Toroidal embeddings of abstractly planar graphs are knotted or linked

TL;DR

This paper explores how certain graphs embedded on a torus can be deformed into planar forms, revealing that complex knots or links are unavoidable, which has implications for molecular structures synthesized on a torus.

Contribution

It provides explicit deformations of torus-embedded graphs, showing they are either knotted or linked, and demonstrates ravels cannot embed on the torus, offering new insights into molecular topology.

Findings

Embedded graphs on a torus are either knotted or linked after deformation.

Ravels cannot be embedded on the torus.

Provides explicit deformation methods for torus embeddings.

Abstract

We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus $T^2 \subset \mathbb{R}^3$ and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecules that are synthesized on a torus.