IUMJ

Title: On some restrictions to the values of the Jones polynomial

Authors: Alexander Stoimenow

Issue: Volume 54 (2005), Issue 2, 557-574

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 $k \le 4$, 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.