Extended Braids and Links

Our aim in this paper is to introduce and study the properties of different kinds of singular braids. Our starting point being the singular braids with double points as introduced in [3] and [5], we define a much bigger class of singular braids which we call geometric singular braids. Geometric singular braids form a monoid and we give an explicit presentation for this monoid. In this paper, we consider only singular braids where the tangents of the strings around a singular point generate a plane. In general this need not be the case. Finally, geometric singular links are defined and it is shown that there is a one-to-one correspondence between geometric singular braids and geometric singular links up to Markov equivalence.