General research of the volume conjecture of knots

纽结的体积猜想指出，有色琼斯多项式的渐近行为决定了纽结补集的单纯体积。这一猜想首先由R.Kashaev对双曲纽结提出，并由H.Murakami和J.Murakami推广到一般纽结。H.Murakami，J.Murakami，M.Okamoto，T.Takata和Y. Yokota通过计算机实验将该猜想进一步推广到包含Chern-Simons不变量。最后，S.Gukov和ft Murakami证明了有色Jones多项式的各种极限不仅支配纽结补的体积，而且支配由Dehn手术或纽结补的变形空间上的Neumann-Zagier函数得到的闭3-流形的体积.本文证明Kashaev提出的双曲体积猜想的策略是：（1）将色Jones多项式表示为环面上的积分;（2）利用莫尔斯理论应用鞍点方法。我们已经在国外的许多研讨会上报道了这些，我正在写一篇关于纽结补数的Neumann-Zagier函数的论文，这是将有色琼斯多项式的极限与体积联系起来所必需的，一篇描述实现任何纽结（1）的算法的论文和一篇关于同时实现某些纽结（2）的论文。另一方面，共同研究员J. Murakami观察到，Whitehead环和Borromean环的Akutsu-Deguchi-Ohtsuki不变量的渐近行为给出了它们定义的轨道的体积。这是S.Gukov和H. Murakami猜想的有趣推广。

The volume conjecture of knots states that the asymptotic behavior of the colored 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 was further generalized to involve the Chern-Simons invariant through computer experiments made by H.Murakami, J.Murakami, M.Okamoto, T.Takata and Y.Yokota. Finally, S.Gukov and ft Murakami conjectured that various limit of the colored Jones polynomial dominates not only the volume of the knot complement but also the volumes of the closed 3-manifolds obtained by Dehn surgeries, or the Neumann-Zagier function on the deformation space of the knot complement. Our strategy to prove the hyperbolic volume conjecture, the first one proposed by Kashaev, is :(1) express the colored Jones polynomial as an integral over a torus(2) apply the saddle point method by using Morse theoryAlong this strategy, with Kashaev at University of Geneva, we achieved (1) for any knot, and (2) for some knots. We have reported these in many workshops abroad, and I am writing a paper on the Neumann-Zagier function of knot complements, which is necessary to connect the limit of the colored Jones polynomial with the volume, a paper to describe the algorithm to realize (1) for any knot and a paper on (2) for some knots simulteneoualy. On the other hand, the co-researcher J. Murakami observed that the asymptotic behavior of the Akutsu-Deguchi-Ohtsuki invariant of Whitehead link and Borromean rings gives the volumes of the orbifolds they define. This is an interesting generalization of the conjecture made by S.Gukov and H.Murakami.