Applications of Seiberg-Witten theory to knot theory

Seiberg-Witten 理论在纽结理论中的应用

基本信息

批准号：
09640135
负责人：
UENO Kimio
金额：
$ 1.79万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640135/
关键词：
knot link three-manifold quantum invariant Vassiliev invariant finite type invariant Casson-Walker invariant knot cobordism 結び目絡み目 Vassiliew不変量量子不変量 Reidemeister torsion Seiberg-Witten不変量 Seiberg-Witten理論結び目理論 4次元多様体 3

项目摘要

I will describe my results paper by paper.In the paper (i) I gave a new elementary, combinatorial definition of the HOMFLY polynomial. In (ii) I looked at the multivariable Alexander polynomial of links from the view point of Vassiliev invariants and define a recursive definition of weight systems derived from it. In (iii) we studied a knot cobordism invariant, 4-dimensional clasp number, introduced by T.Shibuya. He proved that it is greater than or equal to the 4-dimensional genus and raised a problem whether there are knots which do not satisfy the equality. We gave such an example in this paper. In (iv) I studied the quantum SU(2)-invariant of 3-manifolds associated with the gammath root of unity. If gamma is even, it is defined for a class of the first cohomology group modulo 2. In this paper I calculated it for rational homology three-spheres and for the trivial cohomology class and showed that it is a cyclotomic integer and moreover it determines the Casson-Walker invariant. In (v) we introduced a filtration to the vector space spanned by all the Seifert matrices corresponding to the filtration to the vector space spanned by all the knot, which was introduced by V.Vassiliev. Moreover we clarified its relation to the Alexander polynomial. In (vi) I gave an example of hyperbolic three-manifold with trivial finite type invariants (introduced by T.Ohtsuki) up to arbitrarily given degree. In (vii) I followed (iv) and obtained a similar result in the case of non-trivial cohomology classes. Unfortunately I only showed that the invariant is a cyclotomic integer and a relation to the Casson-Walker invariant is now being investigated.

我将用论文描述我的结果。在（ii）中，我从Vassiliev不变性的角度观察了链接的多变量多项式，并定义了从中得出的权重系统的递归定义。在（iii）中，我们研究了T.Shibuya引入的一个结的核心不变的4维扣。他证明了它大于或等于4维属，并提出了一个问题，无论是否有不满足平等的结。我们在本文中给出了这样的例子。在（iv）中，我研究了与统一根部相关的3个manifolds的量子su（2）。如果伽玛甚至是伽玛，则定义为第一类共同体学组模拟2。在本文中，我将其计算为理性同源性三个小时和琐碎的共同体学类别，并表明它是一个环形整数，并且更多的是Casson-Walker Warker Guntariant。在（v）中，我们引入了与所有结的过滤相对应的对相对的滤波器跨越的载体矩阵所跨越的矢量空间的过滤，该矩阵与所有结所跨越的矢量空间相对应，该矩阵由V.Vassiliev引入。此外，我们阐明了它与亚历山大多项式的关系。在（vi）中，我给出了一个柔软的三序示例，其中有限的有限型不变式（由T.Ohtsuki引入）任意地给出了学位。在（vii）中，我遵循（iv），并在非平凡的共同体学类别中获得了类似的结果。不幸的是，我只表明不变的是一个环体整数，现在正在研究与卡森·沃克不变的关系。