by appropriate Dehn surgery.Recently, some experiments reveal that the volumes of hyperbolic tetrahedra are related to quantum 6j-symbols, which are used to define the quantum invariants of closed 3-manifolds. Conbining these results with ours, we now have a strong motivation to generalize the volume conjecture to closed 3-manifolds.