Singular Artin monoids and the Vassiliev homomorphism
by
Noelle Antony
University of Sydney

Baez and Birman introduced the singular braid monoid on n+1 strings, denoted by SB_n+1, which Birman uses in understanding knot invariants. In effect, the positive singular braid monoid on n+1 strings is the type A_n case of an infinite class of monoids, the positive singular Artin monoids, which were studied by Corran as a motivating example of a class of homogeneously presented monoids. In 1993 Birman conjectured (and Paris has recently confirmed) that SB_n+1 embeds in CB_n+1, the group algebra of the braid group, under a map known as the Vassiliev homomorphism or desingularisation map. It is curious that this map extends to a homomorphism from any singular Artin monoid to the group algebra of its corresponding Artin group; a fact which was observed by Corran. The question of the faithfulness of the Vassiliev homomorphism applied to singular Artin monoids would thus seem to naturally arise; although we do not solve it we address some results regarding the problem using purely combinatorial methods.

Date received: September 20, 2003