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A proof using Böhme's Lemma that no Petersen family graph...
Joel Foisy, Catherine Jacobs, Trinity Paquin, Morgan Schalizki, · 2023-02-25 · via math.CO updates on arXiv.org

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.