There exist no Minimally Knotted Planar Spatial Graphs on the Torus

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.

4 pages, 1 figure

Cited by in corpus (3)

• Toroidal embeddings of abstractly planar graphs are knotted or linked
• On chirality of toroidal embeddings of polyhedral graphs
• Canonical decompositions and algorithmic recognition of spatial graphs