Topological Symmetry Groups of the Generalized Petersen Graphs

TL;DR

This work determines which groups can be realized as the topological symmetry group $\mathrm{TSG}(\Gamma)$ or the orientation-preserving version $\mathrm{TSG}_+(\Gamma)$ for embeddings of generalized Petersen graphs $P(n,k)$ in $S^3$, excluding seven exceptional pairs. By combining automorphism rigidity for 3-connected graphs, Smith theory on finite-order homeomorphisms, and explicit geometric embeddings (often with knotted edges) plus the Edge Embedding Lemma and Subgroup Theorem, the authors classify realizable and positively realizable subgroups across non-exceptional and several exceptional cases. They show that in non-exceptional cases, $\mathrm{Aut}(P(n,k))$ is typically $D_n$ (or a related semidirect product) and that all subgroups are realizable (and often positively realizable); for the exceptional graphs $(4,1)$, $(8,3)$, $(10,2)$, and $(10,3)$, they determine precisely which subgroups are positively realizable and provide constructive embeddings to realize them. The results yield a comprehensive map of possible topological symmetries for this important graph family and set the stage for treating $(12,5)$ and $(24,5)$ in a follow-up study.

Abstract

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.

Figures (11)

• Figure 1: The graph $P(6,2)$ and an embedding of $P(7,2)$.
• Figure 2: The exceptional graphs $P(4,1)$, $P(5,2)$, $P(8,3)$, $P(10, 2)$, and $P(10,3)$.
• Figure 3: The embedded vertices of $P(10,3)$, except for $u_5$ and $v_5$ which are on the (green) core in the front of the solid torus.
• Figure 4: The embedding $\Gamma$ of $P(4,1)$ with $\mathrm{TSG}_+(\Gamma) = S_4$ is obtained from the left image by identifying vertices with the same labels and pairs of adjacent branched edges. On the right, we see that $\overline{v_1v_2v_3v_4}$ is the connected sum of four copies of $J$ and four trefoils.
• Figure 5: The embedding $\Gamma$ of $P(4,1)$ with $\mathrm{TSG}_+(\Gamma) =A_4 \times \mathbb{Z}_2$ is obtained from the projection on the left by identifying vertices with the same labels. On the right, we have the projection on a single face.

Theorems & Definitions (73)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Automorphism Rigidity Theorem Flap
• Theorem 1.4: Smith Theory
• Theorem 1.5: Group Rigidity Theorem
• Theorem 1.6: Involution Theorem FNT
• Theorem 1.7
• Lemma 1.8
• Theorem 2.1
• proof