STUDY OF TOPOLOGICAL INVARIANTS FOR KNOTS AND THEIR APPLICATIONS

结拓扑不变量的研究及其应用

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340027/
• 关键词: knot; link; Cn-move; finite type invariant; HOMFLY polynomial; Links-Gould polynomial; Kauffman polynomial; virtual knot; スケイン関係式; 2変数アレキサンダー多項式; 2本組み紐仮想結び目; 仮想結び目群; リボントーラス結び目; 2次元球面結び目; 交換子部分群; リンクス-グールド不変量; 2本橋絡み目; 両手性; 係数多項式; デルタ変形

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.

主要研究结果如下：1.研究了增量运动对杆的一阶HOMFLY系数多项式的影响。然后我们将其推广到CN运动。2.我们研究了作为量子不变量的Links-Gould(LG)多项式。利用石井发现的斜切关系，我们发现了一种构造共享同一LG多项式的纽结或链环的方法。然后给出任意多个具有相同LG多项式的2-桥结和环。这些2-桥结和环还共享相同的HOMFLY、Kauffman和2-变量Alexander(对于环)多项式。3.给出了涉及链数的Kauffman环多项式的前4个系数多项式的公式以及单分量和2-分量子链的Kauffman多项式的系数多项式的公式。4.给出了2-辫子虚结或环的群的表示，我们考虑了某些特殊的2-辫子虚结子族的群。已知虚拟结组的集合与带状环结组的集合相同。利用我们的例子讨论了虚结点群与带状2-球结点群、2-球结点群、环面结点群和3-球结点群等其他结点群之间的关系。5.给出了2个索链的HOMFLYPT多项式的绞合关系。利用这一点，我们已经证明了在第2部分中提到的2-桥结或环的集合也共享相同的2-CASE HOMFLY多项式。

项目成果

Delta move and polynomial invariants of links

链接的 Delta 移动和多项式不变量

• 发表时间: 2005
• 期刊: Topology and its Applications 146/147
• 作者: Kanenobu;Taizo;Nikkuni;Ryo
• 通讯作者: Ryo

Enveloping monoidal quandles,

包络幺半群，

• 发表时间: 2005
• 期刊: Top. appl 146-147
• 作者: S.Kamada;Y.Matsumoto
• 通讯作者: Y.Matsumoto

Kawauchi, Akio: "Topological imitation of a colored link with the same Dehn surgery manifold"Top.Appl.Proceedings of the Second Joint Japan-Mexico Meeting in Topology (Matsue, 2002). (出版予定).

Kawauchi, Akio：“具有相同 Dehn 手术流形的彩色连接的拓扑模仿”Top.Appl.第二届日本-墨西哥拓扑学联合会议论文集（松江，2002 年）（待出版）。

Cn-moves and the HOMFLY polynomials of links

Cn 移动和链接的 HOMFLY 多项式

• 发表时间: 2004
• 期刊: Bol. Soc. Mat. Mexicana (3), Special Issue 10
• 作者: Kanenobu;Taizo
• 通讯作者: Taizo

A relation between the Links-Gould invariant and the Kauffman polynomial

Links-Gould 不变量与 Kauffman 多项式之间的关系

• 发表时间: 2007
• 期刊: Topology Appl. 154
• 作者: 坂口 茂;Rolando Magnanini;Shigeru Sakaguchi;坂口 茂;Masanao Ozawa;小澤正直;小澤正直;小澤正直;小澤正直;小澤正直;小澤正直;小澤正直;小澤正直;Seiichi Kamada;Takao Matumoto;Seiichi Kamada;Seiichi Kamada;Takao Matumoto;Seiichi Kamada;Seiichi Kamada;Takao Matumoto;Seiichi Kamada;松本 堯生;松本 堯生;A.Kodama;S.Kato;J.Noguchi;Y.Matsumoto;A.Kawauchi;Y.Komori;T.Kanenobu
• 通讯作者: T.Kanenobu