Topological Symmetry Groups for Small Complete Graphs

TL;DR

This paper characterizes all possible symmetry groups of embeddings of complete graphs with up to six vertices in three-dimensional space, focusing on their topological symmetry groups.

Contribution

It provides a complete classification of the topological symmetry groups for embeddings of $K_n$ in $S^3$ for all $n \

Findings

Identifies all groups realizable as topological symmetry groups for $K_n$, $n \

Distinguishes between orientation-preserving and general topological symmetry groups.

Provides explicit characterizations for each $n \

Abstract

For each $n\leq 6$, we characterize all the groups which can occur as either the orientation preserving topological symmetry group or the topological symmetry group of some embedding of $K_n$ in $S^3$.