Knot mutation: 4-genus of knots and algebraic concordance
type: Item
instance of: scholarly article
title: Knot mutation: 4-genus of knots and algebraic concordance
zbMATH Open document ID: 1103.57005
author: Se-Goo Kim
author: Charles Livingston
published in: Pacific Journal of Mathematics
publication date: 2006-07-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0310100
review text: The 4-genus of a knot \(K\), \(g_4(K)\), is the least genus of an embedded surface bounded by \(K\) in the 4-ball. One of the results of this paper is the following: Theorem. For every pair of nonnegative integers \(m\) and \(n\), there is a knot \(K\) with mutant \(K^*\) satisfying \(g_4(K)=m\) and \(g_4(K^*)=n\). This is shown by examples which are constructed using \textit{C. Kearton}'s observation [Proc. Am. Math. Soc. 105, 206--208 (1989; Zbl 0675.57005)]. Next, the authors develop a crossing change formula for the algebraic concordance class of a knot, and show that it is presented by mutation. Finally, the authors apply the crossing change formula to give a new proof of \textit{D. D. Long}'s theorem [Math. Proc. Camb. Philos. Soc. 95, 309--312 (1984; Zbl 0547.57006)] that strongly positive amphicheiral knots are algebraically slice.
reviewed by: Hiroshi Goda
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 5039118
zbMATH Keywords: mutation
zbMATH Keywords: knot concordance
zbMATH Keywords: amphicheiral
zbMATH Keywords: 4-genus
zbMATH Keywords: knot genus
MaRDI profile type: Publication
OpenAlex ID: W2101496508
arXiv ID: math/0310100
DOI: 10.2140/PJM.2005.220.87
description: scientific article