Upper bound for the alternation number of a torus knot
type: Item
instance of: scholarly article
title: Upper bound for the alternation number of a torus knot
zbMATH Open document ID: 1191.57007
author: Taizo Kanenobu
published in: Topology and its Applications
publication date: 2009-12-03T00:00:00Z
review text: \textit{A. Kawauchi} [Topology Appl. 157, No.~1, 274--279 (2010; Zbl 1188.57002)] introduced a numerical invariant for a knot or link, called the alternation number, which is the minimal number of crossing changes needed to deform the knot or link into an alternating one. In the present paper, the author gives estimations for the alternation numbers of torus knots of \(3\)-, \(4\)- and \(5\)-braids, and of some special types. In particular, the alternation numbers are determined for torus knots \(T(3, l)\), \(l \equiv 1, 2\pmod 6\) and \(T(4, 5)\), for which the author uses the inequality relating the alternation number, signature and Rasmussen \(s\)-invariant of a knot, developed by \textit{T. Abe} [J. Knot Theory Ramifications 18, No. 3, 363--379 (2009; Zbl 1180.57005)]. It is also shown that, for any positive integer \(k\), there exist infinitely many \(3\)-braid knots whose alternation number is \(k\).
reviewed by: Sangbum Cho
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 5641774
zbMATH Keywords: alternation number
zbMATH Keywords: torus knot
zbMATH Keywords: 3-braid knot
zbMATH Keywords: signature
zbMATH Keywords: Rasmussen \(s\)-invariant
MaRDI profile type: Publication
full work available at URL: j.topol.2009.04.025
OpenAlex ID: W2077552593
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DOI: 10.1016/J.TOPOL.2009.04.025
description: scientific article; zbMATH DE number 5641774