Braid presentation of spatial graphs

TL;DR

This paper introduces a braid presentation framework for spatial graphs, extending the concept from links, and demonstrates that every spatial graph admits such a presentation, highlighting differences from classical link braid indices.

Contribution

It generalizes braid presentations to spatial graphs and proves that all spatial graphs can be represented in this form, unlike the case for links where braid index equals minimal Seifert circles.

Findings

Every spatial graph has a braid presentation.

The braid index equality for links does not extend to spatial graphs.

Braid presentations provide a new way to study spatial graphs.

Abstract

We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy does not hold for spatial graphs.