Topological Symmetry Groups of the Petersen graphs

The {\em topological symmetry group} of an embedding $\Gamma$ of an abstract graph $\gamma$ in $S^3$ is the group of automorphisms of $\gamma$ which can be realized by homeomorphisms of the pair $(S^3, \Gamma)$. These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups which can be realized as topological symmetry groups for each of the graphs in the Petersen Family. Along the way, we also complete the classification of the realizable topological symmetry groups for $K_{3,3}$.