There exist no minimally knotted planar spatial graphs on the torus

Abstract

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.

Toroidal embeddings of abstractly planar graphs are knotted or linked

“…It is not possible to weaken the assumptions of Theorem 1 as shown by giving counter examples in [25].…”

“…The argument gives an explicit deformation from embeddings of abstractly planar graphs on the torus that contain neither a nontrivial knot nor a nonsplit link into the plane. A much shorter but less intuitive proof that relies on deep theorems of topological graph theory is given in [25]. The argument on hand not only presents a self-sufficient argument but will hopefully also give the reader a better feeling for the nature of graphs that are embedded on the torus.…”

Canonical decompositions and algorithmic recognition of spatial graphs

“…As |Γ| ∩ X V is the disjoint union of the properly embedded arcs e ∩ X V , with e ∈ E, the regular neighbourhood N e of each e ∩ X V is a 3 -ball and (N e , e ∩ X V ) is an unknotted ball pair [15,Corollary 3.27]. 3 We now set…”

“…The PL category is the natural home for results in computational topology, such as our main theorem and the machinery in Matveev’s text. It is a standard framework in the field [3, 10, 16, 17], although a theory of smooth, rather than PL, spatial graphs has also been introduced by the first author and Herrmann [4, 5].…”

On chirality of toroidal embeddings of polyhedral graphs

“…For the proof we rely on the fact that all planar toroidal spatial graphs contain a nontrivial knot or a nonsplit link: Theorem 1.3 (Existence of knots and links [1], [2]). Let G be an planar graph and f : G → R 3 be a nontrivial embedding of G with image G. If G is contained in the torus T 2 , it contains a subgraph which is a nontrivial knot or a nonsplit link.…”