On a conjecture of K. Murasugi
type: Item
instance of: scholarly article
title: On a conjecture of K. Murasugi
zbMATH Open document ID: 0563.57003
author: William S. Massey
author: L. Traldi
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let L denote a link in the 3-sphere having \(\mu\) components, \(\mu\geq 2\), and group G. It is a tantalizing problem in the study of such links to determine the structure of the lower central series quotient groups, \(G_ q/G_{q+1}\), of G, and to relate their structure to other known invariants of L. Several years ago, K. Murasugi conjectured that if every pair of components of L has linking number \(\pm 1\), then \(G_ q/G_{q+1}\) is isomorphic to \(F_ q/F_{q+1}\) for all \(q\geq 2\), where F is a free group of rank \(\mu\)-1. A strengthened version of Murasugi's conjecture was proved by \textit{T. Maeda} in his Ph. D. dissertation [Univ. of Toronto, 1983]. A key step in Maeda's proof was the demonstration that \(G_ q/G_{q+1}\cong F_ q/F_{q+1}\) for all \(q\geq 2\) if and only if \(G''G_ q/G''G_{q+1}\cong F''F_ q/F''F_{q+1}\) for all \(q\geq 2\). In the present paper, the authors give several conditions on the link L that are necessary and sufficient for the existence of isomorphisms \(G_ q/G_{q+1}\cong F_ q/F_{q+1}\) for all \(q\geq 2\). Perhaps the most striking of these conditions is the following: if \(G_ 2/G_ 3\cong F_ 2/F_ 3,\) then \(G_ q/G_{q+1}\cong F_ q/F_{q+1}\) for all \(q\geq 2\). In the proofs, essential use is made of the result of Maeda's mentioned above.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 3897836
zbMATH Keywords: classical links
zbMATH Keywords: Chen groups
zbMATH Keywords: I-adic completions
zbMATH Keywords: link in the 3-sphere
zbMATH Keywords: lower central series
zbMATH Keywords: linking number
MaRDI profile type: Publication
full work available at URL: pjm.1986.124.193
OpenAlex ID: W2130146629
Wikidata QID: Q123248418
DOI: 10.2140/PJM.1986.124.193
description: scientific article; zbMATH DE number 3897836