Toroidal Embeddings of non-Intrinsically-Linked Graphs

TL;DR

This paper investigates the conditions under which graphs embeddable on the torus and linklessly in three-dimensional space can also be embedded linklessly within the standard torus, providing results for graphs of order up to 9.

Contribution

It proves that for graphs of order 9 or less, being embeddable on the torus and linkless in 3D implies they can also be embedded linklessly in the standard torus.

Findings

Graphs of order ≤ 9 that are torus-embeddable and linkless in 3D can be embedded linklessly in the standard torus.

The paper establishes a link between topological embedding properties and standard torus embeddings for small graphs.

Abstract

If a graph $G$ can be embedded on the torus, and be embedded linklessly in $\mathbb{R}^3$, it's not known whether or not we can always find a linkless embedding of $G$ contained in the standard (unknotted) torus; We show that, for orders 9 and below, any graph which is both embeddable on the torus, and linklessly in $\mathbb{R}^3$, can be embedded linklessly in the standard torus.