On the S-equivalence of some general sets of matrices
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.
scientific article; zbMATH DE number 3846484