Augmented group systems and \(n\)-knots
Let \(K \subset S^ 3\) be any 1-knot, \(X(K)\) its exterior and \(G=\pi_ 1(X(K),*)\) the group of \(K\), where \(* \in \partial X (K)\). Let \(t \in G\) denote the class of an oriented meridian on \(\partial X (K)\). Choose any incompressible Seifert surface \(S\) for \(K\) such that \(* \in S\), and define \(\mu(S)\) to be \(\bigcap_{k \in Z} t^ k \pi_ 1(S) t^{-k}\). \textit{Osamu Kakimizu} [Math. Ann. 284, No. 3, 515-528 (1989; Zbl 0651.57007)] proved that \(\mu(S)\) does not depend on the choice of \(S\), and hence this subgroup of \(G\) is an invariant of \(K\). We prove that \(\mu(S)\) is a special case of an invariant \(\Gamma_ \infty\) defined for any augmented group system \(\Gamma=(G,h,t)\) consisting of a finitely presented group \(G\), an epimorphism \(h:G \to Z\), and a distinguished element \(t \in G\) such that \(h(t)=1\). Consequently, Kakimizu's invariant can be defined for any \(n\)-knot, \(n \geq 1\). We describe a ``satellite construction'' for triples \(\Gamma\), analogous to a knot-theoretic construction, and we prove a structure theorem for the invariant of the result. We present examples of satellite \(n\)-knots that can be distinguished using \(\Gamma_ \infty\).
scientific article; zbMATH DE number 537270
scientific article