Positive knots, closed braids and the Jones polynomial

Details Positive knots, closed braids and the Jones polynomial Positive knots, closed braids and the Jones polynomial

1998-05-18

arxiv.org/abs/math/9805078v3

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2(2) (2003), 237--285.

Mathematics

Geometric Topology

36 pages; revision 7 Sep 99: added discussion of Brandt-Lickorish-Millett-Ho polynomial, Jones polynomial of the untwisted W

Scientific paper

Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no positive knot with trivial polynomial. We also discuss an extension of the Bennequin inequality, showing that the unknotting number of a positive knot not less than its genus, which recovers some recent unknotting number results of A'Campo, Kawamura and Tanaka, and give applications to the Jones polynomial of a positive knot.

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