Knot groups whose bases are abelian
type: Item
instance of: scholarly article
title: Knot groups whose bases are abelian
zbMATH Open document ID: 0591.57009
DOI: 10.1016/0022-4049(86)90050-2
author: Katsuyuki Yoshikawa
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Any knot group may be obtained as an HNN extension with finitely presentable base and associated subgroups. Here n-knot groups which are HNN extensions of finitely generated abelian groups are studied. (All knot groups with abelian commutator subgroup have this structure, by a theorem of \textit{R. Bieri} and \textit{R. Strebel} [Comment. Math. Helv. 53, 258-278 (1978; Zbl 0373.20035)].) Theorem 3.2 characterizes such n-knot groups when \(n\geq 3\); the result is analogous to the characterization of metabelian n-knot groups given by \textit{J. Levine} and \textit{C. Weber} [Lect. Notes Math. 685, 243-273 (1978; Zbl 0404.57019)] and includes it as a special case. In Section 4 it is shown that when \(n=2\) either the commutator subgroup is finite cyclic or \({\mathbb{Z}}^ 3\) or the base is \({\mathbb{Z}}\oplus ({\mathbb{Z}}/\lambda {\mathbb{Z}})\) for some \(\lambda >0\). In all known examples of the latter type \(\lambda =1\). The final section completes the description of abelian commutator subgroups of n-knot groups (n\(\geq 3)\) of rank 1 or 2 begun by \textit{J. C. Hausmann} and \textit{M. Kervaire} [Enseign. Math., II. Sér. 24, 111-123 (1978; Zbl 0414.57012)].
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 3949315
zbMATH Keywords: HNN extensions of finitely generated abelian groups
zbMATH Keywords: abelian commutator subgroups of n-knot groups
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90050-2
OpenAlex ID: W2045784422
cites work: Q4129638
cites work: Q5565097
cites work: The group \(G'/G''\) of a knot group \(G\)
cites work: Summands of Separable Abelian Groups: Dedicated to Paul Turán on his 60th Birthday
cites work: An Exact Sequence Calculation for the Second Homotopy of a Knot
cites work: Sous-groupes dérivés des groupes de nœuds
cites work: Embedding Theorems for Groups
cites work: High dimensional knot groups which are not two-knot groups
cites work: Les nœuds de dimensions supérieures
cites work: Q5345629
cites work: Unknotting spheres in codimension two
cites work: Knot Modules. I
cites work: Q4189858
cites work: Q4145882
cites work: Knot Groups. Annals of Mathematics Studies. (AM-56)
cites work: On the commutator subgroup of a knot group
cites work: Q3961021
cites work: On 2-Knot Groups with Abelian Commutator Subgroups
Recommended article: Q4027116
Recommended article: Partially abelian representations of knot groups
Recommended article: Abelian normal subgroups of two-knot groups
Recommended article: On groups associated to a knot
Recommended article: Q4280842
Recommended article: Knot complements and groups
Recommended article: Knot groups and residual nilpotence
Recommended article: ON GENERALIZED KNOT GROUPS
Recommended article: Q5403186
Recommended article: Knot-theoretic ternary groups
description: scientific article; zbMATH DE number 3949315