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 , 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.
Cite this article
Alexander Stoimenow, On polynomials and surfaces of variously positive links. J. Eur. Math. Soc. 7 (2005), no. 4, pp. 477–509DOI 10.4171/JEMS/36