On the S-equivalence of some general sets of matrices
type: Item
instance of: scholarly article
title: On the S-equivalence of some general sets of matrices
zbMATH Open document ID: 0533.57011
DOI: 10.1216/RMJ-1983-13-3-541
author: Patrick W. Keef
published in: Rocky Mountain Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: This paper studies the concept of S-equivalence in a more general setting than previously done. S-equivalence is a relation among square matrices generated by congruence and certain specific types of enlargements. The main invariant of the S-equivalence class of a matrix A over a domain R is the \(R[t,t^{-1}]\)-module with presentation matrix \(tA-A^ T\), carrying on its torsion submodule a pairing into \(R(t)/R[t,t^ 1]\), defined from A. This module with pairing in the Seifert-system of A. These notions arise naturally in knot theory - A is a Seifert matrix of the knot, and satisfies the condition: \(A-A^ T\) is unimodular. The Seifert system is the homology of the infinite cyclic covering of the complement with the Blanchfield pairing. In general, the author defines A to be knot-like if \(\det(tA-A^ T)\) is a primitive polynomial over R. The main result is that, when R is a principal ideal domain, two knot- like matrices are S-equivalent if and only if their Seifert systems are isomorphic; moreover, when the matrices are unimodular, S-equivalence is the same as congruence. This result was first proved by Trotter, in the context of Seifert matrices of knots. The techniques used here are an extension and simplification of Trotter's.
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 11E16
zbMATH DE Number: 3846484
zbMATH Keywords: S-equivalence
zbMATH Keywords: square matrices
zbMATH Keywords: Seifert matrix
zbMATH Keywords: knot
zbMATH Keywords: Seifert system
zbMATH Keywords: homology
zbMATH Keywords: infinite cyclic covering
zbMATH Keywords: pairing
zbMATH Keywords: knot-like matrices
reviewed by: Jerome Levine
MaRDI profile type: Publication
full work available at URL: rmj-1983-13-3-541
OpenAlex ID: W2014893147
description: scientific article; zbMATH DE number 3846484