Erratum to: Toroidal embeddings of abstractly planar graphs are knotted or linked

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 ⊂ R3 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.

Minimally Knotted Spatial Graphs are Totally Knotted

Applying Jaco’s Handle Addition Lemma, we give a condition for a 3-manifold to have an incompressible boundary. As an application, we show that the boundary of the exterior of a minimally knotted planar graph is incompressible.

Chordal embeddings of planar graphs

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.

Geodesic Embeddings and Planar Graphs

Knotted Hamiltonian cycles in spatial embeddings of complete graphs

We show the complete graph on n vertices contains a knotted Hamiltonian cycle in every spatial embedding, for n > 7. Moreover, we show that for n > 8, the minimum number of knotted Hamiltonian cycles in every embedding of Kn is at least (n−1)(n−2) . . . (9)(8). We also prove K8 contains at least 3 knotted Hamiltonian cycles in every spatial embedding.