Graphs in the 3--sphere with maximum symmetry

Abstract

We consider the orientation-preserving actions of finite groups $G$ on pairs $(S^3, Γ)$, where $Γ$ is a connected graph of genus $g>1$, embedded in $S^3$. For each $g$ we give the maximum order $m_g$ of such $G$ acting on $(S^3, Γ)$ for all such $Γ\subset S^3$. Indeed we will classify all graphs $Γ\subset S^3$ which realize these $m_g$ in different levels: as abstract graphs and as spatial graphs, as well as their group actions. Such maximum orders without the condition "orientation-preserving" are also addressed.