On Singular Braids

In Vassiliev theory, there is a natural monoid homomorphism from n-strand singular braids to the group algebra of n-strand braid group. J. Birman conjectured that this monoid homomorphism is injective. We show that the monoid homomorphism is injective on braids with up to three singularities and that Birman's conjecture is equivalent to that singular braids are distinguishable by Vassiliev braid invariants.