Linking numbers and Chen groups
type: Item
instance of: scholarly article
title: Linking numbers and Chen groups
zbMATH Open document ID: 0677.57007
DOI: 10.1016/0166-8641(89)90098-9
author: L. Traldi
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Let \(\{G_ q\}\) be the lower central series of a group G, where \(G_ 1=G\) and \(G_{q+1}=[G,G_ q]\) for \(q\geq 1\). The Chen groups of the link group \(G_ L\) are the quotient groups \(G(q)=G''G_ q/G''G_{q+1}\), where \(G''\) is the second commutator subgroup of G. Since \(G_ L\) is finitely generated, G(q) is a finitely generated abelian group. The Chen groups were first studied by K. T. Chen in 1951, and later (1970) the reviewer determined these groups for 2-component link groups. Although the structures of the Chen groups for more than 2-components link groups are algorithmically determined, the precise description of the structures of these abelian groups in terms of known link invariants is not known. Difficulties occur when particularly the linking number between some 2- component links vanishes. One of the main theorems of this paper claims that if a link is sufficiently linked, then G(q) is completely determined by the second lower central quotient \(G_ 2/G_ 3\). To be more precise, let \(L=K_ 1\cup K_ 2\cup...\cup K_{\mu}\) be a \(\mu\)-component link and let \(\Gamma\) be the planar graph with \(\mu\) vertices, where two vertives \(\nu_ i\) and \(\nu_ j\) are joined by an edge if the linking number between \(K_ i\) and \(K_ j\) is not 0. Then the author proves: Theorem 1. Suppose \(\mu\geq 3\). If \(\Gamma\) is connected, then G(q) is a finitely generated abelian group of rank \((q-1)\left( \begin{matrix} \mu +q-3\\ q\end{matrix} \right)\) and the torsion group is isomorphic to \(T^ r\), where T is the torsion subgroup of \(G_ 2/G_ 3\) and \(r=\left( \begin{matrix} \mu +q-3\\ q-3\end{matrix} \right).\) The special case \(T=0\) has been proved previously in the joint paper with W. S. Massey.
reviewed by: K. Murasugi
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 4109657
zbMATH Keywords: lower central series
zbMATH Keywords: Chen groups of the link group
zbMATH Keywords: second commutator subgroup
zbMATH Keywords: linking number
MaRDI profile type: Publication
full work available at URL: 0166-8641(89)90098-9
OpenAlex ID: W2015105204
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description: scientific article; zbMATH DE number 4109657