Linking numbers and Chen groups
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.
scientific article; zbMATH DE number 4109657