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研究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个神秘关系，即著名的纽结体积猜想。我们首先从琼斯不变量的积分近似中提取特征复变函数。然后，对于3-球面中的双曲纽结，我们发现了一个令人惊讶的对应关系之间的定相方程的功能和双曲方程推导出一个理想的三角形的补充。此外，我们观察到，一个临界值的功能给出了体积和陈-西蒙斯不变量的补充。在此基础上，在2000年，我们通过在上述函数中加入三角剖分的方法，得到了一族势函数，该势函数支配着补的双曲结构的变形空间.也就是说，势函数的定态相方程与非完全双曲结构的双曲性方程一致，并且势函数的临界值不仅给出了通过适当的Dehn手术从补得到的闭合双曲3-流形的体积，而且给出了Chern-Simons不变量。一些实验表明，双曲四面体的体积与量子6 j-符号有关，量子6 j-符号用于定义闭三维流形的量子不变量。结合这些结果与我们的，我们现在有一个强烈的动机，推广体积猜想到封闭的三维流形。

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.