A proof using B\"ohme's Lemma that no Petersen family graph has a flat embedding
Combinatorics
2023-02-28 v1 Geometric Topology
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 B\"{o}hme's Lemma and the Jordan-Brouwer Separation Theorem.
Cite
@article{arxiv.2302.12880,
title = {A proof using B\"ohme's Lemma that no Petersen family graph has a flat embedding},
author = {Joel Foisy and Catherine Jacobs and Trinity Paquin and Morgan Schalizki and Henry Stringer},
journal= {arXiv preprint arXiv:2302.12880},
year = {2023}
}