On the generalization of union of knots
type: Item
instance of: scholarly article
title: On the generalization of union of knots
zbMATH Open document ID: 0532.57004
author: Ken Sakai
published in: Hokkaido Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: A generalization of the sum of two knots \(k_ 1\) and \(k_ 2\) in \(S^ 3\) is defined as follows: \(k_ 1+_ n k_ 2\) consists of the class of knots having a representative which is the usual sum of \(k_ 1\) and \(k_ 2\) modified by ''clasping'' a pair of subarcs of the sum along an arc which meets the separating 2-sphere of the sum in n points; for \(n=1\) this is the Kinoshita-Terasaka union of knots. The point of the definition is to attack the well-known conjecture that knots with unknotting number 1 are prime; indeed, this conjecture would follow from the truth of the conjecture that the trivial knot cannot be obtained as a member of \(k_ 1+_ n k_ 2\) for nontrivial knots \(k_ 1\) and \(k_ 2\). The latter conjecture is known to be true for \(n=0\) and \(n=1\); this paper provides a proof for the case \(n=2\) following a cut-and-paste argument similar to \textit{H. Terasaka}'s proof of the case \(n=1\) [Osaka Math. J. 12, 113-144 (1960; Zbl 0118.392)].
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 3844646
zbMATH Keywords: sum of knots
zbMATH Keywords: prime knot
zbMATH Keywords: unknotting number
zbMATH Keywords: Kinoshita-Terasaka union of knots
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381757773
OpenAlex ID: W2047564052
DOI: 10.14492/HOKMJ/1381757773
description: scientific article; zbMATH DE number 3844646