A Gap in Stanfield's Proof of Sachs' Linear Linkless Embedding Conjecture

TL;DR

This paper identifies a significant flaw in Stanfield's proof of Sachs' conjecture, which states that all linklessly embeddable graphs can be linearly embedded in three-dimensional space.

Contribution

It critically examines and highlights a serious gap in an existing proof of a well-known conjecture in topological graph theory.

Findings

Identifies a gap in Stanfield's proof of Sachs' conjecture

Questions the validity of the existing proof for the conjecture

Highlights the need for a revised proof or new approach

Abstract

This is a short note describing what I believe is a serious gap in Stanfield's proof of Sachs' conjecture that every linklessly embeddable graph has a linear linkless embedding in $\mathbb{R}^3$.