Classification of topological symmetry groups of \(K_n\) (Q2862931)

Let \(\gamma\) be an abstract graph and \(\Gamma\) the image of an embedding of \(\gamma\) in the three sphere \(S^{3}\). The topological symmetry group \(\Gamma\) denoted by \(TSG(\Gamma)\) is the subgroup of the automorphism group \(Aut(\gamma)\) which is induced by homeomorphisms of the pair \((S^{3}, \Gamma)\). This paper completes the classification of the topological symmetry groups of the complete graphs \(K^{n}\) by characterizing the \(K^{n}\) that can have a cyclic group, a dihedral group \(D_{m}\), or a subgroup of \(D_{m} \times D_{m}\) for odd \(m\) as its topological symmetry group. This leads to the ability to algorithmically determine the topological symmetry group of \(K^{n}\) for any \(n\).