Mathematical Research Letters

Volume 8, Issue 5, September 2001 pp. 629-635.

The Bennequin number of $n$-trivial closed $n$-braids is negative

Authors: Oliver T. Dasbach and Xiao-Song Lin
Author institution: Oklahoma State University, and University of California, Riverside

Summary: 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.

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