Sachs' Linkless Embedding Conjecture
Sachs' Linkless Embedding Conjecture
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萨克斯的无链接嵌入猜想
DOI：
10.1006/jctb.1995.1032
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
R. Thomas
中科院分区：
文献类型：
--
作者：
N. Robertson;P. Seymour;R. Thomas
本文证明了Sachs猜想：一个图可以嵌入3-空间中，使得它不包含非平凡链（在纽结理论意义下）当且仅当它不包含由K6通过Y-Δ和Δ-Y交换得到的七个图中的任何一个作为子图。我们还展示了以下内容：（i）一个图允许这样一个“无链”嵌入当且仅当它允许一个“嵌板”嵌入，一个这样的图的每一个电路界定一个圆盘从图的其余部分不相交。这是波美的一个猜想。(ii)一个嵌入是嵌格的当且仅当对每个子图，它在3-空间中的补图都有自由基本群。这扩展了Scharlemann和Thompson的定理，他们证明了平面图。(iii)如果同一个图的两个镶嵌是“不同的”，也就是说，不通过3-空间的方向保持同胚相关，那么存在一个子图，它是K5或K3，3的一个细分，使得这个子图的两个诱导嵌入仍然是不同的。
Abstract We prove Sachs′ conjecture that a graph can be embedded in 3-space so that it contains no non-trivial link (in the sense of knot theory) if and only if it contains as a minor none of the seven graphs obtainable from K 6 by Y − Δ and Δ − Y exchanges. We also show the following: (i) A graph admits such a "linkless" embedding if and only if it admits a "panelled" embedding, one such that every circuit of the graph bounds a disc disjoint from the remainder of the graph. This was a conjecture of Bohme. (ii) An embedding is panelled if and only if for every subgraph, its complement in 3-space has free fundamental group. This extends a theorem of Scharlemann and Thompson, who proved it for planar graphs. (iii) If two panelled embeddings of the same graph are "different," that is, are not related by an orientation-preserving homeomorphism of the 3-space, then there is a subgraph which is a subdivision of K 5 or K 3, 3 such that the two induced embeddings of this subgraph are still different.