On knots with trivial Alexander polynomial
type: Item
instance of: scholarly article
title: On knots with trivial Alexander polynomial
zbMATH Open document ID: 1095.57007
published in: Journal of Differential Geometry
publication date: 2006-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0206023
review text: A knot \(K\) in \(3\)-space (or, more generally, in an integral homology sphere) is said to have minimal Seifert rank if it admits a Seifert surface whose genus equals the rank of the Seifert form. In this case, the Alexander polynomial of \(K\) is trivial. The authors prove that the converse does not hold: There are knots with trivial Alexander polynomial which do not have minimal Seifert rank. The proof uses the \(2\)-loop part \(Q(K)\) of the rational version of the Kontsevich integral of \(K\), defined by the first author and \textit{A. Kricker} [Geom. Topol. 8, 115--204 (2004; Zbl 1075.57004)], and the results on the effects on \(Q(K)\) of null-moves (which are \(Y\)-clasper surgeries with null-homologous leaves in the knot complement), obtained by the first author and \textit{L. Rozansky} [Topology 43, No. 5, 1183--1210 (2004; Zbl 1052.57011)]. It is also shown that \(Q(K)\) of a knot \(K\) of minimal Seifert rank is completely determined by three finite type invariants of \(K\) of degrees \(3,5,5\). The starting point of the paper is a wrong lemma of \textit{M. H. Freedman} [Invent. Math. 68, 195--226 (1982; Zbl 0504.57016)] contradicting the main result of the paper under review. In the appendix, the authors give a direct argument for the fact that knots with trivial Alexander polynomial are topologically slice, avoiding Freedman's wrong lemma.
reviewed by: Kazuo Habiro
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 5033768
zbMATH Keywords: knot
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: Seifert form
zbMATH Keywords: Kontsevich integral
zbMATH Keywords: null move
zbMATH Keywords: clasper
Wikidata QID: Q115200669
author: Stavros Garoufalidis
author: Peter Teichner
MaRDI profile type: Publication
OpenAlex ID: W1573957391
arXiv ID: math/0206023
DOI: 10.4310/JDG/1099587731
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description: scientific article