Toroidal embeddings of abstractly planar graphs are knotted or linked

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.

An erroneous figure has been replaced, the funding for the second author added. Terminology adapted to standard terminology. Some minor changes for better readability. This version should be preferred over the published version. 13 pages, 14 figures

References in corpus (2)

• On chirality of toroidal embeddings of polyhedral graphs
• There exist no Minimally Knotted Planar Spatial Graphs on the Torus

Cited by in corpus (2)

• On chirality of toroidal embeddings of polyhedral graphs
• Gauss diagram formulas of Vassiliev invariants of spatial 2-bouquet graphs