On Vassiliev invariants of braid groups of the sphere

Abstract

We construct a universal Vassiliev invariant for braid groups of the sphere and the mapping class groups of the sphere with $n$ punctures. The case of a sphere is different from the classical braid groups or braids of oriented surfaces of genus strictly greater than zero, since Vassiliev invariants in a group without 2-torsion do not distinguish elements of braid group of a sphere.