Exceptional Dehn surgery on hyperbolic 3-manifolds

双曲 3 流形的出色 Dehn 手术

• 批准号: 14540082
• 负责人: TERAGAITO Masakazu
• 金额: $ 1.09万
• 依托单位: HIROSHIMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540082/
• 关键词: knot; Dehn surgery; 3-manifold; torus; Klein bottle; crosscap number; 国際情報交換; 国際研究者交流; アメリカ

In this project, we focused on toroidal surgery among exceptional Dehn surgery on hyperbolic 3-manifolds, and obtained several results. It is well known that a toroidal surgery of a hyperbolic knot in the 3-sphere corresponds to either an integer or a half-integer. First, we determined the simplest hyperbolic knot with non-integral toroidal surgery from a view of bridge index. That is, we showed that the (-2,3,7)-pretzel knot is the only 3-bridge knot with non integral toroidal surgery. Moreover, this knot is the only one with such surgery among pretzel knots. Second, we proposed a conjecture that any integral toroidal slope for a hyperbolic knot is bounded by the genus of the knot multiplied by four. In the first year, we solved this conjecture affirmatively for two important classes of knots, genus one knots and alternating knots. In the second year, we tried to prove the full conjecture, but we could not do this. But we have almost done. According to the minimal intersection number between an essential torus and the core of the attached solid torus, we got the expected upper bound, unless that number is not equal to four. Also, a toroidal surgery open creates a Klein bottle simultaneously. We got the best possible upper bound for Klein bottle surgery on all knots by using genera of knots. By using of this argument, we determined the crosscap numbers of torus knots and 2-bridge knots

在这个项目中，我们专注于双曲三维流形上的特殊Dehn手术中的环面手术，并得到了一些结果。众所周知，三维球面上双曲纽结的环面手术对应于整数或半整数。首先，我们从桥指数的角度确定了最简单的非积分环面手术双曲结。也就是说，我们证明了（-2，3，7）-椒盐卷饼结是唯一一种非整体环形手术的3桥结。而且，这个结是椒盐卷饼结中唯一一个有这种手术的结。其次，我们提出了一个猜想，即任何积分的双曲纽结的环面斜率是有界的纽结乘以四的亏格。在第一年，我们解决了这个猜想肯定的两个重要类的纽结，亏格1纽结和交替纽结。在第二年，我们试图证明完整的猜想，但我们做不到这一点。但我们几乎完成了。根据本质环面与其所依附的实环面的核的最小交数，得到了期望的上界，除非该交数不等于4。此外，一个环形手术打开创建一个克莱因瓶同时。利用纽结的类，得到了所有纽结的Klein瓶手术的最佳可能上界。利用这一论点，我们确定了环面纽结和2-桥纽结的交叉数

项目成果

Hiroshi Goda: "Some Topics on Hyperbolic geometry and Heegaard splittings of 3-manifolds"Interdisciplinary Information Sciences. 9・1. 1-9 (2003)

Hiroshi Goda：“关于双曲几何和 3 流形的 Heegaard 分裂的一些话题”跨学科信息科学 9・1（2003）。

Haruko Ishigami: "The simplest hyperbolic knots with non-integral toroidal surgery"Journal of Knot Theory and its Ramifications. 12・8. 1023-1039 (2003)

Haruko Ishigami：“最简单的双曲结与非整体环形手术”结理论及其分支杂志 12・8（2003）。

Masakazu Teragaito: "Toroidal surgeries on hyperbolic knots, II"Asian Journal of Mathematics. 7(1). 139-146 (2003)

Masakazu Teragaito：“双曲线结的环形手术，II”亚洲数学杂志。

Masakazu Teragaito: "Klein bottle surgery and genera of knots"Pacific Journal of Mathematics. (発表予定).

寺外正和：《克莱因瓶术与属结》太平洋数学杂志（待出版）。

Masakazu Teragaito: "Crosscap numbers of torus knots"Topology and its Applications. 138・1-3. 219-238 (2004)

Masakazu Teragaito：“环面结的交叉数”拓扑及其应用138・1-3（2004）。