Braid Monoid Research Articles

Quasi-commuting extensions of groups

We prove that every monoid obtained from a group by adding new generators submitted to relations of a certain simple type embeds in a group of fractions. This applies in particular to the singular braid monoid.

On the monoid of singular braids

We show that the singular braid monoid on 3 strands embeds in the integral group ring of the braid group proving a special case of a conjecture originating from J.S. Birman (1993). The statement is proven by developing general results connected to the conjecture, including an extension of Artin's braiding theorem to a special class of singular braids. We prove two corollaries which are also in strong connection with the conjecture and discuss the difficulties arising when the number of strands is greater than 3.

A Note on Words in Braid Monoids

The purpose of this note is to give a self-contained (apart from simple facts about Coxeter groups) and we hope a bit shorter and more understandable account of some results of [C1,C2] on normal forms of braids which are themselves based on the papers [D1,T]. In particular a motivation was to give a proof of Proposition 5.1 that we use in [B-M]. Some proofs and results from Section 2 onwards seem to be new. I thank several people for improvements from earlier versions of the manuscript: M. Geck for pointing out some errors, F. Digne for pointing out that some results don't need the braid group to be of finite type, and J.-Y. Hée for suggesting (and providing) further improvements in that direction.

THE SINGULAR BRAID MONOID EMBEDS IN A GROUP

We prove that the singular braid monoid of [2] and [5] embeds in a group. This group has a geometric interpretation as singular braids with two type of singularities which cancel.

ON VASSILIEV INVARIANTS FOR LINKS IN HANDLEBODIES

The notion of Vassiliev algebra in case of handlebodies is developed. Analogues of the results of J. Baez for links in handlebodies are proved. This implies that there is a one-to-one correspondence between the special class of finite type invariants of links in hanlebodies and the homogeneous Markov traces on Vassiliev algebras. This approach uses the singular braid monoid and braid group in a handlebody and the generalizations of the theorem of J. Alexander and the theorem of A. A. Markov for singular links and braids and the relative version of Markov's theorem.

The wellordering on positive braids

This paper studies Artin's braid monoids using combinatorial methods. More precisely, we investigate the linear ordering defined by Dehornoy. Laver has proved that the restriction of this ordering to positive braids is a wellordering. In order to study this order, we develop a natural wellordering ⪡ on the free monoid on infinitely many generators by representing words as trees. Our construction leads to a (new) normal form for (positive) braids. Our main result is that the restriction of our order ⪡ to the normal braid words coincides with the restriction of Dehornoy's ordering to positive braids. Our method gives an alternative proof of Laver's result using purely combinatorial arguments and gives the order type, namely ω ω ω .