Topological Symmetry Groups of the Generalized Petersen Graphs

Abstract

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

24 pages, 11 figures