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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 is 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.
@article{ASNSP_2003_5_2_2_237_0,
author = {Stoimenow, Alexander},
title = {Positive knots, closed braids and the {Jones} polynomial},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {237--285},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 2},
number = {2},
year = {2003},
mrnumber = {2004964},
zbl = {1170.57300},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_2_237_0/}
}
TY - JOUR AU - Stoimenow, Alexander TI - Positive knots, closed braids and the Jones polynomial JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2003 SP - 237 EP - 285 VL - 2 IS - 2 PB - Scuola normale superiore UR - http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_2_237_0/ LA - en ID - ASNSP_2003_5_2_2_237_0 ER -
%0 Journal Article %A Stoimenow, Alexander %T Positive knots, closed braids and the Jones polynomial %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2003 %P 237-285 %V 2 %N 2 %I Scuola normale superiore %U http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_2_237_0/ %G en %F ASNSP_2003_5_2_2_237_0
Stoimenow, Alexander. Positive knots, closed braids and the Jones polynomial. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 2, pp. 237-285. http://geodesic.mathdoc.fr/item/ASNSP_2003_5_2_2_237_0/