Topological Symmetry Groups of the Generalized Petersen Graphs

The topological symmetry group \(\mathrm{TSG}(\Gamma)\) of an embedding \(\Gamma\) of a graph in \(S^3\) is the subgroup of the automorphism group of the graph which is induced by homeomorphisms of \((S^3,\Gamma)\). If we restrict to orientation preserving homeomorphisms then we obtain the orientation preserving topological symmetry group \(\mathrm{TSG}_+(\Gamma)\). In this paper we determine all groups that can be \(\mathrm{TSG}(\Gamma)\) or \(\mathrm{TSG}_+(\Gamma)\) for some embedding \(\Gamma\) of a generalized Petersen graph other than the exceptional graphs \(P(12,5)\) and \(P(24, 5)\) (which will be addressed in a separate paper.