Positive knots, closed braids and the Jones polynomial
Abstract
Description
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.
36 pages; revision 7 Sep 99: added discussion of Brandt-Lickorish-Millett-Ho polynomial, Jones polynomial of the untwisted Whitehead double and signature as well as some Vassiliev invariant inequalites for positive braids. Reorganized several sections; revision 1 Mar 01: minor changes
36 pages; revision 7 Sep 99: added discussion of Brandt-Lickorish-Millett-Ho polynomial, Jones polynomial of the untwisted Whitehead double and signature as well as some Vassiliev invariant inequalites for positive braids. Reorganized several sections; revision 1 Mar 01: minor changes