Extended braids and links (Q2725549)

This paper considers the notion of a singular braid. Singular braids are allowed to have a finite number of multiple points where several strands come together and then depart in a perhaps different sequence. The special case where only double points occur has been considered previously by many people since these braids have an immediate application to the Vassiliev theory of finite-type invariants of knots and links. NEWLINENEWLINENEWLINESingular braids form a monoid under stacking and the first question considered is whether they imbed into a group. This has been proved recently by the authors for the double-point case but is here proved in general as a consequence of a general theorem by one of the authors giving sufficient conditions for imbeddability of a monoid in a group. In other words, one can adjoin formal inverses to the elementary braids consisting of one multiple point to obtain a group containing the singular braid monoid. The elements of this group then have a simple diagrammatic representation. NEWLINENEWLINENEWLINEThe rest of the paper is devoted to the relation between singular braids and singular links. The analogue of the Alexander theorem -- that every singular link arises as the closure of some singular braid -- and the Markov theorem -- that any two singular braids which give rise to isotopic singular links are related by a sequence of certain explicit moves -- are proved by arguments similar to those which have been used to prove the classical theorems.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00034].