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.
Publication Source (Journal or Book title)
Mathematical Research Letters
Dasbach, O., & Xiao-Song, L. (2001). The Bennequin number of n-trivial closed n-braids is negative. Mathematical Research Letters, 8 (5-6), 629-635. https://doi.org/10.4310/MRL.2001.v8.n5.a4