Vassiliev invariants of quasipositive knots

Abstract

Quasipositive knots are transverse intersections of complex plane curves with the standard sphere $S^3 \subset {\mathbb C}^2$. It is known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants: for any oriented knot $K$ and any natural number $n$ there exists a quasipositive knot $Q$ whose Vassiliev invariants of order less than or equal to $n$ coincide with those of $K$.