Publications et prépublications
en liaison avec les tresses
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| Abstract:
A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with
infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe
computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the complete
Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3-manifolds, and for each of them found finite covers which are Haken. There are
interesting and unexplained patternsin the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring
the virtual Haken property under Dehn filling. In particular, we show that if a 3-manifold with torus boundary has a Seifert fibered Dehn filling with
hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the
figure-8 knot is virtually Haken.
MSC : 57M. |
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| Abstract:
In this paper we give an algorithm for solving a main case of the conjugacy problem in the braid groups. We also prove that half-twists satisfy a
special root property which allows us to reduce the solution for the conjugacy problem in half-twists into the free group. Using this algorithm one is
able to check conjugacy of a given braid to one of E. Artin's generators in any power, and compute its root. Moreover, the braid element which
conjugates a given half-twist to one of E. Artin's generators in any power can be restored. The result is applicable to calculations of braid monodromy
of branch curves and verification of Hurwitz equivalence of braid monodromy factorizations, which are essential in order to determine braid
monodromy type of algebraic surfaces and symplectic 4-manifolds.
MSC : |
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| Abstract:
There are recent cryptographic protocols that are based on Multiple Simultaneous Conjugacy Problems in braid groups. We improve an algorithm,
due to Sang Jin Lee and Eonkyung Lee, to solve these problems, by applying a method developed by the author and Nuno Franco, originally intended
to solve the Conjugacy Search Problem in braid groups.
MSC : 20F36, 20F10, 94A60. |
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| Abstract:
From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $\rho: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on
$n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation shall be called the Artin type representation associated to
the pair $(H,h)$. The goal of the present paper is to study different aspects of these representations.
Firstly, we associate to each braid $\beta$ a group $\Gamma_{(H,h)} (\beta)$ and prove that the operator $\Gamma_{(H,h)}$ determines a group
invariant of oriented links. We then give a topological construction of the Artin type representations and of the link invariant $\Gamma_{(H,h)}$, and
we prove that the Artin type representations are faithful. The last part of the paper is dedicated to the study of some semidirect products $G
\rtimes_\rho B_n$, where $\rho: B_n \to \Aut(G)$ is an Artin type representation. In particular, we show that $G \rtimes_\rho B_n$ is a Garside group
if $H$ is a Garside group and $h$ is a Garside element of $H$.
MSC : 20F36, 57M27. |
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| Abstract:
We show that each of the Artin groups of type $B_n$ and $D_n$ can be presented as a semidirect product $F \rtimes {\cal B}_n$, where $F$
is a free group and ${\cal B}_n$ is the $n$-string braid group. We explain how these semidirect product structures arise quite naturally from
fibrations, and observe that, in each case, the action of the braid group ${\cal B}_n$ on the free group $F$ is classical. We prove that, for each
of the semidirect products, the group of automorphisms which leave invariant the normal subgroup $F$ is small: namely, ${\rm
Out}(A(B_n),F)$ has order 2, and ${\rm Out}(A(D_n),F)$ has order 4 if $n$ is even and 2 if $n$ is odd. It is known that the Artin group of
type $D_n$ may be viewed as an index 2 subgroup of the $n$-string braid group over some orbifold. Applying the same techniques, we show
that this latter group has an outer automorphism group of order 2. Finally, we determine the automorphism groups of all Artin groups or rank 2.
MSC : |
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| Abstract:
A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation
for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain
fibration. Our goal in this paper is to understand this monodromy representation using standard tools from the theory of hyperplane arrangements. In
particular, we prove that the representation of Bigelow and Krammer is a sub-representation of the monodromy representation which we consider, but
that it cannot be the whole representation.
MSC : 20F36, 52C35, 52C30, 32S22. |
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| Abstract:
We determine the epimorphisms $A \to W$ from the Artin group $A$ of type $\Gamma$ onto the Coxeter group $W$ of type $\Gamma$, in case
$\Gamma$ is an irreducible Coxeter graph of spherical type, and we prove that the kernel of the standard epimorphism is a characteristic subgroup of
$A$. This generalizes an over 50 years old result of Artin.
MSC : 20F36, 20F55. |
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| Abstract:
Birman, Ko & Lee have introduced a new monoid---with an explicit presentation---whose group of fractions is the n-strand braid group. Building
on a new approach by Digne, Michel & himself, Bessis has defined a dual braid monoid for every finite Coxeter type Artin-Tits group extending
the type A case. Here, we give an explicit presentation for this dual braid monoid in the case of types B and D, and we study the combinatorics of the
underlying Garside structures.
MSC : 20F36, 20F05, 20F10. |
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| Abstract:
We study random walks on the three-strand braid group $B_3$, and in particular compute the drift, or average topological complexity of a random
braid, as well as the probability of trivial entanglement. These results involve the study of magnetic random walks on hyperbolic graphs (hyperbolic
Harper-Hofstadter problem), what enables to build a representation of $B_3$ as generalized magnetic translation operators for the problem of a
quantum particle on the hyperbolic plane.
Nucl. Phys. B 621 (2002), 675-688. MSC : |
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| Abstract:
We exhibit 3-generator Artin groups which have finite 2-dimensional Eilenberg-Mac Lane spaces, but which do not act properly discontinuously by
semi-simple isometries on a 2-dimensional CAT(0) complex. We prove that infinitely many of these groups are the fundamental groups of compact,
non-positively curved 3-complexes. These examples show that the geometric dimension of a CAT(0) group may be strictly less than its CAT(0)
dimension.
MSC : 20F36, 20F67. |
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| Abstract:
In the decade since the discovery that Artin's braid groups enjoy a left-invariant linear ordering, several quite different approaches have been applied
to understand this phenomenon. This book is an account of those approaches, involving self-distributive algebra, uniform finite trees, combinatorial
group theory, mapping class groups, laminations and hyperbolic geometry.
MSC : |
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| Abstract:
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit
skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman and Kuperberg's $G_2$ quantum invariants. Our method
consists in the study of Markov traces on a suitable tower of quotients of cubic Hecke algebras extending Jones approach.
MSC : 16S15, 57M27, 81R15. |
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| Abstract:
We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee. This
algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus link groups among
others).
MSC : 20F36, 20F10. |
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