Talks and lectures

Braids in Paris

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Michel Broué

Complex reflection groups and associated Braids

Gilbert Levitt

Automorphisms of free groups and train tracks

Title : Braids, self-distributivity, and Garside categories

Abstract : Recently identified by Krammer and others, the notion of a Garside category allows to revisit the connection between braidsand the self-distributive law, and to summarize it as follows: associated with the self-distributivity law is a certain leftGarside category, of which the standard Garside structure of braid groups is a projection. This approach leads in particularto a simple restatement of the Embedding Conjecture, the main open question in the area.

Title : On logarithmic knot invariant

Abstract : I would like to explain knot invariants coming from the universal R-matrix of the restricted quantum group. The restricted quantum group is a version of U_q(sl_2) generated by E, F and K as usual, where q is specialized to be N-th root of unity and extra relations E^N = F^n = 0 and K^2N = 1 are added. It is a non-semisimple finite dimensional Hopf algebra. We can construct an invariant coming from the radical part, and I would like to call it the logarithmic knot invariant since it corresponds to the logarithmic conformal field theory. I also would like to explain its relation to the colored Alexander invariant, which is a higher version of the classical Alexander invariant. As an application of logarithmic invariant, I would like to show its relation to certain three manifold. Kashaev found a relation between certain quantum knot invariant and the hyperbolic volume of the knot complement, which is now called the volume conjecture. Such relation seems to be generalized for closed hyperbolic 3-manifold by using the logarithmic invariant.

Title : A representation of the braid group from Lefschetz fibrations

Abstract : In the first part of the talk we construct a $1$-cocycle on the braid group $B_m$ by examining its action on intersection matrices of vanishing cycles in a Lefschetz fibration with $m$ critical fibers. The cocycle is injective and has the same target as the Magnus cocycle, but is not cohomologous to the latter. By reduction, it gives rise to a representation with values in $GL_m(Z[t,t^{-1}])$, unfortunately less interesting than the Burau representation. In the second part of the talk we explain our main motivation, coming from an attempt to define complex Floer homology. Joint work with D. Salamon and G. Massuyeau.

Title : Asymptotical behaviour of quantum hyperbolic invariants and the Kashaev-Jones specialization

Abstract : I will explain some geometric properties and integral formula of the leading order term of the asymptotic expansion of quantum hyperbolic invariants, discussing amazing consequences for links in the three-sphere and the relationship with the colored Jones polynomials.

Title : Chord slides and universal invariants.

Abstract : In joint work with J. E. Andersen, J. B. Meilhan, and R. C. Penner, a universal invariant was constructed for 3-manifolds obtained by surgery on links in surface with boundary S cross an interval. This invariant admits an action of the Ptolemy groupoid of S, the essence of which is captured via the elementary moves of chord slides on a linear chord diagram. In this talk, I will introduce the chord slide groupoid generated by these elementary moves and illustrate its interactions with the above mentioned universal invariant.

Title : Ring groups and wickets

Abstract : We interpret the Hilden subgroup of the mapping class group of a surface in terms of motions of "wickets" in upper-half space. This group can also be viewed as a subgroup of a braid group. We will also use the wicket viewpoint to relate Hilden's group to the "ring-group," or the group of motions of circles in 3-space. Further, we will discuss a K(pi,1)-space for the wicket group as well as some related spaces among which we show certain homotopy equivalences. Among other applications, our methods give finite presentations for several related groups. This is joint work with Allen Hatcher.

Title : On Lawrence-Krammer representations

Abstract : Lawrence-Krammer representations are linear representations of Artin groups of small type, known to be faithful in the spherical cases (or more generally when restricted to the corresponding Artin monoids). We propose to define them with no assumption on the commutative ring of scalars. This first enables us to rephrase in a simple way some known results on those objects : faithfulness criterion, classification in the spherical cases. We then establish their classification in the affine cases. Finally, we generalize the construction of "twisted" Lawrence-Krammer representations, made by F. Digne in the spherical cases, to every Artin monoid that appears as the submonoids of fixed points of an Artin monoid of small type under the action of graph automorphisms.

Title : Blocks of generic and cyclotomic Hecke algebras

Abstract : The definition of Rouquier for the families of characters of Weyl groups in terms of blocks of the Iwahori-Hecke algebra has made possible the generalization of this notion to the case of complex reflection groups. We show that the ``Rouquier blocks' of the cyclotomic Hecke algebras of a complex reflection group depend on numerical data defined by the generic Hecke algebra, which is a quotient of the group algebra of the associated braid group.

Title : On the asymptotics of 6j-symbols of U_q(sl_2)

Abstract : We will give an overview on the problem of understanding the asymptotical behaviour of 6j-symbols and relate it with outstanding conjectures in quantum topology as Witten's asymptotic expansion conjecture and the volume conjecture. More specifically, we will relate the asymptotical behaviour of the 6j-symbols appearing in the computation of Turaev-Viro invariants, to that predicted by a (suitably adapted) version of Witten's expansion.

Title : Invariants de type fini des surfaces dans R^3

Abstract : Depuis les travaux fondateurs de Fox et Milnor, il y a 50 ans, les noeuds bordants [slice knots] et les noeuds rubans [ribbon knots] sont devenus un sujet classique et bien étudié de la théorie des noeuds en dimension 3 et 4. Contrairement au polynôme d'Alexander, le polynôme de Jones ne semble pas refléter ces conditions topologiques. L'objectif de cet exposé est de présenter quelques éléments pour comprendre le polynôme de Jones des entrelacs rubans et plus généralement des surfaces rubans dans R^3. Pour cette étude du polynôme de Jones le bon point de vue est le développement en t=-1, contrairement au développement usuel en t=1. Les coefficients, commencant en degré 0 par le déterminant, ne sont pas des invariants de Vassiliev-Goussarov, par contre ils s'avèrent d'être de type fini par rapport aux changements de croisements entre rubans. Ces résultats motivent de développer la théorie des invariants de type fini pour les surfaces à bord, plongées ou immergées dans R^3. L'approche étendue aux surfaces contient tous les invariants de type fini des entrelacs, et bien plus encore comme témoigne notre exemple phare ci-dessus. L'espoir est d'ainsi réconcilier les invariants quantiques avec les surfaces en dimension 3.

Title : Twisted conjugacy in braid groups.

Abstract : Joint work with Enric Ventura. Given an automorphism f of a group, two elements a and b are said to be twisted conjugate if there exists some c such that $f(c)^{-1}ac=b$. The twisted conjugacy problem asks for an algorithm to determine whether two elements are twisted conjugate, and to find the conjugating element in the affirmative case. We present a solution for the twisted conjugacy problem in braid groups. This solution, toghether with a result by Bogopolski-Martino-Ventura, yields a solution to the usual conjugacy problem for some extensions of braid groups, in particular, for braid-by-free groups.

Title : Improved linear time inversion heuristic for the Burau representation

Abstract : Though not explicitly stated, Daan Krammer's faithfulness proof for the Lawrence-Krammer (LK) representation (rep) of $B_4$ contains an algorithm that computes the unique preimage braid of a given LK matrix directly in dual (Birman-Ko-Lee) Garside normal form. We use his ideas to develop an inversion heuristic for the Burau rep, which also computes a preimage braid of a given Burau matrix directly in dual (BKL) Garside normal form. The succes rates of this Burau inversion heuristic are significantly better than the success rates of the Hughes heuristic, the best known linear time (linear in word or canonical length) inversion heuristic for the Burau rep. Nevertheless, the success rates are far away from the self-correcting Lee-Park algorithm. But no bounds for the complexity of this algorithm are known so far, and, e.g, it seems to be to slow to be applicable for cryptanalytic purposes in braid-based cryptography.

Title : Braid monodromies and its applications in low dimensional topology

Abstract : I'd like to explain how braid monodromies appear in study of braided surfaces, 2-dimensional braid, Lefschetz fibrations, etc. in low dimensional topology. Such a braid monodromy is expressed by a system of elements of the braid group or the mapping class group (in general, elements of quandles).

Title : A Coxeter type B generalization of Thompson's group V

Abstract : We introduce a new finitely presented infinite simple group that contains Thompson's group V. It is modelled on the spherical Coxeter groups of type B in the same way as V is modelled on the symmetric groups. We compute its Schur multiplier. As a corollary, we prove it is non isomorphic to V.

Title : $L^2$-torsion invariants of a surface bundle over the circle

Abstract : The half part of this talk will be given by Takayuki Morifuji. The other half will be given by myself. $L^2$-torsion is a generalization of the classical Reidemeister-Ray-Singer torsion for unitary representations. For the regular representation, it is known that it gives the hyperbolic volume. In this talk, we introduce a series of $L^2$-torison invariants of a surface bundle over the circle by using the nilpotent tower of the surface group and explain a relation to the Magnus representation of the mapping class group. We also describe some explicit formulas for them and discuss the approximation problem for the $L2$-torsion of 3-manifolds.

Title : Conjugacy classes of symmetric braids

Abstract : We show that if two n-braids in the centralizer of a periodic braid are conjugate in the n-braid group, then they are conjugate in the centralizer of the periodic braid. This implies that some monomorphisms between Artin groups induce injective functions on the set of conjugacy classes.

Title : Relations between Homfly and Kauffman satellite invariants

Abstract : The general Homfly satellite invariant of a knot is a linear combination of invariants parametrised by a pair of partitions $lambda$ and $mu$, while the Kauffman satellite invariants depend on a single partition $lambda$. This talk gives a relation between the Homfly invariant with $mu=lambda$ and the corresponding Kauffman invariant with $lambda$, when the invariants are taken to lie in the ring ${f Z}_2[v^{pm1},s^{pm1}]$ modulo denominators $s^r-s^{-r}$. It is an extension of a result of Rudolph from 1987 dealing with the case where $|lambda|=1$.

Title : Complexity of (hyperbolic) links and knotted graphs in arbitrary 3-manifolds

Abstract : As a variation on Matveev's definition of the complexity of a 3-manifold M, one can introduce a similar notion for a pair (M,G) where G is a trivalent graph embedded in M. Some or all the components of G may have no vertices, so the case of links is included. I will review some general theorems proved concerning such a complexity and I will illustrate the results of a computer-aided census of low-complexity (hyperbolic) graphs (joint with Heard, Hodgson, and Martelli). I will also relate on work in progress with Pervova on the complexity of links. If time permits I will mention some results, also joint with Pervova, about colored Turaev-Viro invariants for links, established using the same basic machinery as that underlying complexity theory.

Title : The Cyclotomic Birman-Murakami-Wenzl Algebras and Cylindrical Tangles

Abstract : The algebraic definition of the Birman-Murakami-Wenzl (BMW) algebras uses generators and relations originally inspired by the Kauffman link invariant. They are closely connected with the Artin braid group of type A, Iwahori-Hecke algebras of the symmetric group, and many diagram algebras (e.g., they may be thought of as a deformation of the Brauer algebras). Geometrically, the BMW algebra is isomorphic to the Kauffman Tangle algebra. Its representations and cellularity have been extensively studied in the literature. These algebras also feature in the theory of quantum groups, statistical mechanics and topological quantum field theory. In view of these relationships between the BMW algebras and several objects of "type A", several authors have since naturally generalized the BMW algebras for other types of Artin groups. Motivated by knot theory associated with the Artin braid group of type B, H"aring-Oldenburg introduced the cyclotomic BMW algebras as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also called Ariki-Koike algebras). In this talk, we investigate the structure of these algebras and show they have a diagrammatic interpretation as a certain cylindrical analogue of the Kauffman tangle algebras. In particular, we provide a basis which may be explicitly described both algebraically and diagrammatically in terms of "cylindrical" tangles. This basis turns out to be cellular, in the sense of Graham and Lehrer. We also mention an application of cyclotomic BMW algebras to invariants of links in the solid torus. This talk is a presentation of the results in my Ph.D. thesis, completed end of 2007 at the University of Sydney, Australia.