Topological Symmetry Groups of the Generalized Petersen Graphs

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