# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1902.06091