# A proof using B\"ohme's Lemma that no Petersen family graph has a flat embedding

## TL;DR

This paper provides an alternative proof, using Böhme's Lemma and the Jordan-Brouwer Separation Theorem, to show that no Petersen family graph admits a flat embedding, reinforcing their intrinsic linking properties.

## Contribution

The paper introduces a novel proof technique for non-flatness of Petersen family graphs, differing from previous linking number and counting arguments.

## Key findings

- All Petersen family graphs are intrinsically linked.
- No Petersen family graph admits a flat embedding.
- The proof leverages Böhme's Lemma and the Jordan-Brouwer Separation Theorem.

## 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.