Toroidal embeddings of abstractly planar graphs are knotted or linked

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.