Invariants for links in thickened surfaces and its applications

加厚曲面中的连杆不变量及其应用

• 批准号: 11640059
• 负责人: KANETO Takeshi
• 金额: $ 2.24万
• 依托单位: University of Tsukuba
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640059/
• 关键词: thickened surface; knot; link; alternating link; Tait conjecture; bracket; link invariant; minimal crossing number

Tait conjectures on crossing numbers of alternating links in the 3-dimensional space means the following two conjectures :TC1) If two connected, reduced and alternating link diagrams represent equivalent (ie, ambient isotopic) links, their crossing numbers will be the same.TC2) The crossing number of a connected, reduced and alternating link diagram D will be minimal among all crossing numbers of link diagrams which represent links equivalent to that D does.These conjectures had been unsolved for more than 100 years untill they were positively solved in 1987 after Jones polynomial appeared. In 1996, N.Kamada made a pioneer-work to establish anologous results for alternating links in thickened surfaces. She showed an anologous result to TC1) under a certain assumption and conjectured that the assumption can be removed in her paper. In this research project, we solved her conjecture positively . And, by applying the idea of the proof of her conjecture, we established an anologous result … More to TC2), which is a final goal to seeking anologous results to Tait conjectures in knot-link theory in thickened surfaces because TC2) implies TC1. Our these results was showed for more general both surfaces and link diagrames than those in her paper. In the proofs of our these results, we use the following two fundamental lemmta :ENGULFING LEMMA : For two link diagrams on a surface, one of their regular neiborhoods in the the surface can contain the other by isotopic deformation in the surface under a certain condition,DUAL STATE LEMMA (thickened surface version) : For a pair of dual states of a link diagram D on a surface, the sum of numbers of connected components of two (no crossing) diagrams obtained from D by resolving all crossings according to each state does not exceed to the number of boundary components of the regular neighhood of D in the surface under a certain condition.These lemmata, themselves, are impotant results in our resaerch. Each invesigator got intereting own results as the fundamental research related to this project. Less

三维空间中交错链环交叉数的Tait定理是指下列两个定理：TC 1）如果两个连通的、约化的、交错的链环图表示等价的（即周围同位素的）链环，则它们的交叉数相同; TC 2）在所有表示等价链环的链环图中，连通的、约化的、交错的链环图D的交叉数是最小的。1996年，N.Kamada进行了开创性的工作，建立了加厚表面中交替链接的结果。她在一定的假设条件下给出了一个与TC 1）一致的结果，并证明了在她的论文中可以去掉这个假设。在这个研究项目中，我们积极地解决了她的猜想。并且，通过应用她的猜想的证明的思想，我们建立了一个证明性的结果 ...更多信息 到TC 2），因为TC 2）蕴涵TC 1，所以这是在加厚曲面中寻求与纽结-链环理论中Tait定理一致的结果的最终目标。我们的这些结果比她的论文中的结果更一般的曲面和链接图。在证明这些结果时，我们使用了以下两个基本引理：Engulfing引理：对于曲面上的两个链图，在一定条件下，它们在曲面上的正则邻域之一可以通过曲面的同位变形包含另一个，对偶态引理（加厚表面版本）：对于曲面上的链路图D的一对对偶状态，两个连通分支的个数之和（禁止穿越）通过根据每个状态分解所有交叉从D获得的图不超过在一定条件下D在表面中的正则邻域的边界分量的数目。这些引理本身就是我们研究中的重要结果。作为与本课题相关的基础研究，各研究者都取得了自己感兴趣的成果。少

项目成果

O.Kerner and K.Yamagata: "Auslander-Reiten components containg cores"Representation Theory of Algebras. (to appear).

O.Kerner 和 K.Yamagata：“包含核心的 Auslander-Reiten 组件”代数表示论。

Hisao Kato K.Kawamura et al.: "Measures and topological dynamics on Menger manifolds"Topology and its Appl.. 103. 249-282 (2000)

Hisao Kato K.Kawamura 等：“Menger 流形上的测量和拓扑动力学”拓扑及其应用.. 103. 249-282 (2000)

Takao Hoshina with K.Yamazaki: "Weak C-embedding, weak P-embedding and product spaces"Topology and its Appl.. (To appear).

Takao Hoshina 与 K.Yamazaki：“弱 C 嵌入、弱 P 嵌入和乘积空间”拓扑及其应用（待发表）。

T.Banakh with K.Kawamura and K.Sakai: "Direct limits of the Banach-Mazur compacta"Bull.London Math.Soc.. 32. 709-717 (2000)

T.Banakh 与 K.Kawamura 和 K.Sakai：“Banach-Mazur 契约的直接限制”Bull.London Math.Soc.. 32. 709-717 (2000)

Takao Hoshina (with K.Yamazaki): "Weak C-embedding, weak P-embedding and product spaces"Topology and its Appl.. To appear.

Takao Hoshina（与 K.Yamazaki）：“弱 C-嵌入、弱 P-嵌入和乘积空间”拓扑及其应用.. 出现。