Cyclic surgery on knots
Cyclic surgery on knots
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结节循环手术
DOI：
10.1090/s0002-9939-1989-0984820-6
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发表时间：
1989
期刊：
影响因子：
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通讯作者：
Shi Cheng Wang
中科院分区：
文献类型：
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作者：
Shi Cheng Wang
我们得到一个必要条件，在此条件下，一个非简单的纽结（即卫星纽结）允许一个非平凡的手术产生一个透镜空间。有趣的推论是：（1）如果一个非简单纽结的非平凡手术可以得到一个透镜空间，那么它的基本群的阶数不小于23;（2）任何非简单纽结最多允许一个产生一个透镜空间的非平凡手术。一些拓扑学家对在纽结补集上做手术以获得透镜空间的问题感兴趣。本说明专门就这一问题提供更多的资料。我们的主要结果如下：定理1。如果一个透镜空间可以从一个非简单纽结（即卫星纽结）J（K）上的一个非平凡手术中获得，则K是一个环面纽结，J是K的一个管状邻域中的一个闭辫。定理1有两个有趣的推论。D. Gabai证明，人们不能得到一个透镜空间的基本组的顺序1（即S3）从一个非平凡的外科手术的非简单的结（事实上非简单的结有属性P）。J. Bailey和D. Rolfsen [BR]（也是R. Fintushel和R. Stern [FS]和C. Gordon [G]）证明了通过对非简单纽结的非平凡手术可以得到具有23阶基本群的透镜空间。第一个推论说23是最小的可能。推论1.如果一个透镜空间可以从一个非简单纽结（即卫星纽结）的非平凡手术中获得，那么它的基本群的阶数不小于23。M.卡勒角M. Gordon、J. Luecke和P. Shalen [CGLS]证明，在非环面结上最多有两次非平凡手术，可产生透镜间隙。下面的推论2说，一个允许两个非平凡手术产生透镜空间的非环面结一定是双曲结。（贝尔格发现了无穷多个这样的非环面结。）编辑于1988年2月4日收到，并于8月17日修订。1988. 1980 Alathernatics S/iHject分类（I1985修订版）。第57话第10话(? 1989年美国数学学会0002-9939/89 $1.00 + $.25每页
We get a necessary condition under which a nonsimple knot (i.e. a satellite knot) admits a nontrivial surgery producing a lens space. Interesting corollaries are: ( 1) if a lens space can be obtained from a nontrivial surgery on a nonsimple knot, then the order of its fundamental group is not smaller than 23; (2) any nonsimple knot admits at most one nontrivial surgery which produces a lens space. Some topologists are interested in the problem of doing surgery on a knot complement to get a lens space. This note is devoted to providing more information on this problem. Our main result is the following: Theorem 1. If a lens space can be obtainedfrom a nontrivial surgery on a nonsimple knot (i.e. a satellite knot) J(K), then K is a torus knot and J is a closed braid in a tubular neighborhood of K. There are two interesting corollaries of Theorem 1. D. Gabai proved that one cannot get a lens space with fundamental group of order 1 (i.e. S3) from a nontrivial surgery on a nonsimple knot (in fact nonsimple knots have property P). J. Bailey and D. Rolfsen [BR] (also R. Fintushel and R. Stern [FS] and C. Gordon [G]) proved that some lens space with fundamental group of order 23 can be obtained from a nontrivial surgery on a nonsimple knot. The first corollary says that 23 is the smallest possible. Corollary 1. If a lens space can be obtained from a nontrivial surgery on a nonsimple knot (i.e. a satellite knot), then the order of its fundamental group is not smaller than 23. M. Culler, C. M. Gordon, J. Luecke, and P. Shalen [CGLS] proved that there are at most two nontrivial surgeries on a nontorus knot which produce lens spaces. Corollary 2 below says that a nontorus knot which admits two nontrivial surgeries producing lens spaces must be a hyperbolic knot. (Berge found infinitely many such nontorus knots.) Received by the editors February 4, 1988, and in revised form, August 17. 1988. 1980 Alathernatics S/ihject Classification (I1985 Rewision). Primary 57N 10. (? 1989 American Mathematical Society 0002-9939/89 $ 1.00 + $.25 per page