The Bennequin number of \(n\)-trivial closed \(n\)-braids is negative (Q1598417)

A knot is \(n\)-trivial if all Vassiliev invariants up to degree \(n\) vanish on it. Given an \(n\)-braid \(b\) with exponent sum \(e\), \(e-n\) is the Bennequin number of that \(n\)-braid. In this article the authors prove that the Bennequin number of \(n\)-trivial closed \(n\)-braids is negative. They also prove that if all Vassiliev invariants up to degree \(c\) vanish for a knot with crossing number \(c\), then it has a trivial HOMFLY polynomial. These theorems are shown using a framed version of the HOMFLY polynomial \(\Gamma (b) \in \mathcal{Z} [\mu\) ,\(z ]\) defined for a braid \(b\in B_{n}\). The polynomial can be converted into the traditional HOMFLY polynomial \(P(v,z)\) as follows: For a knot \(\widehat{b}\) that is the closure of the braid \(b \in B_{n}\) substitute \(\mu = (1-v^{2})/2\) into \((1-\mu z)^{(e-n+1)/2} \Gamma (b) (\mu ,z)\) to obtain \(P(b) (v,z)\).