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The topological symmetry group of a canonically embedded complete graph in \(S^3\). (English) Zbl 0891.05024
Let \(G\) be a (finite and simple) graph with vertex set \(V(G)\) and automorphism group \(\operatorname{Aut}(G)\). If \(f:G\to \mathbb{S}^3\) is an embedding of \(G\) into the 3-sphere \(\mathbb{S}^3\), then the topological symmetry group \(\text{TSG}(f)\) of \(f\) is defined by \[ \begin{split} \text{TSG} (f)=\{h\in \operatorname{Aut}(G) \mid\text{there is a homeomorphism }\phi: \mathbb{S}^3\to \mathbb{S}^3 \\ \text{with }\phi(f(G)) =f(G)\text{ such that } f\circ h= \phi\circ f|_{V(G)}\}. \end{split} \] The present paper faces the problem of determining \(\text{TSG} (f_n)\), where \(f_n\) is a particular embedding of the complete graph \(K_n\) with \(n\) vertices into \(\mathbb{S}^3\), called canonical bud presentation, see T. Endo and T. Otsuki [Hokkaido Math. J. 23, No. 3, 383-398 (1994; Zbl 0814.57007)] and T. Otsuki [J. Comb. Theory, Ser. B 68, No. 1, 23-35, Art. No. 0054 (1996; Zbl 0858.05038)]. In particular, it is proved that, if the order \(n\) is at least seven, then the topological symmetry group of \(f_n\) is isomorphic to the dihedral group of order \(2n\).
Note that, since the cases \(n\leq 5\) and \(n=6\) have already been solved by Y. Yoshimata [Topological symmetry group of standard spatial graph of \(K_5\) (Japanese), Master Thesis, Tokyo Woman’s Christian Univ. (1992)] and by K. Kobayashi and C. Toba [Proc. TGRC-KOSEF 3, 153-171 (1993)] respectively, the previous result yields a complete answer to the considered problem.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
57M15 Relations of low-dimensional topology with graph theory
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