The Alexander polynomials of immersed concordant links
type: Item
instance of: scholarly article
title: The Alexander polynomials of immersed concordant links
zbMATH Open document ID: 1315.57012
author: Akio Kawauchi
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2015-01-09T00:00:00Z
review text: In this low-dimensional topology article, the author relates the 4-dimensional clasp distance of two links to their beta-distance and their torsion Alexander polynomials. The main result unifies results of Kanenobu and the author. Let \(L\) and \(L'\) be two links in the 3-sphere \(S^3\) with \(r\geq2\) components. The 4-dimensional clasp distance \(c^4(L,L')\) is given by minimizing the number of transversal double points of \(r\) immersed annuli in \(S^3\times[0,1]\) that fill \(L\times\{0\}\cup L'\times\{1\}\). Let \(H_L\) be the link \(\Lambda\)-module \(H_L\), where \(\Lambda\) denotes the ring of Laurent polynomials with integer coefficients in \(r\) indeterminates. The 4-dimensional clasp distance is bounded from above by the Gordian distance \(u(L,L')\). The beta-rank \(\beta(L)\) is defined to be the \(\Lambda\)-rank of \(H_L\), the (torsion) Alexander polynomial \(\Delta_L\) (\(\Delta_L^T\)) is the characteristic polynomial of (the torsion of) \(H_L\), and the beta-distance \(\beta(L,L')\) is defined to be \(|\beta(L')-\beta(L)|\). Note that \(\Delta_L=0\) if \(\beta(L)>0\) and \(\Delta_L=\Delta_L^T\) if \(\beta(L)=0\). The main result of this article establishes that \[ \beta(L,L')\leq c^4(L,L')\leq u(L,L'), \] for all links \(L,L'\) with the same number of components \(r\geq 2\). Furthermore, if \(\beta(L,L')=c^4(L,L')\), then there exists \(f\in\Lambda\) such that \(\Delta_L^T\Delta_{L'}^T\) equals \(g(L,L')ff^*\) up to multiplication by units, where \(f^*\in\Lambda\) is the element obtained from \(f\) by replacing the indeterminates by their inverses. Here, \(g(L,L')\in\Lambda\) is the so called gap factor and can be described explicitly. This generalizes the \textit{Concordance Theorem} by the author [Osaka J. Math. 15, 151--159 (1978; Zbl 0401.57013)]. As a corollary, the author obtains a \textit{4-dimensional Clasp Number One Theorem}, which in turn recovers the \textit{Unlinking Number One Theorem} of \textit{Kanenobu} (Annonced in an abstract of t…
reviewed by: Peter Feller
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q60
zbMATH DE Number: 6384878
zbMATH Keywords: link
zbMATH Keywords: immersed concordance
zbMATH Keywords: 4-dimensional clasp distance
zbMATH Keywords: Gordian distance
zbMATH Keywords: beta-distance
zbMATH Keywords: torsion Alexander polynomial
MaRDI profile type: Publication
full work available at URL: s40590-014-0023-9
OpenAlex ID: W2079213016
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DOI: 10.1007/S40590-014-0023-9
description: scientific article; zbMATH DE number 6384878