Taut foliations of torus knot complements
Taut foliations of torus knot complements
复制标题
圆环结的紧致叶状补充
DOI：
10.1016/j.aim.2004.10.016
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发表时间：
2007
影响因子：
--
通讯作者：
Yasuharu Nakae
中科院分区：
文献类型：
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作者：
Yasuharu Nakae
我们证明了对于任何环面纽结K（r，s），|R|>s> 0，则存在K（r，s）的补的一族拉紧叶理，当r> 0时，它实现（−∞，1）中的所有边界斜率，或当r< 0时，它实现（−1，∞）中的所有边界斜率。这个定理是通过构造Roberts论文（5）中使用的分支曲面和叠层来证明的。将这种构造应用于纤维纽结K，我们还证明了存在K的索纽结K的补的一族拉紧叶理，它实现了（−∞，1）或（−1，∞）中的所有边界斜率。进一步地，我们将Roberts定理部分地推广到链的情形.本文讨论了环面纽结的补图的张紧叶理。一个3-流形的紧叶理是一个余维为1的叶理，使得有一个圆与每个叶横切。关于3-流形的叶理有很多研究，其中很多研究表明，叶理的结构很好地反映了流形的拓扑结构。Novikov（3）证明了：如果一个非S2 × S1的3-流形有一个不含Reeb分支的叶理，则它的基本群是无限的，第二同伦群π2是平凡的，它的叶都是π1-内射的. Rosenberg（8）证明了如果一个3-流形有一个没有Reeb分支的叶理，那么这个流形是不可约的。将Novikov和Rosenberg的定理与帕尔梅拉（4）的定理相结合，我们可以看到，如果一个3-流形具有一个不含Reeb分支的叶，则它的泛覆盖同胚于R3。因此，“Reebless”叶理的存在性在三维流形的研究中起着重要的作用。事实上，一个Reeb分量没有与所有叶子相交的横圆，因此一个绷紧的叶理没有Reeb分量。因此，就拓扑性质而言，一个绷紧的叶理接管了“无Reebless”叶理的果实。
We show that for any torus knot K(r, s), |r| >s> 0, there is a family of taut foliations of the complement of K(r, s), which realizes all boundary slopes in (−∞, 1) when r> 0, or (−1, ∞) when r< 0. This theorem is proved by a construction of branched surfaces and laminations which are used in the Roberts paper (5). Applying this construction to a fibered knot K, we also show that there exists a family of taut foliations of the complement of the cable knot K of K which realizes all boundary slopes in (−∞, 1) or (−1, ∞). Further, we partially extend the theorem of Roberts to a link case. In this paper, we discuss taut foliations of the complement of a torus knot. A taut foliation of a 3-manifold is a codimension one foliation such that there is a circle which intersects every leaf transversely. There are a lot of studies on foliations of a 3-manifold, many of these indicate that the structure of foliations reflects well the topology of a manifold. Novikov (3) showed that if a 3-manifold other than S 2 × S 1 possesses a foliation without Reeb components, its fundamental group is infinite, the second homotopy group π2 is trivial and its leaves are all π1-injective. Rosenberg (8) showed that if a 3-manifold possesses a foliation without Reeb components, then the manifold is irreducible. Combining theorems of Novikov and Rosenberg with that of Palmeira (4), one can see that if a 3-manifold possesses a foli- ation without Reeb components its universal cover is homeomorphic to R 3 . Therefore the existence of "Reebless" foliations plays an important role in studies of a 3-manifold. In fact, a Reeb component has no transverse circle which intersects all leaves, and hence a taut foliation has no Reeb compo- nent. Thus a taut foliation takes over the fruits of "Reebless" foliations with respect to the topological properties.
DOI：
10.1017/s030500410003824x
发表时间：
1964-10
影响因子：
0.8
作者：
W. Lickorish
通讯作者：
W. Lickorish
DOI：
--
发表时间：
1979-02
期刊：
--
影响因子：
--
作者：
諏訪 立雄
通讯作者：
諏訪 立雄