arXiv 0906.22292009-06-11math.GT

每个图在$S^3$中都有一个不含非双曲纽结的嵌入

Every graph has an embedding in $S^3$ containing no non-hyperbolic knot

Erica Flapan, Hugh Howards

备注：: 12 pages 4 figures

摘要：

在$S^3$中，节点的性质仅取决于其外部拓扑，与之相反，在$S^3$中，图的内在结构与图的所有嵌入的外部拓扑之间存在丰富的相互作用。例如，在[2]中证明了完全图$K_7$在$S^3$中的每个嵌入都包含一个非平凡的纽结。后来证明，对于每个$m \in N$，存在一个完全图$K_n$，使得$K_n$在$S_3$中的每个嵌入都包含一个结$Q$，其最小交叉数至少为$m$。因此，在$S^3$中，一个足够大的完全图的每一个嵌入中都存在任意复杂的节点。我们在这里证明了对比的结果，每个图有一个嵌入在$S^3$，使得每个非平凡的结嵌入是双曲的。我们的定理意味着每个图都有一个嵌入在$S^3$，其中不包含合成或卫星结。

英文摘要：

In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in $S^3$ . For example, it was shown in [2] that every embedding of the complete graph $K_7$ in $S^3$ contains a non-trivial knot. Later in it was shown that for every $m \in N$, there is a complete graph $K_n$ such that every embedding of $K_n$ in $S_3$ contains a knot $Q$ whose minimal crossing number is at least $m$. Thus there are arbitrarily complicated knots in every embedding of a sufficiently large complete graph in $S^3$. We prove here the contrasting result that every graph has an embedding in $S^3$ such that every non-trivial knot in that embedding is hyperbolic. Our theorem implies that every graph has an embedding in $S^3$ which contains no composite or satellite knots.

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