Satellite operators as group actions on knot concordance
type: Item
instance of: scholarly article
title: Satellite operators as group actions on knot concordance
zbMATH Open document ID: 1351.57007
author: Arunima Ray
author: Christopher W. Davis
published in: Algebraic \& Geometric Topology
publication date: 2016-05-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.4632
reviewed by: Livio Liechti
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6577071
zbMATH Keywords: knot
zbMATH Keywords: satellite knot
zbMATH Keywords: knot concordance
zbMATH Keywords: group action
zbMATH Keywords: satellite operator
zbMATH Keywords: homology cylinder
MaRDI profile type: Publication
OpenAlex ID: W3102406197
arXiv ID: 1306.4632
DOI: 10.2140/AGT.2016.16.945
review text: A knot in the solid torus, called a pattern, induces a map on knots via the satellite construction. The corresponding satellite operators induced on concordance classes of knots have received broad attention. The article under review introduces a group action of strong homology cylinders (up to strong homology cobordism) on concordance classes of knots in homology spheres. The main result states that the satellite operators of strong winding number \(\pm1\) patterns are a restriction of this group action.NEWLINENEWLINEThere is a version of the result in the exotic, the topological and the \(R\)-homology category, where \(R\) is any localization of \(\mathbb{Z}\). The proof uses surgery techniques and some basic algebraic topology. Freedman's proof of the 4-dimensional topological Poincaré conjecture is used to inject the concordance classes of knots into the concordance classes of knots in homology spheres.NEWLINENEWLINEThe article under review nicely presents how a better understanding of satellite operators can be obtained from the group action described in the main theorem. For example, injectivity of the satellite operators of strong winding number \(\pm1\) patterns is deduced, a result which has previously been obtained by \textit{T. D. Cochran} and the authors [J. Topol. 7, No. 4, 948--964 (2014; Zbl 1312.57006)]. Furthermore, a criterion for the satellite operator of a strong winding number \(\pm1\) pattern to be surjective is given. In combination, infinitely many patterns (different from connected-sum patterns) with bijective satellite operators are constructed.NEWLINENEWLINEIn this context, it is important to mention a conjecture of Akbulut asserting that there exists a winding number \(\pm1\) pattern such that none of the corresponding satellite knots is slice (Problem 1.45 on [\textit{R. Kirby}, Proc. Symp. Pure Math., Vol. 32, Part 2, 273--312 (1978; Zbl 0394.57002)]). Such an example has recently been provided by Levine in the preprint \textit{Non-surj…
description: scientific article; zbMATH DE number 6577071