4-moves and the Dabkowski-Sahi invariant for knots

TL;DR

This paper investigates the Dabkowski-Sahi 4-move invariant for links, developing computational techniques and demonstrating its limitations in distinguishing certain knots from the unknot, thus impacting the 4-move conjecture.

Contribution

The paper introduces new methods for computing the 4-move invariant and shows its inability to detect certain nontrivial knots, advancing understanding of the invariant's properties.

Findings

The invariant equals that of the unknot for several classes of knots.

The invariant cannot detect some counterexamples to the 4-move conjecture.

Techniques for computing the invariant are developed and demonstrated.

Abstract

We study the 4-move invariant \crl\ for links in the 3-sphere developed by Dabkowski and Sahi, which is defined as a quotient of the fundamental group of the link complement. We develop techniques for computing this invariant and show that for several classes of knots it is equal to the invariant for the unknot; therefore, in these cases the invariant cannot detect a counterexample to the 4-move conjecture.