- Bentalha Asma
quasipositivity.
A quasipositive
braid is a product of conjuguates of generators.It was shown that a necessary
condition for an isotopy type of algebraic curve to be realizable is that
a certain braid must be quasipositive.This condition becomes sufficient if
we deal with a larger class of curves, the pseudoholomorphics ones.Theses
questions are a part of the 16th Hilbert problem. In this paper, we give
a computational algorithm which decides if a braid is quasipositive or not.We
also get interesting algebraic results for all Artin groups.
- Bruguières Alain
Tresses pures, doubles tressages et invariants
topologiques.'
De manière surprenante, il apparait que les invariants d'entrelacs en rubans
(et d'enchevêtrements plus généraux, dits 'turbans') associsé a une
categorie tressée (plus précisément : en rubans) ne dependent en fait pas du
tressage, mais seulement du double tressage (ou du 'twist'). Nous
introduisons une axiomatique des doubles tressages, ou 'enlacements', de
sorte que la categorie des tresses pures est la categorie 'enlacée
universelle. Une categorie monoidale enlacee avec duaux C permet de
construire des invariants de 'stringlinks'. Moyennant certaines conditions
supplémentaires, C définit des invariants d'entrelacs en rubans et de
'turbans'.
Il se trouve que la presentation de type Kirby de la categorie des
cobordismes entre surfaces fermees, utilisée pour construire la TQFT de
Turaev, utilise des enchevêtrements qui sont tous 'turbans'. Les categories
enlacées devraient donc donner lieu à de nouveaux invariants topologiques en
basse dimension.
- Dehornoy Patrick
Geometric presentations of Thompson's groups
Starting from the observation that Thompson's groups F and V are the geometry groups respectively of associativity, and of associativity together with commutativity, we deduce new presentations of these groups. These presentations naturally lead to introducing a new subgroup of V and a torsion free
extension of the latter. We prove that these new groups are the geometry groups of associativity together with the law x(yz) = y(xz), and of associativity together with a twisted version of this law involving self-distributivity, respectively.
- Digne François
The word problem in Artin groups of affine type A.
The standard presentation of infinite Coxeter groups does not
provide normal forms for the elements of the associated Artin group.
Using a dual presentation analogous to the Birman-Ko-Lee presentation of
the braid group we get a nice monoid whose group of fractions is the
affine Artin group of type An. This monoid has Garside
properties and, so, gives normal forms for the elements and a solution to some centralizer
problems.
- Eliahou Shalom
Sur le diamêtre du graphe des rotations
L'exposé porte sur un théorème et une conjecture
de Sleator, Tarjan et Thurston concernant le diamêtre du graphe
des rotations des arbres binaires à n sommets internes.
Je montrerai comment traduire le problème en termes de permutations,
et décrirai une famille de paires d'arbres binaires qui,
expérimentalement, semblent être diamétralement oppos&eavute;s.
- Funar Luis
Extensions of the Thompson group by the infinite braid group.
This is joint work with Christophe Kapoudjian,
reporting on recent progress concerning some
group which links Thompson groups and
the mapping class groups, following
Greenberg, Sergiescu, Dehornoy, Brin.
Our main question is whether the universal
mapping class group of infinite genus is finitely presented.
In a previous work by the same authors we answered
affirmatively the case of the universal mapping class
group of genus zero.
At an intermediary stage between the genus zero and the
infinite genus there is a new extension of the
Thompson group T by the infinite braid group
(which is split over Thompson group F) which is distinct from
the groups considered by Brin and Dehornoy, though as
it seems a simplified version of the Greenberg-Sergiescu
acyclic extension. It has an elementary definition as
a group of homeomorphisms acting on a infinitely punctured
surface.
Our main result is that this group (and its various
generalizations) is finitely presented.
- Kassel Christian
From Sturmian morphisms to the braid group B(4) .
(joint work with Christophe Reutenauer, UQAM, Montreal)
The group Aut(F2) of automorphisms of the free group F2 on two generators a and b contains the monoid St of positive automorphisms consisting of those automorphisms that send a and b onto words involving only positive powers of a and b. Positive automorphisms preserve an important class of infinite words in a and b,
called the Sturmian sequences. (Sturmian sequences occur in various fields
such as number theory, ergodic theory, dynamical systems, computer science,
crystallography.) We are interested in a submonoid St0 of index two in St, which we call the special Sturmian monoid. We show that this submonoid of Aut(F2) can be realized naturally as a submonoid of the braid group B_4 of braids with four strands. We use this to relate the groups B_4 and Aut(F2) in a precise way. We also give a new criterion for two words u and v to form a basis of F2.
- Kapoudjian Christophe
Extensions des groupes de Thompson.
On discutera les diverses relations connues à ce jour entre
les groupes de Thompson (V ou T) et les groupes de tresses,
ou apparentés aux tresses. On développera notamment le cas de
l'extension de V par un groupe de "quasi-tresses", résultant
d'une action explicite de V sur un espace de configuration de
points.
- Kourline Vitaly An explicit description of compressed logarithms
of all Drinfeld associators.
We have completely realized the programme announced at the previous
braid meeting in Autran (June 1-4, 2004).
A Drinfeld associator is a key tool in computing
the Kontsevich integral of knots.
Roughly speaking, a Drinfeld associator is a series in
two non-commuting variables, satisfying highly non-trivial
algebraic equations: hexagons and pentagon.
The logarithm of a Drinfeld associator lives in the Lie algebra
L3 generated by the symbols a,b,c modulo [a,b]=[b,c]=[c,a].
We have found the images of the logarithm of all Drinfeld associators
in the quotient L3/[[L3,L3],[L3,L3]].
- Marco Miguel
Equivalency of braid monodromy of line arrangements.
The goal of this talk is to apply braid monodomy technics to the topological
study of line arrangements. We construct two ordered arrangements conjugated
in Q(\sqrt 5) having different braid monodromies (the notion of equivalent
braid monodromies will be explained). Finite representation of pure braid
groups in the main tool to check the non-equivalency of braid monodromies.
- Wiest Bert
On the complexity of braids.
We define a measure of complexity of a braid which is
natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion
of the length of a braid by introducing generators which are Garside-like half-twists involving
strings i through j.
The geometrical complexity is some natural measure of the
amount of distortion of the n times punctured disk caused
by a homeomorphism. Our main result is that the two notions
of complexity are comparable. We also show how to recover a
braid from its curve diagram in polynomial time, and we prove
that every braid has a sigma1-consistent representative
of linearly bounded length. The key role in the proofs is
played by a technique introduced by Agol, Hass, and Thurston.
Eddy Godelle
Dernière mise à jour : jeudi 9 septembre 2004.
|