Traldi, Lorenzo Linking numbers and Chen groups. (English) Zbl 0677.57007 Topology Appl. 31, No. 1, 55-71 (1989). 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. Reviewer: K.Murasugi Cited in 1 Document MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) Keywords:lower central series; Chen groups of the link group; second commutator subgroup; linking number PDFBibTeX XMLCite \textit{L. Traldi}, Topology Appl. 31, No. 1, 55--71 (1989; Zbl 0677.57007) Full Text: DOI References: [1] Chen, K.-T., Integration in free groups, Ann. Math., 54, 2, 147-162 (1951) · Zbl 0045.30102 [2] Chen, K.-T., Proc. Amer. Math. Soc., 3, 993 (1952) · Zbl 0049.40402 [3] Crowell, R. H., Torsion in link modules, J. Math. Mech., 14, 289-298 (1965) · Zbl 0134.43102 [4] Keng, Hua Loo, Introduction to Number Theory (1982), Springer: Springer Berlin/Heidelberg/New York · Zbl 0483.10001 [5] MacLane, S., Homology (1967), Springer: Springer Berlin/Heidelberg/New York · Zbl 0133.26502 [6] Maeda, T., Lower central series of link groups, (Doctoral Dissertation (1983), Univ. of Toronto: Univ. of Toronto Toronto, Canada) [7] Magnus, W.; Karrass, A.; Solitar, D., Combinatorial Group Theory (1966), Wiley-Interscience: Wiley-Interscience New York [8] Massey, W. S., Completion of link modules, Duke Math. J., 47, 399-420 (1980) · Zbl 0464.57001 [9] Massey, W. S.; Traldi, L., On a conjecture of K. Murasugi, Pacific J. Math., 124, 193-213 (1986) · Zbl 0563.57003 [10] Murasugi, K., On Milnor’s invariant for links. II. The Chen group, Trans. Amer. Math. Soc., 148, 41-61 (1970) · Zbl 0197.20702 [11] Traldi, L., Milnor’s invariants and the completions of link modules, Trans. Amer. Math. Soc., 284, 401-424 (1984) · Zbl 0525.57004 [12] Traldi, L.; Sakuma, M., Linking numbers and the groups of links, (Math. Sem. Notes, 11 (1983), Kobe Univ), 119-132 · Zbl 0552.57002 [13] Zariski, O.; Samuel, P., Commutative Algebra, Vol. II (1960), Van Nostrand: Van Nostrand Princeton, NJ · Zbl 0121.27801 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.