STUDY OF TOPOLOGICAL INVARIANTS FOR KNOTS AND THEIR APPLICATIONS

结拓扑不变量的研究及其应用

基本信息

批准号：
14340027
负责人：
KANENOBU Taizo
金额：
$ 5.18万
依托单位：
Osaka City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

There are five main results.1.We investigated how a delta move influences the first HOMFLY coefficient polynomials of a link. Then we generalized this to a Cn-move.2.We studied the Links-Gould (LG) polynomial, which is a quantum invariant. Using a skein relation discovered by Ishii, we found a way to construct knots or links sharing the same LG polynomial. Then we gave arbitrarily many 2-bridge knots and links with the same LG polynomial. These 2-bridge knots and links also share the same HOMFLY, Kauffman, and 2-variable Alexander (in case of links) polynomials.3.We give formulas for the first four coefficient polynomials of the Kauffman's link polynomial involving linking numbers and the coefficient polynomials of the Kauffman polynomials of the one- and two-component sublinks.4.Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of certain special families of 2-braid virtual knots. It is known that the collection of the virtual knot groups is the same as that of the ribbon torus-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon 2-sphere-knot groups, 2-sphere-knot groups, torus-knot groups, and 3-sphere-knot groups.5.We give a skein relation for the HOMFLYPT polynomials of 2 cable links. Using this, we have shown that the collection of 2-bridge knots or links mentioned in part 2 also share the same 2-cable HOMFLY polynomial.

主要结果有五个：1.我们研究了增量移动如何影响链接的第一个 HOMFLY 系数多项式。然后我们将其推广到 Cn-move。2.我们研究了 Links-Gould (LG) 多项式，它是一个量子不变量。利用 Ishii 发现的绞纱关系，我们找到了一种构造共享相同 LG 多项式的结或链接的方法。然后我们给出任意多个具有相同 LG 多项式的 2 桥结和链接。这些 2 桥结和链接也共享相同的 HOMFLY、Kauffman 和 2 变量 Alexander（在链接的情况下）多项式。 3.我们给出涉及链接数和系数的 Kauffman 链接多项式的前四个系数多项式的公式一分量和二分量子链接的考夫曼多项式的多项式。4.给出 2 编织虚拟结的群的演示或链接中，我们考虑某些特殊族的2-辫虚拟结的群。已知虚拟结组的集合与丝带环面结组的集合相同。使用我们的示例，我们讨论虚拟结组和其他结组（例如带状 2-球体结组、2-球体结组、环面结组和 3-球体结组）之间的关系。5.我们给出2 个电缆链路的 HOMFLYPT 多项式的绞纱关系。利用这一点，我们证明了第 2 部分中提到的 2 桥结或链接的集合也共享相同的 2 电缆 HOMFLY 多项式。