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.

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