Topological Symmetry Groups of the Generalized Petersen Graphs

The topological symmetry group $\mathrm{TSG}(Γ)$ of an embedding $Γ$ of a graph in $S^3$ is the subgroup of the automorphism group of the graph which is induced by homeomorphisms of $(S^3,Γ)$. If we restrict to orientation preserving homeomorphisms then we obtain the orientation preserving topological symmetry group $\mathrm{TSG}_+(Γ)$. In this paper we determine all groups that can be $\mathrm{TSG}(Γ)$ or $\mathrm{TSG}_+(Γ)$ for some embedding $Γ$ 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.