A new condition on the Jones polynomial of a fibered positive link

TL;DR

This paper establishes a new upper bound on the maximum degree of the Jones polynomial for fibered positive links, aiding in classifying knots up to 12 crossings as positive or not.

Contribution

It introduces a novel upper bound relating degrees of the Jones polynomial for fibered positive links, enabling classification of knots with up to 12 crossings.

Findings

Maximum degree of Jones polynomial is at most four times the minimum degree for fibered positive knots.

Seven knots with unknown positivity status are shown to be non-positive.

The classification aligns with independent results by Stoimenow.

Abstract

We give a new upper bound on the maximum degree of the Jones polynomial of a fibered positive link. In particular, we prove that the maximum degree of the Jones polynomial of a fibered positive knot is at most four times the minimum degree. Using this result, we can complete the classification of all knots of crossing number $\leq 12$ as positive or not positive, by showing that the seven remaining knots for which positivity was unknown are not positive. That classification was also done independently at around the same time by Stoimenow.