Notes on spatial representations of graphs (Q1344196)

A spatial representation \(\overline{G}\) of \(G\) is a tame imbedding of \(G\) into \(\mathbb{R}^ 3\); it is locally unknotted, with respect to cycle basis \(\{C_ 1,\dots,C_ n\}\) of \(H_ 1(G;Z)\), if there are internally disjoint disks \(D_ 1,\dots,D_ n\) in \(\mathbb{R}^ 3\) so that, for \(i = 1,\dots,n\), \(D_ i \cap \overline{G} = \partial D_ i = \overline{C}_ i\). \textit{K. Kobayashi} [Hokkaido Math. J. 21, No. 1, 117-140 (1992; Zbl 0765.57003)] showed that all complete graphs have locally unknotted spatial representations. The present authors extend this result to arbitrary graphs. They also show that any two minimal book presentations of a fixed complete graph are ambient isotopic in \(\mathbb{R}^ 3\) up to sheet translations, verifying a conjecture of Kobayashi.