The unknotting numbers of \(10_{139}\) and \(10_{152}\) are 4
type: Item
instance of: scholarly article
title: The unknotting numbers of \(10_{139}\) and \(10_{152}\) are 4
zbMATH Open document ID: 0909.57003
author: Tomomi Kawamura
published in: Osaka Journal of Mathematics
publication date: 1999-04-08T00:00:00Z
review text: In the 1960s, Milnor conjectured that the unknotting number of any algebraic knot would be equal to the genus of the Milnor fiber. In the 1990s, Kronheimer and Mrowka proved this conjecture using gauge theory. In particular, the unknotting number of \((p,q)\)-torus knot is \(((p-1) (q-1))/2\). In 1995, \textit{D. Auckly} [Surgery, knots, and the Seiberg-Witten equations, Lectures for the 1995 TGRCI-W, Preprint] gave an alternative proof of this conjecture for certain torus knots using Seiberg-Witten theory. In this paper we shall compute unknotting numbers of certain knots using Seiberg-Witten theory. By extending the argument in [D. Auckly, loc. cit.], the author obtains the following Theorem. Let \(K\) be an oriented knot in \(S^3\times \{1\}\) and \(L_{m,n}\) the link in \(S^3\times \{0\}\). If there is a compact connected oriented surface \(\widehat F\) in \(S^3\times [0,1]\) such that \(\partial \widehat F=L_{m,n} \sqcup K\), then \(u(K)\geq (m-1)(n-1)- g(\widehat F)\) where \(f(\widehat F)\) denotes the genus of \(\widehat F\). By means of that Theorem the author determines the unknotting numbers of certain knots. In particular, in the present paper, he shows the following result: Theorem. (1) The unknotting number of the knot \(10_{139}\) is 4. (2) The unknotting number of the knot \(10_{152}\) is 4.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 1216669
zbMATH Keywords: unknotting number
zbMATH Keywords: Seiberg-Witten theory
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1216669