On polynomials and surfaces of variously positive links

Abstract

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is $1$, with a similar relation for links. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomial-ribbon genus conjectures for a quasipositive knot. Then we show that the Alexander polynomial completely detects the minimal genus and fiber property of canonical Seifert surfaces associated to almost positive (and almost alternating) link diagrams.