Positive knots, closed braids and the Jones polynomial

TL;DR

This paper establishes new inequalities relating Vassiliev invariants, genus, and Jones polynomial degrees for positive knots, and demonstrates finiteness results for positive knots sharing the same polynomial invariants.

Contribution

It introduces novel inequalities for positive knots using Gauss diagram formulas and extends the Bennequin inequality, leading to finiteness results and applications to the Jones polynomial.

Findings

Finiteness of positive knots with fixed polynomial invariants.

Extended Bennequin inequality relating unknotting number and genus.

New bounds on Vassiliev invariants for positive knots.

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 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.