An integral invariant from the knot group
type: Item
instance of: scholarly article
title: An integral invariant from the knot group
zbMATH Open document ID: 1221.57005
author: Teruhisa Kadokami
author: Zhiqing Yang
published in: Osaka Journal of Mathematics
publication date: 2011-02-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1292854314
review text: The Nakanishi-index \(m(K)\) of a knot \(K\) in \(S^3\) is the minimal number of generators of the Alexander module of \(K\). The authors propose the name MQ-index for the invariant \(a(K)\) considered by Ma and Qiu, which is the minimal number of elements of the knot group of \(K\) that generate normally the commutator subgroup. Ma and Qiu showed that \(m(K)\leq a(K)\leq u(K)\), where \(u(K)\) is the unknotting number. The authors show that \(a(K)\leq \min\{r(K)-1, u(K)\} \leq \min\{t(K), u(K)\}\), where \(r(k)\) is the rank of the knot group and \(t(K)\) is the tunnel number. It is known that the tunnel number is not additive with respect to connected sums of knots, but this is not known for the unknotting number. The authors show that \(a(K)\) is not additive by proving that for the connected sum \(K_{p,q} \) of two torus knots \(T(2,p)\) and \(T(2,q)\) with \(3\leq p\leq |q|\), \,\(a(K_{p,q})=1 \) if and only if \(p\) and \(q\) are relatively prime, and this is true if and only if \(m(K_{p,q})=1\).
reviewed by: Wolfgang Heil
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 5848637
zbMATH Keywords: knot invariant
zbMATH Keywords: unknotting number
zbMATH Keywords: tunnel number
describes a project that uses: KnotInfo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5848637