Study of Knot theory from Geometric Structure

从几何结构研究纽结理论

• 批准号: 17540059
• 负责人: UCHIDA Yoshiaki
• 金额: $ 1.88万
• 依托单位: Yamagata University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540059/
• 关键词: knot; covering space; Seifert fiberd space; link; Riemann surface; Dedekind sum; Legendrian knot; strong trivial; マルコフ同値; 不変量; 結び目理論; 多様体; 被膜空間

1. Uchida studies (1) the three-fold irregular branched covering space of three-dimensional sphere branched over three-bridge knot and three-braid knot And he showed that the covering space is a lens space of type L (n, 1) and L (n, m), respectively. (2) lf a torus knot has a three-fold irregular branched covering space, then its type is T (2x, 3y), where x and y are co-prime integers. Its covering space is a Seifert fibered space of type M (β_1/2x, β_1/x, β_2/Y), where β_1 and β_2 are integers with 2xβ_2+3yβ_1=±1. Moreover for T (2, x) and T (3, x), he proved this theorem by using a knot diagram.2. Ashikaga studies topology and algebraic geometry. (1) He describes the recent development of study of degenerate families of Riemann surfaces. This field is located at the boundary area where topology, algebraic geometry complex analysis and others get complicated each other. (2) He propose a certain formula of Dedekind sum by using two mutually distinct methods. The one is an elementary number theoretic method and the other is a geometric method. Moreover he re-prove the reciprocity law by the formula.3. Torisu gives (1) some examples of links having 2-adjacency relation and he also reports a recent study of 2-bridge links with 2-adjacency relation. (2) He gives a necessary condition for a two-bridge knot or link S (p, q) to be 2-adjacent to another two-bridge knot or link S (r, s). In particular he shows that if the trivial knot or link is 2-adjacent to S (p, q), then S (p, q) is trivial, that if S (p, q) is 2-adjacent to its mirror image, then S (p, q) is amphicheiral, and that for a prime integer p, if S (p, q) is 2-adjacent to S (r, s), then S (p, q)=S (r, s) or S (r, s)=S (1, 0). (3) He present a off Lain family of strongly 1-trivial Montesinos knots, and show that if a well known conjecture on Seifert surgery is valid, then the family contains all strongly 1-trivial Montesinos knots.

1. Uchida研究了（1）三维球面在三桥纽结和三辫纽结上分枝的三重不规则分枝覆盖空间，并证明了覆盖空间是L（n，1）和L（n，m）型的透镜空间。(2)若环面纽结有一个三重不规则分支覆盖空间，则它的类型是T（2x，3 y），其中x和y是互质整数.它的覆盖空间是M（β_1/2x，β_1/x，β_2/Y）型Seifert纤维空间，其中β_1和β_2是整数，2xβ_2+ 3 y β_1=±1。此外，对于T（2，x）和T（3，x），他证明了这一定理，通过使用一个结。足利研究拓扑和代数几何。(1)他介绍了最近的发展研究退化家庭的黎曼曲面。该领域处于拓扑学、代数几何、复分析等学科相互复杂化的边界区域。(2)他用两种截然不同的方法给出了Dedekind和的一个公式。一种是初等数论方法，另一种是几何方法。并利用公式重新证明了互反律. Torisu给出了（1）具有2-邻接关系的链路的一些例子，并报告了最近对具有2-邻接关系的2-桥链路的研究。(2)他给出了一个两桥纽结或链环S（p，q）与另一个两桥纽结或链环S（r，s）2-相邻的必要条件。特别地，他证明了如果平凡纽结或环与S（p，q）2-相邻，则S（p，q）是平凡的，如果S（p，q）与其镜像2-相邻，则S（p，q）是双螺旋的，并且对于素数p，如果S（p，q）与S（r，s）2-相邻，则S（p，q）=S（r，s）或S（r，s）=S（1，0）。(3)他提出了一个关闭Lain家庭的强烈1-平凡Montesinos结，并表明，如果一个著名的猜想塞弗特手术是有效的，那么家庭包含所有强烈1-平凡Montesinos结。

项目成果

Branched covering space branched along braid

沿辫子分支的分支覆盖空间

• 发表时间: 2007
• 作者: Toshiaki;ADACHI;鳥巣伊知郎;足立 俊明;鳥巣伊知郎;Toshiaki ADACHI;Yoshiaki Uchida
• 通讯作者: Yoshiaki Uchida

Local signatute of fibered complex surfaces via monodromy and stable reduction

通过单向性和稳定还原的纤维复杂表面的局部签名

• 发表时间: 2006
• 作者: Toshiaki;ADACHI;足利 正
• 通讯作者: 足利 正

符号数・モノドロミー・Dedekind

代码数、单一性、戴德金

• 发表时间: 2006
• 作者: Toshiaki;ADACHI;足利 正;足利 正;足利 正
• 通讯作者: 足利 正

A note on strongly n-trivial links

关于强n-平凡链接的注释

• 发表时间: 2005
• 期刊: Journal of Knot Theory and Its Ramifications 14
• 作者: Ichiro;Torisu
• 通讯作者: Torisu

Deneration of non-Galois covering in projective space bundle

射影空间丛中非伽罗瓦覆盖的简化

• 发表时间: 2007
• 作者: Toshiaki;ADACHI;Yoshiaki Uchida;足立 俊明;Yoshiaki Uchida;Toshiaki ADACHI;Yoshiaki Uchida;Tadashi Ashikaga
• 通讯作者: Tadashi Ashikaga