Exceptional Dehn surgery on hyperbolic 3-manifolds

双曲 3 流形的出色 Dehn 手术

基本信息

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

项目摘要

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，我们专注于在双曲线3型脉冲上的特殊Dehn手术中的环形手术，并获得了几个结果。众所周知，三个球体中双曲线结的环形手术对应于整数或半数。首先，我们从桥梁指数的视图中确定了最简单的双曲线结。也就是说，我们表明（-2,3,7） - 旋转结是唯一进行非整体环形手术的3桥结。此外，这个结是椒盐脆饼结中唯一进行此类手术的结。其次，我们提出了一个猜想，即双曲线结的任何整体环形斜率都被结的属构成，乘以四个。在第一年，我们针对两个重要的结，一个结和交替结属地肯定了这一猜想。在第二年，我们试图证明全部猜想，但我们无法做到这一点。但是我们几乎完成了。根据必需的圆环与附着的固体圆环的核心之间的最小交点，我们得到了预期的上限，除非该数字不等于四个。此外，环形手术开放也会同时创建一个klein瓶。通过使用结属，我们在所有结上都可以在所有结上进行klein瓶手术的最佳上限。通过使用此参数，我们确定了圆环结和2桥结的交叉盖数