Knots with algebraic unknotting number one
type: Item
instance of: scholarly article
title: Knots with algebraic unknotting number one
zbMATH Open document ID: 0809.57004
author: Micah E. Fogel
published in: Pacific Journal of Mathematics
publication date: 1995-03-30T00:00:00Z
review text: In [Quest. Answers Gen. Topology 8, 283-292 (1990; Zbl 0704.57004)], \textit{H. Murakami} introduced the notion of an algebraic unknotting operation on Seifert matrices and defined the algebraic unknotting number \(u_ a (K)\) of a knot \(K\) by using the notion. This satisfies the inequality \(u_ a (K) \leq u(K)\), where \(u(K)\) is the unknotting number of \(K\). In the paper under review, the author gives a characterization of knots with \(u_ a (K) = 1\). Namely, he proves that a knot with Alexander polynomial \(\Delta_ K\) has \(u_ a(K) = 1\) if and only if the Alexander module is cyclic and has a generator \(g\) with \(\beta (g,g) = \pm 1/ \Delta_ K\), where \(\beta\) is the Blanchfield pairing. The only if part had been proved by Murakami. As an application, the author shows that the knot \(8_{10}\), which is thought to have unknotting number 2, has algebraic unknotting number 1. This shows that abelian methods cannot be used to show its unknotting number is actually 2.
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 562513
zbMATH Keywords: algebraic unknotting operation on Seifert matrices
zbMATH Keywords: algebraic unknotting number
zbMATH Keywords: unknotting number
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: Alexander module
zbMATH Keywords: Blanchfield pairing
reviewed by: Makoto Sakuma
MaRDI profile type: Publication
full work available at URL: pjm.1994.163.277
OpenAlex ID: W1996422982
DOI: 10.2140/PJM.1994.163.277
description: scientific article; zbMATH DE number 562513