An estimate of the Rasmussen invariant for links and the determination for certain links
type: Item
instance of: scholarly article
title: An estimate of the Rasmussen invariant for links and the determination for certain links
zbMATH Open document ID: 1330.57008
author: Tomomi Kawamura
published in: Topology and its Applications
publication date: 2015-12-18T00:00:00Z
review text: The Rasmussen invariant is a knot concordance invariant given by \textit{J. Rasmussen} [Invent. Math. 182, No. 2, 419--447 (2010; Zbl 1211.57009)]. The Rasmussen invariant was extended to a link invariant by \textit{A. Beliakova} and \textit{S. Wehrli} [Can. J. Math. 60, No. 6, 1240--1266 (2008; Zbl 1171.57010)]. This invariant is also called the Rasmussen invariant and denoted by \(s\). In the paper under review, the author gives an estimate of the Rasmussen invariant for links. Let \(L\) be a link and \(D_{L}\) be a diagram of \(L\). Let \(D^{0+}_L\) be the diagram obtained by eliminating all negative crossings of \(D_L\) in the same manner as in the Seifert algorithm. Denote by \(l_{0}(D_{L})\) the number of the split components of the link represented by \(D_{L}^{0+}\). Then, the author proves the following: \(s(L) \geq w(D_{L})-O(D_{L})+2l_{0}(D_{L})-1\), where \(w(D_{L})\) is the writhe of \(D_{L}\) and \(O(D_{L})\) is the number of Seifert circles of \(D_{L}\) (Theorem 1.5). This result was independently given by \textit{A. Lobb} [Compos. Math. 147, No. 2, 661--668 (2011; Zbl 1220.57007)]. \textit{T. Abe} [Proc. Am. Math. Soc. 139, No. 7, 2647--2656 (2011; Zbl 1226.57004)] has shown that equality holds if \(D_{L}\) is a homogeneous diagram (cf. Remark 6.3). Moreover, the author computes the Ozsváth-Szabó \(\tau\)-invariants and the Rasmussen invariants for certain pretzel knots.
reviewed by: Keiji Tagami
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57N70
zbMATH DE Number: 6522013
zbMATH Keywords: slice Euler characteristic
zbMATH Keywords: Ozsváth-Szabó \(\tau\)-invariant
zbMATH Keywords: Rasmussen invariant
zbMATH Keywords: Bennequin inequality
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.034
OpenAlex ID: W370059333
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DOI: 10.1016/J.TOPOL.2015.05.034
description: scientific article; zbMATH DE number 6522013