Topological Symmetry Groups of the Generalized Petersen Graphs

Abstract

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.