The singular braid group
by
Mercedes Jordan-Santana
CIMAT (Mexico)

Let C denote the complex plane and let I denote the unit interval. A geometric braid is a collection of n disjoint strands in C ×I where the k-th strand runs from (k, 0) to some point (h, 1), monotonically in the second variable and k, h ∈ {0, 1, ..., n}. Two geometric braids are equivalent if there exists an ambient isotopy, that fixes the end points of the braids and takes one braid to the other. The notation for this set is B_n. We define a multiplication between two braids by concatenation and with this operation we have that B_n is a group.

Singular braids SB_n are defined in the same way as classical braids but we allow two strings to intersect transversally. The multiplication is also defined by concatenation but singular braids do not have an inverse, this is because applying an isotopy to singular braids does not undo the intersection points, and SB_n is not a group. Nevertheless it is possible to create a group SG_n which contains the singular braids: Fenn, Kenman and Rourke found a way to do so by coloring the singularities either in black or in white.

The main goal of this talk is to show that this group has no torsion. I will explain carefully all the definitions including: classical braid, singular braid, bands in a braid, etc. The proof of the free torsion will be combinatorial and very geometric.

Date received: May 13, 2008