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Wen does Dehn surgery yield a Seifert fibered manifold?

Dehn 手术是否能产生 Seifert 纤维歧管？

基本信息

批准号：
10640091
负责人：
MIYAZAKI Katura
金额：
$ 0.96万
依托单位：
Tokyo Denki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640091/
关键词：
knot Dehn surgery 3-manifold Dehn surgery Seifert レンズ空間 Dehn Surgery

项目摘要

W. Thurston's Hyperbolic Surgery Theorem shows that all but a finite number of Dehn surgeries on a hyperbolic knot yield hyperbolic manifolds. It is thus important to study those exceptional surgeries yielding non-hyperbolic manifolds. By the join work with Kimihiko Motegi (Nihon Univ.) the author studied when surgey yields a Seifer fibered manifold or a manifold containing essential tours, typical examples of non-hyperbolic manifolds, and obtained the following results.1. It had been already known that no Dehn surgery on a knot with period greater than 2 does not yield a Seifert fibered manifold. Regarding a knot with period 2, we showed that if such a knot yeilds a Seifert fibered manifold, then the quotient of the knot by its periodic map is a torus knot.2. When does Dehn surgery on a hyperbolic, periodic knot yield a manifold containing an essential torus? We showed that a knot with period greater than 2 admits such a surgery if and only if its genus is 1, its period is 3, and the surgery coefficient is 0; if a knot with period 2 admits such a surgery, then the surgery coefficient is integer.3. What Seifert fibered manifolds are obtined by Dehn surgery? We studied this problem in terms of indices of exceptinal fibers of Seifert fibered manifolds, and showed that : for arbitary integers p, q, r with some pair coprime, a surgery on some hyperbolic knot yields a Seifert fibered manifold with 3 exceptional fibers of indices p, q, r.

W. Thurston的双曲外科手术定理表明，除了有限数量的Dehn手术双曲结产生双曲流形。因此，重要的是研究这些特殊的手术产生非双曲流形。通过与Kimihiko莫泰吉（日本大学）的合作，摘要研究了非双曲流形的典型例子--我们已经知道，对于周期大于2的纽结，没有一个Dehn外科手术不会产生塞弗特纤维流形。对于周期为2的纽结，我们证明了如果这样的纽结是一个Seifert纤维流形，那么这个纽结与它的周期映射的商是一个环面纽结.什么时候德恩手术双曲，周期结产生一个流形包含一个本质环面？我们证明了一个周期大于2的纽结允许这样的手术当且仅当它的亏格是1，它的周期是3，手术系数是0;如果一个周期为2的纽结允许这样的手术，那么手术系数是整数。什么塞弗特纤维流形是由德恩手术阻塞？本文利用Seifert纤维流形的例外纤维指标研究了这一问题，证明了：对于任意整数p，q，r，若它们具有某种互质对，则对某个双曲纽结作一次手术，得到一个Seifert纤维流形，它具有3个指数为p，q，r的例外纤维.