Various inbariants appearing in low-dimensional topology

我正在研究结点的体积猜想和它的各种推广。体积猜想指出，一个结的有色琼斯多项式的某个极限将决定结的补的体积。更准确地说，我们用exp（2Pi^*I/V）替换‘color’ N的有色琼斯多项式的参数，然后考虑N趋于无穷时的极限。至此，我将体积猜想推广如下：如果用exp（a/N）代替参数，变换a，取极限，那么我们不仅可以得到结补的体积，还可以得到由结用Dehn手术得到的三流形的体积和chen - simons不变量。这一学年，我不仅研究了有色琼斯多项式对大N的对数渐近展开式的极限，还研究了几个系数。体积猜想相当于说N的系数决定了体积和chen - simons不变量。作为与S.Gukuv共同工作的结果，我们提出了以下新的猜想：log N的系数将由在SL(2， C).2处的结群的表示所扭转的同调群的维数决定。常数项将由对应于SL（2， C）处结群的表示的Reidemeister扭转来确定。上述猜想为体积猜想及其推广提供了一个新的方面。此外，我们已经通过计算机计算证实了这一猜想对某些节是正确的。

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.