Sachs' linkless embedding conjecture
type: Item
instance of: scholarly article
title: Sachs' linkless embedding conjecture
zbMATH Open document ID: 0832.05032
author: Neil Robertson
author: Robin Thomas
author: Paul Seymour
published in: Journal of Combinatorial Theory. Series B
publication date: 1996-02-20T00:00:00Z
review text: The core part of this paper is a proof of Sachs' conjecture, which states that a graph can be linklessly embedded in the 3-space iff it has no Petersen family minor, i.e., iff it contains as minor none of the seven graphs obtainable from \(K_6\) by \(Y-\Delta\) and \(\Delta- Y\) exchanges. Other results: (1) A proof of a conjecture of Böhme stating that a graph admits a linkless embedding iff it admits a panelled embedding (one in which every circuit of the graph is the boundary of a disc disjoint from the remainder of the graph). (2) An embedding is panelled iff the complement in 3-space of every subgraph has free fundamental group. This extends a theorem of Scharelmann and Thompson for planar graphs. (3) If two panelled embeddings are not related by an orientation-preserving homeomorphism of the 3-space, then there is a subgraph which is a subdivision of \(K_5\) or \(K_{3,3}\) such that the two embeddings of this subgraph are still ``different''.
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 798747
zbMATH Keywords: Petersen family minor
zbMATH Keywords: conjecture of Böhme
zbMATH Keywords: linkless embedding
zbMATH Keywords: panelled embedding
zbMATH Keywords: fundamental group
zbMATH Keywords: subdivision
Wikidata QID: Q56431020
reviewed by: Stelian Mihalas
MaRDI profile type: Publication
full work available at URL: jctb.1995.1032
OpenAlex ID: W2052259760
DOI: 10.1006/JCTB.1995.1032
Recommended article: About the Connes embedding conjecture
Recommended article: Q4462218
Recommended article: On Schechter's linking theorems
Recommended article: The Baum-Connes conjecture
Recommended article: Abhyankars Conjecture and embedding problems
Recommended article: Connes' embedding problem and Tsirelson's problem
Recommended article: Abhyankar-Sathaye embedding conjecture for a geometric case
Recommended article: Q3217278
Recommended article: A slope conjecture for links
description: scientific article; zbMATH DE number 798747