Gordian adjacency for torus knots
type: Item
instance of: scholarly article
title: Gordian adjacency for torus knots
zbMATH Open document ID: 1288.57011
author: Peter Feller
published in: Algebraic \& Geometric Topology
publication date: 2014-03-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.5248
review text: An unknotting sequence for a knot \(K\) is a sequence \(K=K_0,\ldots,K_n\) of knots such that \(K_n\) is the unknot, \(K_i\) differs from \(K_{i-1}\) by a single crossing change for each \(i\), and \(n\) is minimal among such sequences. Then \(K\) is said to have unknotting number \(n\). A second knot \(K'\) is said to be Gordian adjacent to \(K\) if it occurs in some unknotting sequence for \(K\). Note that Gordian adjacency gives a partial order on knots, and is not symmetric. This paper addresses the question of when one torus knot is Gordian adjacent to another. The first result shows that reducing one or both of the defining integers gives Gordian adjacent torus knots. The second specifies when an index 2 torus knot is Gordian adjacent to an index 3 torus knot (the index of a torus knot is the smaller of the two defining integers). Next the author considers the asymptotics of the values of torus knots that are Gordian adjacent, and the final section discusses the relationship between Gordian adjacency and another partial order called algebraic adjacency. The positive results about adjacency are shown by a combination of direct construction and studying isotopies of knots in a thickened torus. The negative results are shown by using Levine--Tristam signatures as an obstruction to Gordian adjacency.
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 14B07
zbMATH DE Number: 6272609
zbMATH Keywords: Gordian distance
zbMATH Keywords: Gordian adjacency
zbMATH Keywords: unknotting number
zbMATH Keywords: torus knots
zbMATH Keywords: plane curve singularities
zbMATH Keywords: algebraic adjacency
reviewed by: Jessica Banks
MaRDI profile type: Publication
OpenAlex ID: W3103455635
arXiv ID: 1301.5248
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DOI: 10.2140/AGT.2014.14.769
description: scientific article