Various inbariants appearing in low-dimensional topology

我正在研究纽结的体积猜想及其各种推广。体积猜想指出，一个纽结的有色琼斯多项式的某个极限将决定该纽结的补集的体积。更准确地说，我们用exp（2 Pi ^*I/V）代替“color”N的有色琼斯多项式的参数，然后考虑N趋于无穷大的极限。到目前为止，我将体积猜想推广如下：如果我们用exp（a/N）代替参数，对a作不同的改变，取极限，那么我们不仅得到了纽结补的体积，而且得到了由纽结通过Dehn手术得到的三维流形的体积和Chern-Simons不变量。研究了有色琼斯多项式对数关于大N的渐近展开式的极限和几个系数。体积猜想等价于说N的系数决定体积和Chern-Simons不变量。作为与S.Gukuv合作的结果，我们提出了如下新猜想：1. log N的系数由纽结群在SL（2，C）上的表示所扭曲的上同调群的维数决定。2.常数项由纽结群在SL（2，C）上的表示所对应的Reidemeister挠率决定。上述猜想为体积猜想及其推广提供了一个新的方面。此外，我们还通过计算机计算证实了这一猜想对某些纽结是成立的。

I am working on the volume conjecture for knots and its various generalizations. The volume conjecture states that a certain limit of the colored Jones polynomial of a knot would determine the volume of the complement of the knot. More precisely, we replace the paramaeter of the colored Jones polynomial of 'color' N with exp(2Pi^*I/V) and then consider the limit where N goes to the infinity. So far I generalized the volume conjecture as follows : If we replace the parameter with exp(a/N), change a variously, and take the limit, then we would have not only the volume of the knot complement but also the volume and the Chern-Simons invariant of the three-manifold obtained from the knot by Dehn surgery.In this academic year, I studied not only the limit but also several coefficients of the asymptotic expansion of the logarithm of the colored Jones polynomial with respect to large N. The volume conjecture is equivalent to saying that the coefficient of N would determine the volume and the Chern-Simons invariant.As a result of a joint work with S.Gukuv, we proposed the following new conjecture :1.The coefficient of log N would be determined by the dimension of the cohomoloty group twisted by a representation of the knot group at SL(2, C).2.The constant term would be determined by the Reidemeister torsion corresponding to a representation of the knot group at SL(2, C).The conjecture above gives a new aspect to the volume conjecture and its generalizations. Moreover, we have confirmed by computer caluculations that this conjecture is true for some knots.