Fibered knots and algebraic singularities
Fibered knots and algebraic singularities
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纤维结和代数奇点
DOI：
10.1016/0040-9383(74)90037-8
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发表时间：
1974
期刊：
影响因子：
--
通讯作者：
Alan Durfee
中科院分区：
文献类型：
--
作者：
Alan Durfee
设 V c C”“, n 2 0 是一个代数超曲面，其中 p EV 是 V 的孤立奇点（或可能是非奇点）。关于 p 的足够小的 (2n+ l) 球体 S’“” 与 V 的交点是一个 (2~-I) 流形 K c S’“+ l，奇点的链接。K 本身的拓扑性质已经被许多作者研究过。另一方面，K 在SZni-’只有在 n= 1 时才被完全理解（参见[9]中的参考文献）；结 K c S3 作为代数曲线奇点的链接出现的充分必要条件是它是满足一些附加条件的复合环面链接。41-3 致力于为纤维结理论奠定基础。在 $1 中，我们将纤维结定义为 a 的嵌入。 S'“”中的高度连通的流形K，其补纤维在S'上。（纤维结与[3]的“S'“+ I”上的简单可旋转结构一致，提供n 2 2。）米尔诺已经证明，孤立的超曲面奇点的链接是纤维结。在 $2 中，我们以标准方式定义与纤维结相关的积分双线性形式（称为 Seifert 形式），计算一些示例，并展示 Seifert 形式如何确定其他结不变量。 $3 的第一个结果是 Levine 工作的简单概括：
LET V c C”“, n 2 0, be an algebraic hypersurface with p EV an isolated singularity (or possibly non-singular point) of V. The intersection with V of a sufficiently small (2n+ l)-sphere S’“” about p is a (2~-I)-manifold K c S’“+ l, the link of the singularity. The topological properties of K itself have been investigated by many authors. On the other hand, the knotting of K in SZni-’IS only completely understood when n= 1 (see the bibliography in [9]); a necessary and sufficient condition that a knot K c S3 occur as the link of a singularity of an algebraic curve is that it be a compound torus link satisfying some additional conditions. Our purpose is to prove a necessary condition in higher dimensions.41-3 are devoted to laying the foundations of the theory of fibered knots. In $1 we define a fibered knot as an embedding of a highly-connected manifold K in S’“” whose complement fibers over S’.(Fibered knots coincide with the “simple spinnable structures on S ‘“+ I” of [3], provided n 2 2.) Milnor has shown that the link of an isolated hypersurface singularity is a fibered knot. In $2 we define in the standard way an integral bilinear form associated to a fibered knot called the Seifert form, compute some examples, and show how the Seifert form determines other knot invariants. The first result of $3 is an easy generalization of the work of Levine: