Spatial Graphs with Local Knots

TL;DR

The paper proves that for any locally knotted edge in a 3-connected graph embedded in three-dimensional space, there exists a unique ball containing all local knots, which aids in understanding the symmetry groups of such graphs.

Contribution

It introduces a unique ball containing all local knots of an edge in a 3-connected graph, advancing the study of topological symmetry groups in spatial graphs.

Findings

Existence of a unique ball containing all local knots for any knotted edge.

Application of this result to analyze topological symmetry groups.

Enhanced understanding of spatial graph embeddings in 3-space.

Abstract

It is shown that for any locally knotted edge of a 3-connected graph in $S^3$, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in $S^3$.