Graphs in the 3-Sphere with Maximum Symmetry

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 ⊂ S 3 . Indeed we will classify all graphs Γ ⊂ 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.