Augmented group systems and \(n\)-knots
type: Item
instance of: scholarly article
title: Augmented group systems and \(n\)-knots
zbMATH Open document ID: 0793.57012
DOI: 10.1007/BF01445122
author: Daniel S. Silver
published in: Mathematische Annalen
publication date: 1994-08-25T00:00:00Z
full work available at URL: https://eudml.org/doc/165102
review text: 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 \(\Gamma_ \infty\).
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 537270
zbMATH Keywords: \(n\)-knot
zbMATH Keywords: satellite construction
zbMATH Keywords: satellite \(n\)-knots
zbMATH Keywords: incompressible Seifert surface
zbMATH Keywords: augmented group system
zbMATH Keywords: finitely presented group
MaRDI profile type: Publication
cites work: Amalgamated products and finitely presented groups
cites work: Almost finitely presented soluble groups
cites work: On maximal fibred submanifolds of a knot exterior
cites work: Groups of higher dimensional satellite knots
cites work: On the failure of the Howson property for a group with a single defining relation
cites work: Q4145882
cites work: Q4135772
cites work: Examples of 3-knots with no minimal Seifert manifolds
cites work: Growth rates of \(n\)-knots
cites work: Q3941589
cites work: On irreducible 3-manifolds which are sufficiently large
OpenAlex ID: W2000984168
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description: scientific article; zbMATH DE number 537270