Positive knots, closed braids and the Jones polynomial

Alexander Stoimenow

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 2, page 237-285
  • ISSN: 0391-173X

Abstract

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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.

How to cite

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Stoimenow, Alexander. "Positive knots, closed braids and the Jones polynomial." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.2 (2003): 237-285. <http://eudml.org/doc/84502>.

@article{Stoimenow2003,
abstract = {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.},
author = {Stoimenow, Alexander},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {237-285},
publisher = {Scuola normale superiore},
title = {Positive knots, closed braids and the Jones polynomial},
url = {http://eudml.org/doc/84502},
volume = {2},
year = {2003},
}

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
PB - Scuola normale superiore
VL - 2
IS - 2
SP - 237
EP - 285
AB - 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.
LA - eng
UR - http://eudml.org/doc/84502
ER -

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