There exist no Minimally Knotted Planar Spatial Graphs on the Torus
  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. 
Senja Barthel
math.GT