Seifert fibred manifolds and Dehn surgery
Seifert fibred manifolds and Dehn surgery
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Seifert 纤维歧管和 Dehn 手术
DOI：
10.1016/0040-9383(96)00009-2
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
Kimihiko Motegixf
中科院分区：
文献类型：
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作者：
Katura Miyazak;Kimihiko Motegixf
什么时候可以得到一个塞弗特流形的Dehn手术结在3球S3？在[l]莫泽澄清说，德恩手术上的非环面结不能产生塞弗特双曲歧管，特别是，不能产生一个透镜空间。然而，某些电缆结，连接和两个环面结和某些双曲结变成反例莫泽猜想[2-S]。Bleiler和Litherland [9]和Wu [lo]完全确定了卫星结手术何时产生透镜间隙。另一方面，循环外科定理[11]和最近的有限外科定理[12，7，13]对产生具有循环或有限基本群的流形的外科斜率给出了一些限制。(The球面空间形式猜想指出每个具有有限基本群的3-流形是塞弗特流形。例如，已知非环面结上的循环手术斜率为整数，而非环面、非缆索结上的有限手术斜率具有至多2的乘数。我们注意到，所有已知的有限手术斜率的例子都是整数（见[12，7，13]）。在本文中，我们将描述那些手术的卫星纽结，使塞弗特复流形（定理1.2和1.4）。作为推论，我们得到了以下关于手术斜率的结果。推论1.1。设K是一个卫星结，它不恰好被连接一次。如果K上的非平凡手术产生一个塞弗特jibred流形，则手术斜率是积分的。此外，最多有四个这样的手术;如果有四个，那么它们是两对连续的整数。
When can one obtain a Seifert fibred manifold by Dehn surgery on a knot in the 3-sphere S3? In [l] Moser conjectured that Dehn surgery on a non-torus knot could not produce a Seifert fibred manifold, and in particular, could not produce a lens space. However certain cable knots, connected sums of two torus knots and certain hyperbolic knots turn out to be counterexamples to Moser’s conjecture [2-S]. Bleiler and Litherland [9] and Wu [lo] completely determined when surgeries on satellite knots produce lens spaces. On the other hand, the cyclic surgery theorem [11] and recent finite surgery theorems [12, 7, 13] gave some restrictions on the surgery slopes which produce manifolds with cyclic or finite fundamental groups.(The spherical space form conjecture states that every 3-manifold with finite fundamental group is Seifert fibred.) For instance, the cyclic surgery slopes on non-torus knots are known to be integers, and finite surgery slopes on non-torus, non-cable knots have denominators at most 2. We note that all the known examples of finite surgery slopes on such knots are integers (see [12, 7, 13]). In this paper we shall describe those surgeries on satellite knots which give Seifert fibred manifolds (Theorems 1.2 and 1.4). As a corollary, we obtain the following result on the surgery slopes.COROLLARY 1.1. Let K be a satellite knot which is not cabled exactly once. If a non-trivial surgery on K yields a Seifert jibred manifold, then the surgery slope is integral. Moreover, there are at most four such surgeries; if there are four, then they are two pairs of successive integers.