A characterization of knots in a spatial graph (Q2706865)

For a graph \(G\), let \(\Gamma\) be the set of cycles of \(G\). Suppose that for each \(\gamma \in \Gamma\), an embedding \(\varphi_{\gamma} \to S^3\) is given. A set \(\{ \varphi_{\gamma} \mid \gamma \in \Gamma \}\) is said to be realizable if there is an embedding \(f : G \to S^3\) such that the restriction \(f|_{\gamma}\) is ambient isotopic to \(\varphi_{\gamma}\) for any \(\gamma \in \Gamma\). A graph \(G\) is said to be adaptable if any set \(\{ \varphi_{\gamma} \mid \gamma \in \Gamma \}\) is realizable. NEWLINENEWLINENEWLINEAny non-planar graph is not adaptable [\textit{T. Motohashi} and \textit{K. Taniyama}, Knots '96, World Scientific. 185-200 (1997; Zbl 0960.57003)] and the set of non-adaptable graphs all of which proper minors are adaptable contains exactly two non-planar graphs, \(K_5\) and \(K_{33}\), [the reviewer, ibid., 115-121(1997; Zbl 0960.57004)] and at least eight planar graphs [\textit{K. Taniyama} and the reviewer, Topology Appl. 112, No. 1, 87-109 (2001; Zbl 0968.57001)]. For seven graphs of the eight planar graphs, the author gives a necessary and sufficient condition for \(\{ \varphi_{\gamma} \mid \gamma \in \Gamma\}\) to be realizable in terms of the second coefficient of the Conway polynomial. The necessary condition was essentially given in [Taniyama and the reviewer, loc. cit.].