Higher-order linking forms for knots (Q851099)

\textit{T. D. Cochran} [Algebr. Geom. Topol. 4, 347--398 (2004; Zbl 1063.57011)] introduced the higher Alexander modules \(\mathcal{A}_{n}\) of a knot, which are obtained from the derived series \(\{G^{(n)}\}\) of a knot group \(G\) by considering a factor group \(G^{(n+1)}/G^{(n+2)}\) as a right module over \(G/G^{(n+1)}\) via conjugation. In the paper under review the author shows that Cochran's ``genetic infection'' technique can be used to construct, for each \(n\), examples of knots which have isomorphic higher Alexander modules \(\mathcal{A}_{m}\) for every \(m\leq n\) and isomorphic linking forms defined on \(\mathcal{A}_{m}\) for every \(m<n\), but nonisomorphic linking forms defined on \(\mathcal{A}_{n}\). The corresponding result in the ``classical'' case (\(n=0\)) is due to \textit{H. F. Trotter} [Invent. Math. 20, 173--207 (1973; Zbl 0269.15009)] and \textit{C. Kearton} [Trans. Am. Math. Soc. 202, 141--160 (1975; Zbl 0305.57016)].