Various inbariants appearing in low-dimensional topology

低维拓扑中出现的各种不变量

• 批准号: 15340019
• 负责人: MURAKAMI Hitoshi
• 金额: $ 7.94万
• 依托单位: Tokyo Institute of Technorogy
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340019/
• 关键词: knot; 3-mainfold; colored Jones polynomial; volume conjecture; quantum invariant; Chern-Simons invariant; Reidemeister torsion; torus結び目; 体積; Jones多項式; Alexander多項式; トーラス結び目; 8の字結び目; Dehn手術

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.

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数的体积。更准确地说，我们用exp(2pi^*i/V)来代替有色琼斯多项式N的参数，然后考虑N到无穷大的极限。到目前为止，我把体积猜想推广为：如果用exp(a/N)代替参数，变换a，取极限，那么我们不仅得到了结点补的体积，还得到了由Dehn手术得到的三维流形的体积和Chern-Simons不变量。在本学年，本文不仅研究了有色Jones多项式的对数关于大N的渐近展开式的极限，而且还研究了它的几个系数。体积猜想等价于说N的系数将决定体积和Chern-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：1.log N的系数将由SL(2，C)上的纽结群的表示所扭曲的上同调群的维来决定。2.常数项由SL(2，C)上的纽结群的表示所对应的Reidemister挠率来确定。C)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节点是正确的。

项目成果

リーマン面の退化族の諸相

黎曼一方堕落部落的各个方面

• 发表时间: 2004
• 期刊: 数学 56巻1号
• 作者: Osaka;N.;小野 薫;M.Kaneko;Takashi Tsuboi;遠藤 久顕
• 通讯作者: 遠藤 久顕

Some limits of the colored Jones polynomials of the figure-eight knot

八字结彩色琼斯多项式的一些极限

• 发表时间: 2004
• 期刊: Kyungpook Math.J. 44 No.13
• 作者: Fumihiro Sato;Yumiko Hironaka;H.Murakami
• 通讯作者: H.Murakami

Tangent circle bundles admit positive open book decompositions along arbitrary links

正切圆束允许沿任意链接进行正开书分解

• 发表时间: 2004
• 期刊: Topology 43
• 作者: T.Kitano;T.Morifuji;M.Takasawa;Akito Futaki;Akihiro Tsuchiya;M.Ishikawa
• 通讯作者: M.Ishikawa

M.Ishikawa: "On positive open book decompositions of 3-manifolds""Intelligence of Low Dimensional Topology", Shodo-Shima. 1-13 (2003)

M.Ishikawa：“关于 3 流形的正开书分解”“低维拓扑智能”，Shodo-Shima。

Positive open book decompositions of Stein fillable 3-manifolds along prescribed links

斯坦因可填充 3 流形沿规定链接的正开书分解

• 发表时间: 2006
• 期刊: Topology 45
• 作者: Ogata;Akira et al.;M.Ishikawa
• 通讯作者: M.Ishikawa