Twisted Alexander invariants detect trivial links

TL;DR

This paper demonstrates that twisted Alexander invariants can detect trivial links, the unknot, the Hopf link, the trefoil, and the figure-8 knot, providing a powerful tool for link classification.

Contribution

It extends the detection capabilities of twisted Alexander polynomials to include the trefoil, figure-8 knot, and split links, improving link detection methods.

Findings

Twisted Alexander polynomials detect the trefoil and figure-8 knot.

They also detect whether a link is split.

Twisted Alexander modules detect trivial links.

Abstract

It follows from earlier work of Silver-Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links.