On Singular Braids

In Vassiliev theory, there is a natural monoid homo-morphism 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 homomor-phism is injective on braids with up to three singularities and that Birman's conjecture is equivalent to that singular braids are dis-tinguishable by Vassiliev braid invariants. The braid group may be deened by generators 1 ; : : :; n?1 and relations j k = k j if jj ? kj > 1; j k j = k j k if jj ? kj = 1: There is a geometric description of braids as strings in space, see 4], which allows us to study knots and links via braid groups. Many well known knot invariants, such as the Alexander polynomial and Jones polynomial, can be obtained from representations of braid groups. Playing a role similar to that of braids in knot theory where every knot is represented by a closed braid, singular braids are very useful in studying Vassiliev's knot invariants because every singular knot can be represented by a closed singular braid. We refer to 1] and 5] for more details. The strings of a singular braid are allowed to intersect, but only in discrete double points, where they deene a unique common tangent plane. As with braids, one identiies singular braids which are isotopic. The isotopy need not preserve levels, but one must move only through singular braids which have monotone strings, and the tangent plane deened by the two strings at a double point is required to vary smoothly 1 2 JUN ZHU (in 3-space) during any isotopy of the singular braids. Multiplication is by concatenation as with braids; a braid with one or more singularities is not invertible. Thus singular braids constitute a monoid. Let SB n denote the monoid of singular braids on n strings; generators for SB n are shown in Figure 1. j j j j+1 j j+1 Figure 1: Generators for singular braids In addition to the braid generators 1 ; ; n?1 we have the corresponding elementary singular braids 1 ; ; n?1. Together these generate SB n. A proof is sketched in 5] that, with the invertibility of the i , and the braid relations given above, the following additional relations serve to deene SB n as a monoid: Notice that the string labels involving …