On a conjecture of K. Murasugi
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.
scientific article; zbMATH DE number 3897836