Complete bipartite graphs whose topological symmetry groups are polyhedral

TL;DR

This paper characterizes the values of n for which the complete bipartite graph K_{n,n} can be embedded in the 3-sphere with a topological symmetry group isomorphic to one of the polyhedral groups A_4, A_5, or S_4.

Contribution

It provides a complete classification of the embeddings of K_{n,n} with specific polyhedral symmetry groups in S^3.

Findings

Identifies all n for which K_{n,n} has the specified symmetries in S^3.

Connects graph embeddings with polyhedral symmetry groups.

Advances understanding of symmetries in topological graph embeddings.

Abstract

We determine for which $n$, the complete bipartite graph $K_{n,n}$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$.