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On spatial graphs isotopic to planar embeddings. (English) Zbl 0962.57002

Suzuki, S. (ed.), KNOTS’96. Proceedings of the international conference and workshop on knot theory, Tokyo, Japan, July 22-26, 1996. Singapore: World Scientific. 1-22 (1997).
From the introduction: We present a rearrangement theorem (Theorem 1) for any spatial embeddings isotopic to planar embeddings, and extend some results obtained in [T. Soma, Topology Appl. 73, No. 1, 23-41 (1996; Zbl 0861.57005)] for trivalent graphs to non-trivalent ones. For our convenience, we denote by \([\Gamma]_{\text{cobor}}\) (resp. by \([\Gamma]_{\text{isotopy}})\) the subset of the set of all PL-embeddings \({\mathcal S}(G)\) consisting of all elements cobordant to (resp. isotopic to) \(\Gamma \in{\mathcal S}(G)\), and call it the cobordism class (resp. the isotopy class) of \(\Gamma\). Here, we note that any isotopy between two embeddings \(\Gamma, \Gamma'\in{\mathcal S}(G)\) is realized by a finite sequence of blowing-downs \(\searrow\) and ups \(\nearrow\), see [K. Taniyama, Topology 33, No. 3, 509-523 (1994; Zbl 0823.57006)] for details.
Theorem 1. Let \(G\) be a graph admitting a planar embedding \(\Gamma_0:G\to\mathbb{R}^2\subset\mathbb{R}^3\). For any \(\Gamma\in [\Gamma_0]_{\text{isotopy}}\), there exists a sequence of blowing-downs such that \(\Gamma\searrow\cdots\searrow\Gamma_0\).
An element \(\Gamma \in{\mathcal S}(G)\) is unknotted if \(\Gamma\) is ambient isotopic to a planar embedding \(\Gamma_0:G\to\mathbb{R}^2\subset\mathbb{R}^3\) and otherwise knotted. For a planar graph \(G\), an embedding \(\Gamma\in{\mathcal S}(G)\) is said to be minimally knotted if \(\Gamma\) itself is knotted, but the restriction \(\Gamma|_H: H\to \mathbb{R}^3\) is unknotted for any proper subgraph \(H\) of \(G\).
We say that a graph \(G\) is a generalized bouquet if \(G\) contains a vertex \(v\) such that \(G-\{v\}\) is acyclic. According to [Taniyama, loc. cit., Theorem A], if \(G\) is a generalized bouquet, then any embedding \(\Gamma:G \to\mathbb{R}^3\) is isotopic to a planar embedding \(\Gamma_0:G\to\mathbb{R}^2\subset\mathbb{R}^3\), i.e. \([\Gamma_0]_{\text{isotopy}}= {\mathcal S}(G)\). Thus, if \(G\) is a generalized bouquet without free edges, then \([\Gamma_0]_{\text{isotopy}}\) contains a minimally knotted embedding. The following theorem implies the converse.
Theorem 2. Suppose that \(G\) is a graph admitting a planar embedding \(\Gamma_0: G\to\mathbb{R}^2 \subset \mathbb{R}^3\). If the isotopy class \([\Gamma_0]_{\text{isotopy}}\) contains a minimally knotted embedding, then \(G\) is a generalized bouquet without free edges.
The authors do not know whether there is a rearrangement theorem for the isotopy class \([\Gamma]_{\text{isotopy}}\) of a knotted embedding \(\Gamma:G \to \mathbb{R}^3\) of any graph \(G\). As a partial answer, we have the following theorem valid for any connected graphs without cut vertices, where a vertex \(v\) of \(G\) is called a cut vertex if \(v\) disconnects the component of \(G\) containing \(v\).
Theorem 3. Let \(G\) be a connected graph without cut vertices, and let \(\Gamma_1, \Gamma_2: G\to\mathbb{R}^3\) be embeddings isotopic to each other. Then, there exists an embedding \(\Gamma_3:G \to\mathbb{R}^3\) and a sequence of blowing-downs followed by blowing-ups such that \(\Gamma_1 \searrow\cdots \searrow \Gamma_3 \nearrow\cdots \nearrow\Gamma_2\).
For the entire collection see [Zbl 0959.57001].

MSC:

57M15 Relations of low-dimensional topology with graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
57Q20 Cobordism in PL-topology
57Q37 Isotopy in PL-topology
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