Topological symmetry groups and mapping class groups for spatial graphs

TL;DR

This paper establishes a precise criterion linking the mapping class group of a 3-sphere with an embedded graph to its topological symmetry group, advancing understanding of spatial graph symmetries.

Contribution

It provides a necessary and sufficient condition for the isomorphism between the mapping class group and the topological symmetry group of spatial graphs.

Findings

Derived a criterion for group isomorphism in spatial graphs

Connected topological symmetry groups with mapping class groups

Enhanced understanding of symmetries in spatial graph embeddings

Abstract

We give a necessary and sufficient condition for the mapping class group of the pair of the 3-sphere and a graph embedded in it to be isomorphic to the topological symmetry group of the embedded graph.