On polynomials and surfaces of variously positive links

Details On polynomials and surfaces of variously positive links On polynomials and surfaces of variously positive links

2002-02-22

arxiv.org/abs/math/0202226v3

Jour. Europ. Math. Soc. 7(4) (2005), 477--509.

Mathematics

Geometric Topology

27 pages, 11 figures; revision 26 Sep 03: added ex. 7, section 4.4 and minor corrections

Scientific paper

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. 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.

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