Graphs in the 3--sphere with maximum symmetry

TL;DR

This paper determines the maximum symmetry group orders acting on connected graphs embedded in the 3-sphere, classifies the graphs achieving these symmetries, and explores both orientation-preserving and general actions.

Contribution

It introduces the maximum order of symmetry groups acting on embedded graphs in the 3-sphere and classifies the graphs realizing these symmetries at various levels.

Findings

Maximum order of symmetry groups for graphs of genus g in S^3

Classification of graphs achieving maximum symmetry

Analysis of orientation-preserving and general group actions

Abstract

We consider the orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Gamma)$, where $\Gamma$ 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 $(S^3, \Gamma)$ for all such $\Gamma\subset S^3$. Indeed we will classify all graphs $\Gamma\subset S^3$ 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.