A Proof Using Böhme's Lemma That no Petersen Family Graph has a Flat Embedding

Abstract

Sachs and Conway-Gordon used linking number and a beautiful counting argument to prove that every graph in the Petersen family is intrinsically linked (have a pair of disjoint cycles that form a nonsplit link in every spatial embedding) and thus each family member has no flat spatial embedding (an embedding for which every cycle bounds a disk with interior disjoint from the graph). We give an alternate proof that every Petersen family graph has no flat embedding by applying Böhme's Lemma and the Jordan-Brouwer Separation Theorem.