Singular Braids and Markov's Theorem

Abstract

In this article we will give a new version of Markov's Theorem, treating singular braids and knots in the sense of Vassiliev's theory of knot invariants. The classical version of Markov's Theorem states that two closed braids represent the same link if and only if the braids are related by a sequence of algebraic operations, known as Markov's moves. Birman has published the first rigorous proof of this regular version in 1975, [Birl] using elementary techniques. Our proof uses a suitable version of these techniques in order to reduce the singular case to the regular case. Birman's proof then completes ours. Some technical points will only be sketched. Full details can be found in [Gem].