Publications et prépublications
en liaison avec les tresses
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| Abstract:
If $W$ is a Coxeter group, one can consider the length $l_R$ on $W$ with respect
to the generating set $R$ consisting of all reflections. Let $c$ be a Coxeter
element in $W$ and let $P_c$ be the set of elements $p\in W$ such that $c$ can be
written $c = pp'$ with $l_R(c) = l_R(p)+l_R(p')$. We define the monoid $M(P_c)$ to
be the monoid generated by a set $\underline P_c$ in one to one correspondence, $p\mapsto\underline{p}$,
with $P_c$ with only relations $\underline{pp'}=\underline{p}.\underline{p'}$ whenever $p$, $p'$ and $pp'$ are in $P_c$.
We conjecture that the group of quotients of $M(P_c)$ is the Artin-Tits group
associated to $W$ and that it has a simple presentation. These
conjectures are known to be true for spherical type Artin-Tits groups. In
the present paper we prove them for Artin-Tits groups of type Ã. Moreover,
for a suitable choice of the Coxeter element we obtain a (quasi-) Garside
monoid, which allows to define normal forms on the Artin-Tits group and
to solve some questions such as to determine the centralizer of a power of
the Coxeter element in the Artin-Tits group.
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| Abstract:
We apply Murasugi-Tristram inequality to real algebraic curves of odd degree on $RP^2$ with a deep nest, i.e. a nest of the depth $k-1$ where $2k+1$ is the degree. For such curves, the ingredients of the Murasugi-Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus links due to Eisenbud and Neumann as the base of the induction and Conway's skein relation as the induction step.
In Appendix B, we give some generalization of the skein relation.
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| Abstract:
In this paper we study some combinatorial aspects of the singular Artin monoids. Firstly, we show that a singular Artin monoid $SA$ can be presented as a semidirect product of a graph monoid with its associated Artin group $A$. Such a decomposition implies that a singular Artin monoid embeds in a group. Secondly, we give a solution to the word problem for the FC type singular Artin monoids. Afterwards, we show that FC type singular Artin monoids have the FRZ property. Briefly speaking, this property says that the centralizer in $SA$ of any non-zero power of a standard singular generator $\tau_s$ coincides with the centralizer of any non-zero power of the corresponding non-singular generator $\sigma_s$. Finally, we prove Birman's conjecture, namely, that the desingularization map $\eta: SA \to \Z [A]$ is injective, for right-angled singular Artin monoids.
MSC : 20F36 |
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| Abstract:
We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing number. We also compute their Jones polynomials.
MSC : 57M25 |
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| Abstract:
We improve and shorten the argument given in (Journal of Algebra, vol.~210 (1998) pp~291--297). In particular, the fact that Artin braid groups are torsion free now follows from Garside's results almost immediately.
MSC : 20F05, 20F36, 20B40 |
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| Abstract:
We study the geodesic growth series of the braid group on three strands, B_3 := MSC : |
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| Abstract:
One of the possible generalizations of the discrete logarithm problem to arbitrary groups is the so-called conjugacy search problem (sometimes erroneously called just the conjugacy problem): given two elements a, b of a group G and the information that a^x=b for some x \in G, find at least one particular element x like that. Here a^x stands for xax^{-1}. The computational difficulty of this problem in some particular groups has been used in several group based cryptosystems. Recently, a few preprints have been in circulation that suggested various "neighbourhood search" type heuristic attacks on the conjugacy search problem. The goal of the present survey is to stress a (probably well known) fact that these heuristic attacks alone are not a threat to the security of a cryptosystem, and, more importantly, to suggest a more credible approach to assessing security of group based cryptosystems. Such an approach should be necessarily based on the concept of the average case complexity (or expected running time) of an algorithm.
These arguments support the following conclusion: although it is generally feasible to base the security of a cryptosystem on the difficulty of the conjugacy search problem, the group G itself (the "platform") has to be chosen very carefully. In particular, experimental as well as theoretical evidence collected so far makes it appear likely that braid groups are not a good choice for the platform. We also reflect on possible replacements.
MSC : 20F36, 68Q17 |
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| Abstract:
A knot in 3-space is transverse if it is nowhere tangent to the standard tight contact structure. Its transversal knot type TX is its equivalence class under transversal isotopy. Two known invariants of TX are its topological knot type X and its Bennequin number b(TX). A transversal knot type is transversally simple if these two invariants suffice to characterize it. In this paper we give several constructions which yield examples, both explicit and non-explicit, of pairs of transversal knots K, K' with the property: K and K' have the same topological knot type and the same Bennequin invariant, but are not transversally equivalent. We also found interesting examples that relate to the maximal Bennequin number for the transverse knots with a fixed topological knot type. The methods are indirect, and use the machinery which was introduced in the companion paper, "Stabilization in the braid groups-I:MTWS".
MSC : 7M50, 57M25 |
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| Abstract:
Let X,Y be closed braid representatives of an oriented link type L, where Y has minimal braid index among all closed braid representatives of L. The main result is the Markov Theorem Without Stabilization. It asserts that there is a complexity function and a finite set of templates such that (possibly after initial complexity-reducing modifications which replace X,Y with X',Y' the following holds: there is a sequence of closed braid representatives X' = X_1,X_2,...., X_r = Y' of L such that each passage X_i to X_{i+1} is strictly complexity reducing and non-increasing on braid index. The templates which define the passages X_i to X_{i+1} include 3 familiar ones: the destabilization, exchange move and flype templates, and in addition, for each braid index m>3 a finite set T(m) of new ones. The number of templates in T(m) is a non-decreasing function of m. We give examples of members of T(m), but not a complete listing. There are applications to contact geometry, which are given in a separate paper, "Stabilization in the braid groups-II:Transversal simplicity of knots".
MSC : 57M25, 57M07, 20F36 |
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| Abstract:
Garside's results and the existense of the greedy normal form for braids are shown to be true for the singular braid monoid. An analogue of the presentation of J. S. Birman, K. H. Ko and S. J. Lee for the braid group is also obtained for this monoid.
MSC : 20F36; 20F38 |
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| Abstract:
We prove that there is no functorial universal finite type invariant for braids in $\Sigma\times I$ if the genus of $\Sigma$ is positive.
MSC : 20F36, 57M27, 57N05, 55P62 |
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| Abstract:
Let $M$ be a closed oriented surface of genus $g\ge 1$, let $B_n(M)$ be the braid group of $M$ on $n$ strings, and let $SB_n(M)$ be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map $\eta: SB_n(M) \to \Z [B_n(M)]$, introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective.
MSC : 20F36, 57M27 |
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| Abstract:
Consider the ring $R:=\Q[\tau,\tau^{-1}]$ of Laurent polynomials in the variable $\tau$. The Artin's Pure Braid Groups (or Generalized Pure Braid Groups) act over $R,$ where the action of every standard generator is the multiplication by $\tau$. In this paper we consider the cohomology of such groups with coefficients in the module $R$ (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We give a sort of \textit{stability} theorem for the cohomologies of the infinite series $A$, $B$ and $D,$ finding that these cohomologies stabilize, with respect to the natural inclusion, at some number of copies of the trivial $R$-module $\Q$. We also give a formula which compute this number of copies.
MSC : 20J06, 20F36 |
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| Abstract:
Let $B_n$ be the Artin braid group on $n$ strings with standard generators $\sigma_1, ..., \sigma_{n-1}$, and let $SB_n$ be the singular braid monoid with generators $\sigma_1^{\pm 1}, ..., \sigma_{n-1}^{\pm 1},\tau_1, ..., \tau_{n-1}$. The desingularization map is the multiplicative homomorphism $\eta: SB_n \to \Bbb Z[B_n]$ defined by $\eta(\sigma_i^{\pm 1}) = \sigma_i^{\pm 1}$ and $\eta (\tau_i) = \sigma_i - \sigma_i^{-1}$, for $1 \le i \le n-1$. The purpose of the present paper is to prove Birman's conjecture, namely, that the desingularization map $\eta$ is injective.
MSC : 20F36 |
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| Abstract:
The goal of this paper is to construct examples of centralizers in the Artin braid groups requiring the number of generators quadratic in the number of strings. These examples disprove a recent conjecture of N. Franco and J. Gonzalez-Meneses.
MSC : 20F36, 57M07, 57M99 |
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| Abstract:
The cycling operation endows the super summit set $S_x$ of any element $x$ of a Garside group $G$ with the structure of a directed graph $\Gamma_x$. We establish that the subset $U_x$ of $S_x$ consisting of the circuits of $\Gamma_x$ can be used instead of $S_x$ for deciding conjugacy to $x$ in $G$, yielding a faster and more practical solution to the conjugacy problem for Garside groups. Moreover, we present a probabilistic approach to the conjugacy search problem in Garside groups. The results are likely to have implications for the security of recently proposed cryptosystems based on the hardness of problems related to the conjugacy (search) problem in braid groups.
MSC : 20F36, 20F10 |
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| Abstract:
We give a new presentation of the braid group $B$ of the complex reflection group $G(e,e,r)$ which is positive and homogeneous, and for which the generators map to reflections in the corresponding complex reflection group. We show that this presentation gives rise to a Garside structure for $B$ with Garside element a kind of generalised Coxeter element, and hence obtain solutions to the word and conjugacy problems for $B$.
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| Abstract:
We prove a conjecture due to Makanin: if a and b are elements of the Artin braid group B_n such that a^k=b^k for some nonzero integer k, then a and b are conjugate. The proof involves the Nielsen-Thurston classification of braids.
MSC : 20F36, 20F65 |
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| Abstract:
In this paper, we prove that each automorphism of a surface braid group is induced by a homeomorphism of the underlying surface, provided that this surface is a closed, connected, orientable surface of genus at least 2, and the number of strings is at least three. This result generalizes previous results for classical braid groups, mapping class groups, and Torelli groups.
MSC : 32G15, 20F38, 30F10, 57M99 |
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| Abstract:
We show that every trivial $3$-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result
extends to every Artin-Tits group of type~$I_2(m)$, but it fails to extend to braids with $4$~strands and more. The proof uses a partition of the Cayley graph and a
continuity argument.
MSC : 20F36 |
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| Abstract:
The mixed braid groups are the subgroups of Artin braid groups whose elements preserve a given partition of the base points. We prove that the centralizer of any braid
can be expressed in terms of semidirect and direct products of mixed braid groups. Then we construct a generating set of the centralizer of any braid on n strands, which
has at most k(k+1)/2 elements if n=2k, and at most k(k+3)/2 elements if n=2k+1. These bounds are shown to be sharp, by an example due to S. J. Lee. We conjecture that
the set of generators given in this paper has the smallest possible number of elements. Finally, we describe how one can compute this generating set.
MSC : 20F36, 20E07, 20F65 |
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| Abstract:
Let g be a complex, simple Lie
algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V
and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group B of type g is a deformation of the action of (a finite extension
of) W on V. The residues of D are the Casimirs of the sl(2)-subalgebras of g corresponding to its roots. The irreducibility of a subspace U of V under these implies that, for
generic values of the parameter, the braid group B acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the
weight spaces of all simple g-modules if g is sl(3) but that this is not the case if g is not isomorphic to sl(2) or sl(3). We use this to disprove a conjecture of Kwon and
Lusztig stating the irreducibility of quantum Weyl group actions of Artin's braid group B_n on the zero weight spaces of all simple U_{q}sl(n)-modules for n greater or
equal to 4. Finally, we study the irreducibility of the action of the Casimirs on the zero weight spaces of self-dual g-modules and obtain complete classification results for
g=sl(n) or g(2) and conjecturally complete results for g orthogonal or symplectic.
MSC : 20F36, 20E07, 20F65 |
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| Abstract:
Algorithm of construction of all knots, links with given number of crosses on diagram of knot, link is offered. This algorithm is based on simple proposition, that there is
a representation of knot (link) as closure of braid with n threads and length of this braid does not exceed n(4n-5)+2.
MSC : 55-04, 51P05, 70G99 |
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| Abstract:
The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in
[Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321-356]) and the classifying bundle is the first James bundle (defined in "James bundles"
ArXiv:math.AT/0301354). We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James-Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for \pi_2 of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links. MSC : 55Q40, 57M25; 57Q45, 57R15, 57R20, 57R40 |
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| Abstract:
In this paper we survey some work on representations of $B_n$ given by the induced action on a homology module of some space. One of these, called the
Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all $n$. We will outline the methods used, applying them to a
closely related representation for which the proof is slightly easier. The main tool is the Blanchfield pairing, a sesquilinear pairing between elements of relative homology.
We discuss two other applications of the Blanchfield pairing, namely a proof that the Burau representation is not faithful for large $n$, and a homological definition of the
Jones polynomial. Finally, we discuss possible applications to the representation theory of the Hecke algebra, and ultimately of the symmetric group over fields of
non-zero characteristic. Proceedings of the ICM, Beijing 2002, vol. 2, 37--46. MSC : 20F36, 20C08. |
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| Abstract:
The goal of this paper is to define a new class of objects which we call triple affine Artin groups and to relate them with Cherednik's double affine Hecke algebras. This
has as immediate consequences new and simple descriptions of double affine Weyl and Artin groups, the double affine Hecke algebras as well as the corresponding elliptic
objects. We also recover in an transparent and elementary way results of Cherednik on automorphisms of double affine Hecke algebras.
MSC : 20C08 |
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| Abstract:
We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs
naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed
graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show
that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type
invariants and quandle cocycle invariants of comtes.
MSC : 57M25, 57M15, 05C20 |
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| Abstract:
A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel
boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.
MSC : 20F36, 57M60. |
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| Abstract:
We study some aspects of the geometric representation theory of the Thompson and Neretin groups, suggested by their analogies with the diffeomorphism groups of the
circle. We prove that the Burau representation of the Artin braid groups extends to a mapping class group $A_T$ related to Thompson's group $T$ by a short exact
sequence $B_{\infty}\hookrightarrow A_T\to T$, where $B_{\infty}$ is the infinite braid group. This {\it non-commutative} extension abelianises to a central extension
$0\to \Z\to A_T/[B_{\infty},B_{\infty}]\to T\to 1$ detecting the {\it discrete} version $\bar{gv}$ of the Bott-Virasoro-Godbillon-Vey class. A morphism from the
above non-commutative extension to a reduced Pressley-Segal extension is then constructed, and the class $\bar{gv}$ is realised as a pull-back of the reduced
Pressley-Segal class. A similar program is carried out for an extension of the Neretin group related to the {\it combinatorial} version of the Bott-Virasoro-Godbillon-Vey
class.
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| Abstract:
Let G be a chordal graph, X(G) the complement of the associated complex arrangement and Gamma(G) the fundamental group of X(G). We show that Gamma(G) is a
limit of colored braid groups over the poset of simplices of G. When G = G_T is the comparability graph associated with a rooted tree T, a case recently investigated by
the first author, the result takes the following very simple form: Gamma(G_T) is a limit over T of colored braid groups.
MSC : 52C35, 55R10, 20F36, 18A30, 05C38 |
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| Abstract:
We prove by explicit construction that graph braid groups and all but three surface groups can be embedded as quasi-convex subgroups of right-angled Artin groups, and
we point out some consequences of these embedding results. We also show that every right-angled Artin group can be embedded in a pure surface braid group. On the
other hand, by generalising to right-angled Artin groups a result of Lyndon for free groups, we show that the Euler characteristic -1 surface group (given by the relation
x^2y^2=z^2) never embeds in a right-angled Artin group. (The other two surface groups which fail to embed are those of the kleinbottle and the projective
plane).
MSC : 20F36, 05C25. |
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| Abstract:
Let $(A_{S},S)$ and $(A_{T},T)$ be two Artin-Tits systems and $\varphi_p : A_S\to A_T$ an LCM-homomorphisms constructed by folding. When $T$ is
indecomposable then we prove that the quasi-centraliser of $\varphi_p(S)$ is equal to the quasi-centraliser of $T$ in $A_T^+$ ; if furthermore $T$ is of spherical type
then the same is true in $A_T$. If $T$ is decomposable and of spherical type, we prove that the quasi-centraliser of $\varphi_p(S)$ in $A_T$ remains a subgroup of the
quasi-centraliser of $T$, and we give a presentation of this subgroup.
MSC : 20F36. |
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| Abstract:
A Garside monoid $G^+$ is a monoid with a finite lattice generating set and a Garside group is the group of fractions of a Garside monoid. The familly of Garside
monoids (groups) contains the Artin monoids (groups) of spherical type. We generalise the well-known notion of parabolic submonoid (subgroup) of an Artin monoids
(groups) into that of parabolic submonoid (subgroup) of a Garside monoid (group). We also define a notion of Garside submonoid (subgroup) of a Garside monoid
(group), which is related to the notion of LCM-homomorphisms between Artin group, and prove that most of the properties extend. If $G^+$ is what we call balanced,
then the quasi-centralisers of Garside submonoids (subgroups) are described using the notion of ``ribbons''~; we introduce and investigate the category of ribbons. In the
last section we apply to Artin groups of spherical type the general resuls of the previous sections.
MSC : 20F05, 20F36, 20L05, 03G10. |
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| Abstract:
We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation
for these invariants. The algebras conjecturally give the relative contact homology of certain Legendrian tori in five-dimensional contact manifolds. We present several
computations and derive a relation between the knot invariant and the Alexander polynomial.
MSC : 57M27, 53D35, 20F36 |
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| Abstract:
We present a topological interpretation of knot and braid contact homology in degree zero, in terms of chords and skein relations. This interpretation allows us to extend
the knot invariant, among other things, to an invariant of embedded graphs and handlebodies in three-space modulo isotopy. We give a related presentation for knot
contact homology in terms of plats, including a calculation for all two-bridge knots.
MSC : 57M27, 53D35, 20F36 |
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| Abstract:
We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show
that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the
proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group. More generally, we define, for any crystallographic root system, a braid group and a parametrized braid group with parameters in a commutative ring. The parametrized braid group is expected to be isomorphic to the semi-direct product of the corresponding Steinberg group with the braid group. The first part of the paper (described above) treats the case of the root system $A_n$; in the second part, we handle the root system {$D_n$}. Other cases will be treated in the sequel. MSC : 20F36, 19Cxx, 20F55. |
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| Abstract:
We suggest a new algorithm for finding a canonical representative of a given braid, and also for the harder problem of finding a $\sigma_1$-consistent representative. We
conjecture that the algorithm is quadratic-time. We present numerical evidence for this conjecture, and prove two results: (1) The algorithm terminates in finite time. (2)
The conjecture holds in the special case of 3-string braids - in fact, we prove that the algorithm finds a minimal-lenght representative for any 3-string braid.
MSC : |
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