A study on knots and transverse knots using braid theory and Floer theory

利用辫子理论和弗洛尔理论研究结子和横向结子

• 批准号: 23740053
• 负责人: MATSUDA Hiroshi
• 金额: $ 2.66万
• 依托单位: Yamagata University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Young Scientists (B)
• 财政年份: 2011
• 资助国家: 日本
• 起止时间: 2011 至 2013
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-23740053/
• 关键词: 横断的結び目; 組み紐; フレアーホモロジー群; 2橋指数

I constructed an example of a pair of closed 4-braids with the following properties; (1) they are related by a Hopf-flype, (2) they are distinct as transverse knots, (3) they have the same self-linking number. I also constructed a similar example of a pair of a closed 3-braid and a closed 7-braid. I determined 2-bridge numbers of torus knots of type (p, q), where p and q are integers. I also determined 2-bridge numbers of knots that had alternating diagrams of closed braids. An invariant of a mapping class group of a surface (fixing its boundary) is defined in bordered Floer theory. When a surface has one boundary component and is of genus 2, I calculated this invariant for elements in Torelli group. Torelli group is a subgroup of a mapping class group of a surface that acts trivially on its first homology group.

我构造了一个具有以下属性的一对闭合 4 辫子的示例； (1) 它们通过 Hopf-flype 相关，(2) 它们作为横向结是不同的，(3) 它们具有相同的自联数。我还构建了一个类似的示例，由一对封闭的 3 辫子和一个封闭的 7 辫子组成。我确定了 (p, q) 类型的环面结的 2 桥数，其中 p 和 q 是整数。我还确定了具有闭合辫子交替图的 2 桥结数量。表面的映射类组的不变量（固定其边界）在有界弗洛尔理论中定义。当表面具有一个边界分量且属于 2 时，我计算了 Torelli 群中元素的这一不变量。 Torelli 群是曲面的映射类群的子群，其作用于其第一同调群。

项目成果

2-bridge numbers of torus knots

2 桥环面结数

• 发表时间: 2011
• 作者: Tanaka;Toshifumi;渡邉忠之;Hiroshi Matsuzoe;Tadayuki Watanabe;田中利史;田中利史;田中利史;松添博;田中利史;Hiroshi Matsuzoe;Hiroshi Matsuzoe;松添博;松添博;Hiroshi Matsuzoe;松添博;松添博;松添博，小原敦美，甘利俊一;Hiroshi Matsuzoe;Matsuda Hiroshi;Shin-ichi Ohta and Karl-Theodor Sturm;Shin-ichi Ohta;松田 浩
• 通讯作者: 松田 浩

A construction of transversely non-simple knot types

横向非简单结类型的构造

• DOI: 10.1142/s0218216512501088
• 发表时间: 2012
• 期刊: Journal of Knot Theory and Its Ramifications
• 影响因子: 0.5
• 作者: Tanaka;Toshifumi;渡邉忠之;Hiroshi Matsuzoe;Tadayuki Watanabe;田中利史;田中利史;田中利史;松添博;田中利史;Hiroshi Matsuzoe;Hiroshi Matsuzoe;松添博;松添博;Hiroshi Matsuzoe;松添博;松添博;松添博，小原敦美，甘利俊一;Hiroshi Matsuzoe;Matsuda Hiroshi
• 通讯作者: Matsuda Hiroshi