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

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 $\le 12$ 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.