On some restrictions to the values of the Jones polynomial

Abstract

We prove that Jones polynomials of positive and almost positive knots have positive minimal degree and extend this result to an inequality for k-almost positive knots. As an application, we classify k-almost positive alternating achiral knots for k ≤ 4, and show a finiteness result for general k. Another consequence is a proof that almost positive and fibered positive links (with the obvious exceptions) are non-alternating (the latter extends the results for torus knots known from Murasugi, Jones, and Menasco-Thistlethwaite), and that if a positive knot is alternating, then all its alternating diagrams are positive.