Braids
(14 May - 13 Jul 2007)
~ Abstracts ~
Braids - definitions and braid
groups
Dale Rolfsen, University of British Columbia, Canada
This will be an introduction to the braid groups and their basic algebraic properties. One reason they are so useful and fascinating is that the braid groups can be defined in many different ways, some very geometric and others more purely algebraic. We will discuss a half-dozen equivalent definitions of these groups, and point out how each point of view reveals certain aspects of this important family of groups. Students wishing to prepare for this tutorial in advance, may download the following notes by the author, which accompanied a similar minicourse in 2006: http://www.math.ubc.ca/~rolfsen/papers/brasil/BrasilBraid.pdf
Configuration spaces and
robotics
Robert Ghrist, university of Illinois, USA
Braids are intimately related to configuration spaces of points. These configuration spaces give a useful model of autonomous agents (or robots) in an environment. Problems of relevance to engineering systems (e.g., motion planning, coordination, cooperation, assembly) are directly related to topological and geometric properties of configuration spaces, including their braid groups. This tutorial series will detail this correspondence, and explore several novel examples of configuration spaces relevant to applications in robotics. No familiarity with robotics will be assumed.
Simplicial objects and homotopy
groups
Jie Wu, National University of Singapore
Simplicial objects are combinatorial models for spaces. One can study homology and homotopy groups using simplicial models. On the other hand, one can study simplicial structure for many interesting objects in many areas of mathematics. It is possible to establish important connections between algebraic topology and other areas of mathematics using the simplicial techniques. For instance, there is a simplicial structure on the sequence of the braid groups by removing-doubling the strands of braids. From establishing such a simplicial structure on braids, one gets a surprising connection between the braid groups and the general homotopy groups of spheres.
Magnetohydrodynamics
Mitchell Berger, University College London, UK
My lectures will cover several applications of braid
theory in astrophsysics, fluid dynamics, and dynamical
systems. The lectures begin with an introduction to
magnetohydrodynamics (MHD), and move on to descriptions of
twisted and braided magnetic field lines in the solar
atmosphere and in fusion energy devices. There will be
material on mixing theory, vortex dynamics in superfluids,
and braided trajectories in Hamiltonian systems. A brief
tutorial on computer visualization of curves in three
dimensions will also be presented.
An algorithm to compute Seifert
Matrices from braids
Julia Collins, University of Edinburgh, UK
A Seifert surface of a knot K is an oriented
surface in S3 which has K as its
boundary. A corresponding Seifert matrix has as its entries
the linking numbers of a set of homology generators of the
surface. Thus a Seifert matrix encodes essential information
about the structure of a knot and, unsurprisingly, can be
used to define powerful invariants, such as the Alexander
polynomial. Algorithms that allow computers to find Seifert
matrices have never before been developed from the
perspective of braids, yet working from the braid
representation of a knot makes such computation easy. The
program that I have developed may also shed light on old
questions about the genus of knots, for example, why it is
that for some knots Seifert’s algorithm does not produce a
minimal genus surface. It is designed to work alongside the
program SeifertView (designed by Jarke J. van Wijk and Arjeh
Cohen) to provide users with a theoretical background to the
surfaces that the program helps to visualise.
Combinatorial description of
homotpy groups of wedge of spheres
Hao Zhao, Nankai University, China
In this paper, we give a combinatorial description of the homotopy groups of wedge of spheres in various dimensions, which generalizes J. Wu’s results on the homotopy groups of wedge of 2-spheres. In particular, the higher homotopy groups of spheres are given as the centers of certain combinatorially described groups with special generators and relations.
Reduction systems and ultra summit
sets of reducible braids
Sang Jin Lee, Konkuk University, Korea
Let D_n be the n-punctured disc in the complex plane such
that the punctures are on the real line. An essential curve
system in D_n is said to be standard if each component is
isotopic to a round circle centered at the real line. Note
that it is easy to decide whether a given braid has a
standard reduction system, but reduction systems of
reducible braids are in general very complicated.
In the talk, we show that for some class of reducible
braids, each element in the ultra summit set has a standard
reduction system, hence finding a reduction system is as
easy as finding an element in the ultra summit set.
Singular Hecke algebras, Markov
traces, and link invariants
Luis Paris, Université de Bourgogne, France
A singular braid is a braid which admits finitely many transversal double-points. The isotopy classes of singular braids on n strands form a monoid (and not a group) called the singular braid monoid and denoted by SBn. Let K be the field of rational functions on a variable q. We define the singular Hecke algebra H(SBn) to be the quotient of the monoid algebra K[SBn] by the so-called Hecke relations: "si^2 = (q-1) si +q", i=1,...,n-1, where s1,...,s(n-1) are the standard generators of the braid group. Following the same approach as Jones for the non-singular Hecke algebras, we define the notion of a Markov trace on the singular Hecke algebras, and show that a Markov trace determines an invariant for singular links. Our main result is that the Markov traces form a graduated vector space, TR, where the dth subspace, TRd, in the graduation, is of dimension d+1. The space TR0 is spanned by the Oceanu trace, and, for d>0, TRd is the space of traces defined on braids with d singular points. Thanks to this result, we can define a universal Markov trace which gives rise to a universal HOMFLY-type invariant for singular links. This invariant turns to be a Laurent polynomial on 4 variables which can be computed by means of generalized skein relations.
The Leray spectral sequence and the
cohomology of configuration spaces
Cristhian Emmanuel Garay López, Centro de Investigación y
Estudios Avanzados del Instituto, Mexico
The computation of the cohomology of the configuration space F(M,k) for a general manifold M is an important open problem. In this talk we will outline the construction of the Leray spectral sequence for a mapping, and then restrict to the natural inclusion of the configuration space of F(M,k) into M^k. In this way we will derive a spectral sequence, which converges to the cohomology of F(M,k).
Free differential calculus and
representations
Peng Zheng, Zhejiang University, China
Free differential calculus can be used to define a number of interesting matrix representations of a free group of finite rank and of various subgroups of the automorphism group of a free group. The Burau representation of Artin's braid group and the Gassner representation of the pure braid group are of particular importance.
Twisted Reidemeister torsion for
twist knots
Huynh Quang Vu, Vietnam National University, Vietnam
This is a joint work with Jerome Dubois, and Yoshikazu Yamaguchi. We give an explicit formula for the SL(2,C)-twisted Reidemeister torsion in the cases of twist knots. For hyperbolic twist knots, we also prove that the twisted Reidemeister torsion at the holonomy representation can be expressed as a rational function evaluated at the cusp shape of the knot.
Orderability of braids
Zhixian Zhu, Zhejiang University, China
This will be an introduction to the orderability of braid groups. So far there are at least six different approaches to the ordering of braids. I will introduce some combinatorial approaches found in the paper "Why are braids orderable?" by Dehornoy-Dynnikov-Rolfsen-Wiest.
Rational homotopy of the mapping
space
Urtzi Buijs Martin, Universidad de Malaga, Spain
Via the Bousfield-Gugenheim functor, and starting from
the Brown-Szczarba model of a function space, we give a
functorial framework to describe basic objects and maps
concerning the rational homotopy of function spaces and its
path components.
Then, we give a complete description of the homotopy Lie
algebra of the components of the free and pointed mapping
space in terms of derivations giving an explicit formula for
the Whitehead product in terms of such derivations and
obtain important consequences from it.
Braid groups cryptography
David Garber, Holon Institute of Technology, Israel
In the last decade, a number of public key cryptosystems
based on combinatorial group theoretic problems in braid
groups have been proposed. Our tutorial is aimed at
presenting these cryptosystems
and some known attacks on them.
We begin with some basic facts on braid groups and on the
Graside normal form of its elements.
We then present some known algorithms for solving the word
problem in the braid group.
After that, we explain and demonstrate the known public-key
cryptosystems based on the braid group (mainly, the
Anshel-Anshel-Goldfeld key exchange, and Diffie-Hellman type
key exchange, but also a cryptosytem based on the root
problem and a cryptosytem based on the shifted conjugacy
problem).
We then discuss some of the known attacks on these
cryptosystems.
No background in cryptography is assumed.
Length-based cryptanalysis of
the braid group and some applications
David Garber, Holon Institute of Technology, Israel
An important problem in combinatorial group theory is:
Given a system of equations in a finitely generated group,
find (in an efficient manner) a solution to this system of
equations. This problem generalizes many problems of
combinatorial group theory (the conjugacy problem, the
membership problem, etc.).
In the last decade, several public-key cryptosystems where
suggested, which rely on the difficulty of this problem in
the braid group.
We will present an efficient algorithmic way for finding a
small ordered list of elements in the subgroup which
contains a solution to the equation with a significant
probability. In many cases, the solution will be the first
in this list. This approach actually shows the vulnerability
of all mentioned cryptosystems. This is a joint work with S.
Kaplan, M. Teicher, B. Tsaban and U. Vishne.
If time permits, we will describe a recent length-based
attack of Ruinskiy, Shamir and Tsaban on a cryptosystem of
Shpilrain and Ushakov which is based on Thompson’s group. It
seems that their
generalized algorithms would be useful in testing the
security
of any future cryptosystem based on combinatorial group
theoretic problems.
On global and braid index
Olubunmi Abidemi Fadipe-Joseph, University of Ilorin,
Nigeria
We investigate the global index of a subfactor. The
braid index of some knots using the trace invariant for
special angles are also determined.
Topological complexity of formal
spaces
Aniceto Murillo Mas, Universidad De Malaga, Spain
After remarking how hard it is to compute the
topological complexity of a motion planning algorithm for a
given configuration space, we calculate it for the class of
formal spaces, i.e., spaces whose rational homotopy type
depend only on its rational cohomology algebra.
On hopf algebras associated to groups
and quandles
Maria Guadalupe Castillo Perez, Bonn Universität,
Germany
I will describe certain Hopf algebras H(G) associated to
groups or to quandles G, namely the Tensor algebra with a
quandle-shuffle product and the usual diagonal. On these
Hopf algebras there exist certain endomorphisms solving the
Yang Baxter equation. One of my goals is to find knot and
link invariants with the help of these solutions.
Furthermore, these algebras H(G) are differential graded
algebras; in the special case of G being the symmetric
group; their homology is related to the homology of moduli
spaces of Riemann surfaces. The other goal is to use these
Hopf algebras to study the homology of moduli spaces of
Riemann surfaces.
Preliminaries in topology and
algebra
E-Jay Ng, National University of Singapore
This tutorial will be a brief overview of the basic
concepts in algebraic topology, including homotopy theory,
the fundamental group, covering spaces, homology and
cohomology. It is not meant to be a detailed exposition and
proof of the results, but a refresher on the basic
definitions, examples, and important theorems. Some
knowledge of basic group theory such as the notion of a
group homomorphism, cosets, normal subgroups and quotient
groups will be assumed.
The braid groups of the projective
plane
Laura Rocio González Ramírez, Centro de Investigación
y Estudios Avanzados del Instituto, Mexico
We recall the definition of braid groups for surfaces
and explain some of their basic properties and their
relation with the ordinary braid groups, for surfaces other
than a 2-sphere or the projective plane. In the case of the
projective plane, we will describe some of the results of
Van Buskirk, and of Goncalves and Guaschi, about the
structure of the pure and full braid groups, their torsion
elements, etc.
String links and orderability
Ekaterina Yurasovskaya, The University of British
Columbia, Canada
I shall discuss classification of links up to link-homotopy
as solved in 1990 by Nathan Habegger and Xiao-Song Lin. The
main tool in classification was a group of link-homotopy
classes of string links - H(k). Since then H(k) itself
became an object of interest in low-dimensional topology. If
time permits, I shall discuss H(k) as an example of
orderable groups appearing in topology.
The integral homology ring
of the based loop space on a flag manifold
Jelena Grbic, University of Manchester, UK
I will show how to calculate the homology ring of a
special family of homogeneous spaces, flag manifolds and
their based loop spaces. Firstly, I'll explain how to
calculate the rational homology ring of the based loop space
on a flag manifold. Secondly, I'll extend that result to the
integral homology.
Patterns generated during the
transition to turbulence
Hua-Shu Dou, Temasek Laboratories, National
University of Singapore
Recently, we proposed a new theory, named as energy
gradient theory, to explain the flow instability and
transition to turbulence. The critical condition calculated
at turbulent transition determined by experiments obtains
very good agreement with the available experimental data for
parallel flows and Taylor-Couette flows. When the theory is
considered for both parallel and curved shear flows, three
important theorems have been deduced. These theorems are:
(1)Potential flow (inviscid and irrotational) is stable. (2)
Inviscid rotational (inviscid and nonzero vorticity) flow is
unstable. (3) Velocity profile with an inflectional point is
unstable when there is no work input or output to the
system, for both inviscid and viscous flows. Following these
results, it is presumed that the classical Rayleigh theorem
is incorrect which states that a necessary condition for
inviscid flow instability is the existence of an inflection
point on the velocity profile. In present study, we
demonstrate rigorously the reason why Rayleigh theorem is
incorrect and give two new theorems. (1) The disturbance
amplified in 2D inviscid flows is necessarily 3D. (2) After
the breakdown of Tollmien-Schlichting waves in 2D parallel
flows, the disturbance becomes a type of spiral waves which
proceed along the streamwise direction. Experimental data
showed that in the transition to turbulence in boundary
layer flows, under small disturbance environment, some
staggered or unstaggered “lambda” shaped pattern could be
generated; while under larger disturbance, streamwise
vortices could be generated. We show that these two types of
cell patterns are the products of flow instability. The
instability of 2D Tollmien-Schlichting waves generates 3D
disturbances in which there is a phase difference between
the velocity disturbances. Thus, a spiral waves in
streamwise direction could be produced after instability.
Therefore, it would display “lambda” shaped patterns (looked
from above) when the disturbance is small. The pattern to be
staggered or unstaggered depends on the phase difference and
the boundary conditions. When the disturbance is large, the
spiral waves are strong, and they would form streamwise
vortices via vortices merging process. It is concluded that
the instability of 2D laminar flow necessarily lead to 3D
flows. Linear instability of 2D laminar flow leads to 3D
flow but to be laminar flow. Only nonlinear instability of
2D (or 3D) laminar flow could result in turbulence. Laminar
flows in 2D could not lead to turbulence in 2D. There is no
2D turbulence in nature.
Batalin -- Vilkovisky algebras,
braid groups and free loop spaces
Miguel Xicotencatl Merino, Centrode Investigacion
Estudios Avanzados, Mexico
If V is a graded algebra, then two classical theorems of
F. Cohen and E. Getzler state that:
(i) V is a Poisson algebra if and only if V is an algebra
over the homology of the pure braid group, and
(ii) V is a Batalin -- Vilkovisky algebra if and only if V
is an algebra over the homology of the framed little disks
operad.
Some important examples are given by the homology of (i)
double loop spaces, (ii) the free loop space of manifold M
(Chas--Sullivan, Cohen--Jones) and (iii) the free loop space
of the classifying space of an orbifold X = [M / G] (Lupercio--Uribe--Xicotencatl).
Representing braids by
hypergeometric integrals
Toshitake Kohno, University of Tokyo, Japan
The purpose of this talk is to give an overview on
linear representations of braid groups arising from
hypergeometric integrals. We start from classical
hypergeometric functions due to Gauss and Appel. We explain
that an idea of uniformization of orbifolds by
hypergeometric functions can be applied to determine the
kernel of Burau and Gassner representations at some special
values. Then, we proceed to describe recent progress on the
correspondence between the monodromy representations of KZ
equation and hypergeometric integrals. We focus on
representing basis of the space of conformal blocks as
hypergeometric integrals on regularizable cycles with
twisted coefficients.
A new construction of Anick's
fibration
Stephen Theriault, University of Aberdeen, UK
After localizing at a prime p larger than 3, Anick showed
there is a homotopy fibration whose base is the loops on a
2n+1 sphere, whose fiber is a 2n-1 sphere, and whose total
space has its bottom two cells connected by a Bockstein.
This fibration ties in very tightly with Cohen, Moore, and
Neisendorfer's programme of studying the homotopy theory of
spheres and Moore spaces. However, Anick's construction was
very long and complex. We present a much more conceptual
construction, which is also valid at the prime 3.
Homotopy and cohomology of
spaces of homomorphisms
Alejandro Adem, University of British Columbia, Canada
Let Q be a discrete group, and G a Lie group. In this
talk we describe basic properties of the space of
homomorphisms Hom(Q,G), including
--cohomology calculations
--stable homotopy decompositions
--simplicial structures
This is joint work with Fred Cohen and Enrique Torres.
On the inverse braid monoid
Volodia Vershinin, Universite Montpellier II, France
Inverse braid monoid describes a structure on braids
where the number of strings is not fixed. So, some strings
among the initial ones may be deleted. In the talk we show
that many properties and objects based on braid groups may
be extended to the inverse braid monoids. Namely we prove an
inclusion into a monoid of partial monomorphisms of a free
group. This gives a solution of the word problem. Another
solution is obtained by an approach similar to that of
Garside. We give also the analogues of Artin presentation
with two generators and Sergiescu graph-presentations.
The paper is posted on the web: arXiv:0704.3002
Densely ordered braid subgroups
Dale Rolfsen, University of British Columbia, Canada
P. Dehornoy showed that the braid groups B(n) can be given a left-invariant total ordering. This ordering is discrete: every braid has a unique predecessor and successor in the order. I will discuss the seemingly paradoxical result that certain normal subgroups G of B(n), with the same ordering, are actually densely ordered: if f, g are in G, with f < g, then there is another h in G with f < h < g. Examples of such subgroups are the commutator subgroup, kernels of the Burau representation (when nontrivial), Brunnian braids, and pure braids which are link-homotopically trivial. This is joint work with Adam Clay.
Coloring n-string braids and
tangles and its application to molecular biology
Junalyn Navarra-Madsen, Texas Woman's University, USA
Fox coloring is an easy-to-visualize knot or link invariant. It can be generalized to n-string tangle classification. We will describe how we utilized coloring and tangle analysis to understand protein-DNA binding and elucidate enzymatic mechanisms.
Braids and differential equations
Robert Ghrist, University of Illinois, Urbana-Champaign,
USA
This talk will describe a topological index for pairs of braids based on a relative Morse theory on spaces of braids. This index has applications to finding solutions to parabolic partial differential equations.
Towards a polynomial solution
to the conjugacy problem in braid groups
Juan Gonzalez-Meneses, Univ. de Seville, Spain
This is a joint work with Joan S. Birman and Volker Gebhardt. We present the latest achievements in a project to find a polynomial solution to the conjugacy decision problem and the conjugacy search problem in braid groups. First, we introduce a new kind of conjugation, that we call sliding, to reduce the length of an element in a Garside group. This replaces classical cyclings and decyclings and simplifies the usual algorithm. In the case of braid groups, using this new tool toghether with the geometric decomposition of braids as periodic, reducible and pseudo-Anosov, one can show that the problem reduces to solving conjugacy search problem in polynomial time for "rigid" braids.
A polynomial-time solution to the
reducibility problem
Ki Hyoung Ko, Korea Advanced Institute of Science and
Technology, Korea
We propose an algorithm for deciding whether a given braid is pseudo-Anosov, reducible, or periodic. The algorithm is based on Garside's weighted decomposition and is polynomial-time in the word-length of an input braid. Moreover, a reduction system of circles can be found completely if the input is a certain type of reducible braids.
Lorenz links
Joan S. Birman, Columbia University, USA
Lorenz links were first studied in a paper that Bob
Williams and I wrote in 1983. They arose as the periodic
orbits in a system of ODE's that are important in the study
of chaos, and are presented naturally as closed braids. They
are of nenewed interest right now because of the work of
Etienne Ghys, who showed that Lorenz'
equations also describe the geodesic flow on the modular
surface.
In new work, Ilya Kofman and I have proved that Lorenz links
coincide as a class with certain repeated twisted torus
links. There are consequences, both ways. Braids play a
crucial role in ourn work.
Stabilizing braid groups and
mapping class groups of 3-manifolds
Nathalie Wahl, University of Chicago, USA
(joint work with Allen Hatcher) Let M be a compact,
connected 3-manifold with a fixed boundary component d_0M.
For each prime manifold P, we consider the mapping class
group of the manifold M_n^P obtained from M by taking a
connected sum with n copies of P. We prove that the ith
homology of this mapping class group is independent of n in
the range n>2i+1. When P is just a ball, the theorem holds
in any dimension and recovers homological stability for the
braid groups.
Cohomological dimension of the
Torelli group
Dan Margalit, University of Utah, USA
The Torelli group is the subgroup of the mapping class group consisting of elements which act trivially on the homology of the surface. In joint work with Mladen Bestvina and Kai-Uwe Bux, we prove that the cohomological dimension of the Torelli group for a closed surface of genus g is equal to 3g-5. This answers a question of Mess, who proved that the dimension is at least 3g-5. To prove the theorem, we introduce a new complex on which the Torelli group acts.
On the configuration space of type B
Nguyen Viet Dung, Vietnamese Academy of Science and
Technology, Vietnam
The purpose of the talk is the configuration space of type B, F_B(R^q,n). Its relation with the ordinary configuartion spaces F(R^q,n) was established via a system of maps f_{h,k}. We will discuss its homotopy groups, especially the fundamental group.
On the genericity of pseudo-Anosov
elements of mapping class groups
Bert Wiest, Université de Rennes 1, France
Everybody agrees that a "generic" element of a mapping class groups is pseudo-Anosov, but not many precise and proven results exist that give an exact sense to this intuition. In this talk, which reports on work in progress and contains no theorems, I want to propose a larger geometric framework in which to place this intuition.
Spaces of group homomorphisms and
group cohomology
Torres Giese Enrique, The University of British Columbia,
Canada
In this talk I will discuss some applications of Spaces
of Group Homomorphisms to Group Cohomology.
I will define a family of spaces that parametrize group
theoretical information of the classifying space of a group.
This is joint work with F. Cohen and A. Adem.
Braids, Twist, Writhe, and Solar
Activity
Mitchell Berger, University College London, UK
It has long been thought that braided magnetic field
lines power the heating of the solar atmosphere. The field
lines live within flux tubes which become entangled due to
motions at the solar surface. In addition to the braiding,
the field lines can be twisted within the tube. Modelling
the geometrical structure involves minimizing magnetic
energy, similar to minimizing the number of crossings. In
addition, the twist and writhe of the tubes must be computed
. New definitions of writhe will be given appropriate to
open curves.
The loop orbifold of the symmetric
product
Miguel Xicotencatl Merino, Centrode Investigacion
Estudios Avanzados, Mexico
It is well known that the Chas - Sullivan product in the
homology of the free loop space of a manifold M gives rise
to a BV-algebra. A nice way to see this is by showing that
the homology of LM is an algebra over the homology of the
framed little disks operad and, in recent work, E. Lupercio,
B. Uribe and my self have generalized this structure to the
case of the free loop space of the classifying space of an
orbifold.
In this work we use the loop orbifold of the symmetric
product to give a formula for the Poincare polynomial of the
free loop space of the Borel construction of the symmetric
product. We also show that the Chas-Sullivan orbifold
product, induces a ring structure in the homology of the
inertia orbifold of th symmetric product.
On spaces of representations
Fred R. Cohen, University of Rochester, USA
This lecture is survey of recent joint work concerning
the structure of spaces of homomorphisms Hom(pi, G) where pi
is a discrete group and G is a Lie group. The main question
in the current lecture is the structure of automorphisms of
certain surfaces and whether/how these translate to faithful
representations in Hom(pi, G).
This work is joint with A. Adem, E. Torres-Giese, S.
Prassidis, J. Lopez and M. Conder.
