There exist no Minimally Knotted Planar Spatial Graphs on the Torus

TL;DR

This paper proves that any nontrivial embedding of a planar graph on a torus necessarily contains a nontrivial knot or link, implying no minimally knotted planar spatial graphs without such features exist on the torus.

Contribution

It establishes a fundamental topological restriction on planar graphs embedded on the torus, showing the impossibility of minimally knotted embeddings without nontrivial knots or links.

Findings

All nontrivial torus embeddings of planar graphs contain knots or links.

Minimally knotted planar spatial graphs without nontrivial knots or links do not exist on the torus.

The result links graph embedding properties with knot theory on the torus.

Abstract

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.