Complete graphs whose topological symmetry groups are polyhedral (Q538637)

In this paper, the authors determine for which \(m\) the complete graph \(K_m\) has an embedding in \(S^3\) whose topological symmetry group is isomorphic to one of the polyhedral groups \(A_4,A_5\) or \(S_4\). In particular, \(K_m\) has an embedding with topological symmetry group isomorphic to \(A_4\) if and only if \(m \equiv 0, 1,4,5,8\) (mod \(12\)), an embedding with topological symmetry group isomorphic to \(A_5\) if and only if \(m \equiv 0, 1, 5, 20\) (mod \(60\)) and an embedding with topological symmetry group isomorphic to \(S_4\) if and only if \(m \equiv 0, 4, 8, 12, 20\) (mod \(24\)). \textit{E. Flapan} et al. [J. Lond. Math. Soc., II. Ser. 73, No. 1, 237--251 (2006; Zbl 1091.57001)] earlier characterized which finite groups can occur as a topological symmetry group of an embedding of a complete graph: namely a finite cyclic group, a dihedral group, a subgroup of \(D_m \times D_m\) for some odd \(m\), or \(A_4, S_4\) or \(A_5\). In another paper [\textit{E. Flapan}, \textit{B. Mellor}, \textit{R. Naimi} and \textit{M. Yoshizawa}, ``A characterization of topological symmetry groups of complete graphs'', preprint (2011)], the authors (together with Yoshizawa) characterize which complete graphs can have a cyclic group, a dihedral group or another subgroup of \(D_m \times D_m\) as their topological symmetry group. The paper uses a nice blend of techniques from topology including covering spaces and knot theory, as well as machinery from Smith Theory [\textit{P. A. Smith}, Ann. Math. (2) 40, 690--711 (1939; Zbl 0021.43002)], the recently proven Geometrization Theorem [\textit{J. W. Morgan} and \textit{F. T-H. Thong}, Ricci flow and geometrization of 3-manifolds. Providence, RI: American Mathematical Society (AMS) (2010; Zbl 1196.53003)] and new results of \textit{F. Bonahon} and \textit{L. Siebenmann} [``New geometric splittings of classical knots, and the classification and symmetries of arborescent knots'', unpublished manuscript (\url{http://www-bdf.usc.edu/~fbonahon/Research/Preprints/BonSieb.pdf})(2010)].