Mathematical Research Letters
Volume 8 (2001)
The Bennequin number of $n$-trivial closed $n$-braids is negative
Pages: 629 – 635
A famous result of Bennequin states that for any braid representative of the unknot the Bennequin number is negative. We will extend this result to all $n$-trivial closed $n$-braids. This is a class of infinitely many knots closed under taking mirror images. Our proof relies on a non-standard parameterization of the HOMFLY polynomial. Another interesting corollary of this parameterization is that if all Vassiliev invariants up to degree $c$ vanish on a knot of crossing number $c$, then this knot has trivial HOMFLY polynomial.