A second order algebraic knot concordance group
type: Item
instance of: scholarly article
title: A second order algebraic knot concordance group
zbMATH Open document ID: 1244.57023
author: Mark Powell
published in: Algebraic \& Geometric Topology
publication date: 2012-05-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.5645
reviewed by: Jonathan A. Hillman
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6035491
zbMATH Keywords: concordance
zbMATH Keywords: COT filtration
zbMATH Keywords: knot
zbMATH Keywords: Poincaré triad
MaRDI profile type: Publication
OpenAlex ID: W2139255712
arXiv ID: 1203.5645
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DOI: 10.2140/AGT.2012.12.685
review text: The classical invariant of knot concordance is the Witt class of the Blanchfield pairing, which is defined in terms of Poincaré duality in the infinite cyclic cover. In higher dimensions this is a complete invariant, but in the classical case there are secondary invariants, such as the Casson-Gordon invariants, which detect non-slice knots with trivial Witt class. Cochran, Orr and Teichner have defined geometric filtrations of the knot concordance group \(\mathcal{C}_1\) whose highest non-trivial quotient is the algebraic concordance group \(\mathcal{AC}_1\). (Here and below ``concordance'' means ``locally flat TOP concordance''.) This paper presents a new algebraic invariant which is a single-stage invariant, is defined in terms of metabelian covers and refines the Witt invariant.NEWLINENEWLINEThe first part of the paper involves defining the invariant. Let \(K\) be a knot in \(S^3\), with exterior \(X\) and group \(\pi=\pi_1(X)\). A pair of parallel meridians in \(\partial{X}\) decomposes \(X\) as a triad \({(X;\partial_1X,\partial_2X;S^1\times{S^0})}\), where \(\partial_1X\cong\partial_2X\cong{S^1\times{D^1}}\). This may be viewed as a relative homology cobordism from \(\partial_1X\) to \(\partial_2X\). A handle decomposition of this cobordism determines an ``algebraic Poincaré triad'' over \(\mathbb{Z}[\pi/\pi'']\). One of the difficulties to be overcome is that the quotient \(\pi/\pi''\) depends on the knot \(K\); the abelianization \(\pi/\pi'\cong\mathbb{Z}\) is the only universal quotient of knot groups. The author successfully defines an equivalence relation on such triads which leads to an algebraically defined group \(\mathcal{AC}_2\). (However this group has not yet been calculated explicitly!)NEWLINENEWLINEIt is then shown that \(1.5\)-solvable knots have trivial image in \(\mathcal{AC}_2\) and that the Witt class map from \(\mathcal{C}_1\) to \(\mathcal{AC}_1\) factors as \(\mathcal{C}_1\to\mathcal{C}_1/\mathcal{F}_{1.5}\to\mathcal{AC}_2\to\mathcal{AC}_…
description: scientific article; zbMATH DE number 6035491