We determine for which m the complete graph Km has an embedding in S3 whose topological symmetry group is isomorphic to one of the polyhedral groups A4, A5 or S4.
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Flapan, E., B. Mellor and R. Naimi, 2011: Complete graphs whose topological symmetry groups are polyhedral. Algebraic and Geometric Topology, 11, 1405-1433.