The stable 4-genus of knots
type: Item
instance of: scholarly article
title: The stable 4-genus of knots
zbMATH Open document ID: 1213.57015
author: Charles Livingston
published in: Algebraic \& Geometric Topology
publication date: 2010-11-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.3054
review text: The 4-genus \(g_4(K)\) of a knot \(K\) in the 3-sphere is the minimal genus of a compact orientable surface in the 4-disk bounded by \(K\). The author introduces the stable 4-genus \(g_{\text{st}}(K)\) of \(K\) as the limit of \(g_4(nK)/n\), for \(n\) tending to \(\infty\), where \(nK\) is the \(n\)-fold knot sum of \(K\). As the 4-genus, the stable 4-genus is bounded below by \(\frac{1}{2}\) of the absolute value of the knot signature, is subadditive, and leads to a seminorm on the rational knot concordance group \(\mathcal C _{\mathbb Q}\). The unit ball with respect to this seminorm is investigated on 2-dimensional subspaces of \(\mathcal C _{\mathbb Q}\), spanned by torus knots, using the Ozsváth-Szabó invariant, the Khovanov-Rasmussen invariant, and the Tristram-Levine signature. The stable 4-genus is not necessarily an integer: By a construction using Casson-Gordon invariants, it is proved that the value of the stable 4-genus can be arbitrarily close to \(\frac{1}{2}\).
reviewed by: Dieter Erle
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N70
Mathematics Subject Classification ID: 57Q60
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 5809277
zbMATH Keywords: 4-genus of a knot
zbMATH Keywords: knot concordance group
zbMATH Keywords: knot signature
zbMATH Keywords: Casson-Gordon invariant
zbMATH Keywords: Ozsváth-Szabó invariant
zbMATH Keywords: Khovanov-Rasmussen invariant
zbMATH Keywords: torus knot
zbMATH Keywords: algebraically slice knot
MaRDI profile type: Publication
OpenAlex ID: W2964147452
arXiv ID: 0904.3054
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DOI: 10.2140/AGT.2010.10.2191
description: scientific article