On the Goertiz matrix of a link
type: Item
instance of: scholarly article
title: On the Goertiz matrix of a link
zbMATH Open document ID: 0536.57003
DOI: 10.1007/BF01304208
author: L. Traldi
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173531
review text: Elaborating on a result of \textit{C. McA. Gordon} and \textit{R. A. Litherland} [Invent. Math. 47, 53-69 (1978; Zbl 0391.57004)], it is shown that the Goeritz matrix associated to a diagram of an oriented classical link \(\ell\) can be modified, by adjoining a number of entries (0 and \(\pm\) 1) on the diagonal, in such a way that (up to integral congruence) the effect of the Reidemeister moves on the resulting matrix H consists solely of the adjunction and deletion of diagonal blocks \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) If \(\Lambda ={\mathbb{Z}}[t_ 1,...,t_ m,t_ 1^{- 1},...,t_ m^{-1}]\) (where m is the multiplicity of \(\ell)\) and \(\tau\) : \(\Lambda\to {\mathbb{Z}}\) maps each \(t_ i\) to -1, the elementary ideals of H are given by \(\tau E_ k(\ell)=E_{k-1}(H)\). For \(k\in {\mathbb{Z}}\) let \(\delta_ k(\ell)=\sup \{\delta \geq 0| 2^{\delta}\quad divideseveryelementof\tau E_ k(\ell)\}\) (with \(\delta_ k(\ell)=\infty\) if \(\tau E_ k(\ell)=0)\); the sequence \((\delta_ k(\ell))\) is shown to be an invariant of the topological concordance class of the unoriented link obtained from \(\ell\), and an example is given of an infinite family of two-component links which can be distinguished by their \(\delta_ 1\) but not by their Murasugi nullities, Murasugi signatures, or Milnor \({\bar \mu}\)-invariants.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 57R90
zbMATH DE Number: 3852028
zbMATH Keywords: Goeritz matrix of an oriented classical link
zbMATH Keywords: concordance class of link
zbMATH Keywords: Milnor mu invariant
zbMATH Keywords: Reidemeister moves
zbMATH Keywords: two-component links
zbMATH Keywords: Murasugi nullities
zbMATH Keywords: Murasugi signatures
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3852028