Interpolating manifolds for knots in S3
Interpolating manifolds for knots in S3
复制标题
S3 中节点的插值流形
DOI：
10.1016/0040-9383(63)90015-6
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发表时间：
1963
期刊：
影响因子：
--
通讯作者：
L. Neuwirth
中科院分区：
文献类型：
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作者：
L. Neuwirth
l-球面在3-球面中的半线性嵌入引起了许多代数和拓扑性质的问题。我们对这些问题之间的相互作用特别感兴趣。有许多定理在纽结理论的文献作出拓扑假设，并得出代数结论（当然，对这些定理的陈述作出代数假设，并得出拓扑结论）。这类定理的例子可以在[1，2，3，4和51]中找到，其中的几何假设包括：交替投影的存在性[1]，E4中半空间中局部平坦圆盘的边界[2]，纽结类型的弯曲性[3]，上交到下交到解结的最小变化次数[4]，亏格[5]。代数结论包括：亚历山大多项式[1，2，51]，生成纽结补的基本群所需的最小生成元数[3]，从纽结补的基本群到2的同态的阿贝尔化核的最小生成元数[4]。当然，这些结果并没有穷尽这些定理的列表：例如，遗漏了史密斯问题的许多部分结果（没有非平凡结是S3的周期同胚的不动点集）。
THE SEMI-LINEAR imbedding of a l-sphere in the 3-sphere raises many questions both algebraic and topological in nature. We are particularly interested in the interplay between these questions. There are many theorems in the knot theory literature which make topological assumptions, and draw algebraic conclusions (of course the contrapositive statements of those theorems make algebraic assumptions, and draw topological conclusions). Examples of this type of theorem may be found in [I, 2, 3, 4 and 51, where the geometric assumptions involve: the existence of an alternating projection [I], the bounding of a locally flat disc in a half space in E4 [2], the crookedness of a knot type [3], the minimal number of changes of overcrossings to undercrossings to unknot [4], the genus [5]. The algebraic conclusions involve: the Alexander polynomial[I, 2, 51, the minimal number of generators needed to generate the fundamental group of the complement of the knot [3], the minimal number of generators of the abelianized kernel of the homomorphism from the fundamental group of the complement of the knot onto 2,[4]. Of course these results by no means exhaust the list of such theorems: left out, for example, are many partial results on the Smith problem (no non-trivial knot is the fixed point set of a periodic homeomorphism of S3).-