Augmented group systems and \(n\)-knots. (English) Zbl 0793.57012

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 \(\bigcap_{k \in Z} t^ k \pi_ 1(S) t^{-k}\). 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 \(\Gamma_ \infty\).


57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
57M05 Fundamental group, presentations, free differential calculus
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[1] Baumslag, G., Shalen, P.B.: Amalgamated products and finitely presented groups. Comment. Math. Helv.65, 243-254 (1990) · Zbl 0711.20015
[2] Bieri, R., Strebel, R.: Almost finitely presented soluble groups. Comment. Math. Helv.53, 258-278 (1978) · Zbl 0373.20035
[3] Kakimiku, O.: On maximal fibred submanifolds of a knot exterior. Math. Ann.284, 515-528 (1989) · Zbl 0651.57007
[4] Kanenobu, T.: Groups of higher dimensional satellite knots. J. Pure Appl. Algebra28, 179-188 (1983) · Zbl 0516.57011
[5] Karrass, A., Solitar, D.: On the failure of the Howson property for a group with a single defining relation. Math. Z.108, 235-236 (1969) · Zbl 0182.03403
[6] Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. Berlin Heidelberg New York: Springer 1977 · Zbl 0368.20023
[7] Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory. New York. Dover 1976 · Zbl 0362.20023
[8] Silver, D.S.: Examples of 3-knots that do not have minimal Seifert manifolds. Math. Proc. Camb. Philos. Soc.110, 417-420 (1991) · Zbl 0742.57013
[9] Silver, D.S.: Growth rates andn-knots. Topology Appl.42, 217-230 (1991) · Zbl 0744.57013
[10] Simon, J.: Compactifications of covering spaces of compact 3-manifolds. Mich. Math. J.23, 245-256 (1976) · Zbl 0338.57003
[11] Robinson, D.J.S.: A course in the theory of groups. Berlin Heidelberg New York: Springer 1982 · Zbl 0483.20001
[12] Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math.87, 56-88 (1968) · Zbl 0157.30603
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