A refinement of the first Vassiliev invariant can distinguish the orientation of knots

TL;DR

This paper refines a known knot invariant to create a simple, effective tool that can distinguish knot orientations and detect non-invertibility, potentially advancing knot classification methods.

Contribution

It introduces a simplified refinement of the Vassiliev invariant that can distinguish knot orientations and detect non-invertibility, extending to all n-component string link satellites.

Findings

Invariant detects non-invertibility of knots

Refinement simplifies the computation of the invariant

Potential to distinguish all classical knots

Abstract

We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the non-invertibility of knots. This makes the natural generalization of our invariant for all n-component string link satellites of a framed long knot to a candidate for distinguishing all classical knots.