A remark on Alexander polynomial criterion for bi-orderability of fibered 3-manifold groups

TL;DR

This paper examines the Alexander polynomial criterion for bi-orderability in fibered 3-manifold groups, showing that twisted Alexander polynomials do not provide additional obstructions beyond the classical polynomial.

Contribution

It demonstrates that twisted Alexander polynomials do not strengthen the bi-orderability obstruction for fibered 3-manifold groups compared to the classical Alexander polynomial.

Findings

Twisted Alexander polynomials do not improve bi-orderability obstructions.

Classical Alexander polynomial remains the primary tool for this criterion.

Analysis of maximal ordered abelian quotients supports these conclusions.

Abstract

We observe that Clay-Rolfsen's obstruction of bi-orderability, which uses the classical Alexander polynomial, is not strengthened by using the twisted Alexander polynomials for finite representations unlike many known applications of the Alexander polynomial. This is shown by studying the maximal ordered abelian quotient of bi-ordered groups.