权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Studies of local moves and finite type invariants in knot theory

结理论中的局部移动和有限类型不变量的研究

基本信息

批准号：
16540083
负责人：
OHYAMA Yoshiyuki
金额：
$ 2.3万
依托单位：
Tokyo Woman's Christian University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540083/
关键词：
knot local move C_n-move finite type invariant Vassiliev invariant Cn-move

项目摘要

In 1990, Vassiliev invariants for knots were defined. They order all knot invariants and they are also called finite type invariants. The first aim of this research is to study the finite type invariants by combinatorial methods. The start point is the following result proved by Goussarov and Habiro independently ; two knots have the same Vassiliev invariants of order less than n if and only if they can be transformed into each other by a finite sequence of C_n-moves.As a joint work with Prof. Yasutaka Nakanishi, we have that for any given pair of a natural number n and a knot K, there exist infinitely many knots whose Vassiliev invariants of order less than or equal to n and Conway polynomials coincide with those of K. In the finite type invariants, the coefficients of the Conway polynomial are not powerful to classify the knots.C_n-moves may change the Vassiliev invariants of order n. As a joint work with Harumi Yamada, we showed that a standard C_n-move can change the coefficient of z^n by 0 or ±2. It is possible to say that we nearly cleared the relation between C_n-moves and the coefficients of the Conway polynomial.We can define the simplicial complex for the set of knots by using C_n-moves and it is called the C_n-Gordian complex of knots. Let K be a knot and K^<C_n> the set of knots obtained from K by a single C_n-move. We showed that there are knots K_1 and K_2 such that they have the same Conway polynomial and the sets of Conway polynomials of K_1^<C_n> and those of K_2^<C_n> do not coincide, as a joint work with Prof. Yasutaka Nakanishi. This theorem are related to the C_n-Gordian complex and the Conway polynomial and we consider an expansion of the result.

在1990年，Vassiliev不变量的纽结被定义。它们对所有纽结不变量排序，也称为有限型不变量。本研究的第一个目的是用组合方法研究有限型不变量。出发点是由Goussarov和Habiro独立证明的下列结果;两个纽结具有相同的小于n阶的Vassiliev不变量当且仅当它们可以通过有限的C_n-移动序列相互转换.作为与Yasutaka Nakanishi教授的联合工作，我们有，对于任何给定的自然数n和纽结K的对，存在无穷多个节点，其Vassiliev不变量的阶小于或等于n，Conway多项式与K的一致。在有限型不变量中，Conway多项式的系数对纽结的分类作用不强，C_n移动可以改变n阶Vassiliev不变量。作为与Harumi Yamada的联合工作，我们证明了标准C_n移动可以使z^n的系数改变0或±2。可以说，我们几乎弄清了C_n-移动与Conway多项式系数之间的关系，可以用C_n-移动定义纽结集的单纯复形，称之为纽结的C_n-Gordian复形。设K是一个纽结，K^是<C_n>由K通过一次C_n移动得到的纽结的集合。本文与Yasutaka Nakanishi教授共同证明了存在纽结K_1和K_2使得它们具有相同的Conway多项式，并且K_1^<C_n>和K_2^的Conway多项式的集合<C_n>不重合。这个定理与C_n-Gordian复形和Conway多项式有关，我们考虑结果的一个推广。