**`Braids in Cortona'**

Cortona (Arezzo), Italy, June 19-22, 2002

International Conference Organized
by

Riccardo Benedetti, Carlo Petronio,
and Mario Salvetti

Abstracts

**MINI-COURSES**

Stephen
Bigelow

"Representations of braid groups"

The representation theory of the
symmetric group is well understood over a

field of characteristic zero. This
corresponds to the fact that the Hecke

algebra is well understood when
the parameter q is not a root of unity. I

will explain how to obtain representations
of the Hecke algebra by letting

the braid group act on the homology
of certain spaces. This gives new

topological insight into the mysterious
behaviour at roots of unity.

Michel
Boileau

"Non-free actions of finite groups
on S^3"

The goal of this minicourse is
to explain the proof of the following result:

a finite subgroup G of Diff^+(S^3)
which acts non-freely on S^3

is conjugated into SO(4). We will
introduce the notion of hyperbolic cone

manifold structures on the orbifold
quotient S^3/G and study their deformations

to get an elliptic structure on
S^3/G.

Dale
Rolfsen

"Braids, knots, 3-manifolds and
orderable groups"

It was discovered by Dehornoy,
about ten years ago, that the

Artin braid groups are left-orderable.
That is, there is a total ordering

of its elements which is invariant
under left-multiplication. The pure

braid groups are bi-orderable,
having an ordering invariant under

multiplication on both sides.
This minicourse will discuss these results

and the algebraic consequences
of orderability of groups. Moreover, it will

be shown that many groups which
arise in topology are orderable. For

example all knot and link groups
are left-orderable and some are

bi-orderable. We will also
discuss orderability of fundamental groups of

3-manifolds and relation with other
questions in 3-manifold theory, such

as the virtual Haken conjecture
and the existence of taut foliations.

Alexander
Varchenko

"Multidimensional hypergeometric
functions and representation theory"

The Knizhnik-Zamolodchikov and
quantum Knizhnik-Zamolodchikov equations

are differential and respectively
difference equations of

mathematical physics and representation
theory with many connections to

statistical mechanics, topology,...
The modern theory of multidimensional

hypergeometric functions is a broad
generalization of the classical theory

of the Gauss hypergeometric and
q-hypergeometric functions. It turned out

that the KZ and qKZ equations can
be realized geometrically: a class of

hypergeometric functions was distinguished
which satisfy the KZ and qKZ

equations. This fact enriches both
sides of this interaction: the

jorepresentation theory and the
theory of hypergeometric functions.

In these lectures the geometric
theory of the KZ and qKZ equations will

be discussed.

**TALKS**

Joan Birman

"Transverse knots"

A classical knot in 3-space is
said to be {\it transverse}

if at every point on the knot the
tangent vector is transverse

to the 2-planes of the standard
tight contact structure in S^3.

Its {\it transversal knot type}
is its equivalence class under

isotopy through transverse knots.
This is a sharper notion than

topological knot type. I
will discuss a theorem proved in new

joint work with William Menasco:

Theorem: There exist infinitely
many examples of pairs of transverse

knots (TK_1, TK_2) such that TK_1
and TK_2 have the same classical knot

type and the same Thurston-Bennequin
invariant, but have distinct

transversal knot types.

Daniel
Cohen

"Gauss-Manin connections for arrangements"

(joint work with Peter Orlik)

We study the Gauss-Manin connection
for the moduli space of an

arrangement of complex hyperplanes
in the cohomology of a complex rank one

local system. For an arrangement
of points in the complex line, the

moduli space is a configuration
space, and the connection is closely

related to the classical Gassner
representation of the pure braid group.

For an arrangement of hyperplanes
in general position (and certain local

systems), the Gauss-Manin connection
was determined by Aomoto and Kita.

We show how this result may be
used to determine the Gauss-Manin

connection for an arbitrary arrangement
(and certain local systems).

Fred
Cohen

"Representations of pure braid groups and
associated vector bundles"

Let G denote a discrete group together with either a real orthogonal or

complex unitary representation. Vector bundles associated to such

representations over the space K(G,1) are considered here in the

special cases for which G is the pure braid group or the fundamental

group of a K(G,1) hyperplane arrangement.

The main results in this case are as follows:

(1) The vector bundle associated to a real orthogonal representation of G

is trivial if and only it is a Spin representation.

(2) All non-trivial bundles which arise from a representation of the pure

braid group are realized through the cohomology algebra (as elucidated in the

lecture). The subgroup of the real K-theory of K(G,1) generated by

orthogonal representations is an elementary abelian 2-group determined by

the cohomology of G.

(3) Properties of some other representations are discussed.

The above is based on joint work with Alex Adem, and Dan Cohen.

(4) A bridge between further families representations of the pure braid

group, homotopy theory, knots, links, and homotopy links will be addressed.

This part is based on joint work with Jie Wu.

Corrado
De Concini

"Some facts on the cohomology of Artin and Coxeter
groups"

In this talk we shall review some results which have been

obtained in recent years together with Salvetti and also with Procesi

and Stumbo, regarding the construction of some explicit complexes

which can be used to compute the cohomology of Artin and Coxeter

groups. Some explicit calculations will be explained.

Patrick
Dehornoy

"Homology of Gaussian groups"

(joint work with Yves Lafont)

We give one (actually two) explicit
method(s) for computing

the homology of a Gaussian group,
or, more generally, of a

monoid where division is Noetherian
and least common multiples

exist when common multiples do.
We construct two exact chain

complexes, one relying on the greedy
normal form, and the

other on using some preordering
after Kobayashi.

Roger Fenn

"The birack: an invariant of (virtual)
knots and links"

(joint work with M. Jordan, L.
Kauffman, G. Wraith)

The birack is a generalisation of the rack and as such occupies a position

with respect to virtual knots and links as the rack does with classical

knots and links. It is also thought that its importance occupies a

similarly exalted position. Using a particular birack composed of

quaternions a polynomial can be defined. This work is currently very much

in progress but computations have been made which show that the above

polynomial is new.

Emmanuel
Giroux

"Open books, closed braids and
contact structures"

We will show that, on any closed
three-manifold V, isotopy classes of contact

structures are in one-to-one correspondence
with open books in V up to isotopy

and positive plumbings. We will
then discuss several applications of this result as

well as some related questions.

John
Guaschi

"Roots of the full twist in surface
braid groups"

(joint work with Daciberg Gonçalves)

Let M be a compact, connected surface
without boundary different from RP^2. We

study the roots of the full twist
braid Delta_n in the surface braid groups B_n(M) of M. If

M is not S^2 then \Delta_n possesses
a k-th root if and only if k divides n

or n-1, while if M=S^2, it possesses
a k-th root if and only if k divides

n, n-1 or n-2. We also show that
the Artin pure braid groups P_n(D^2) and the

sphere pure braid groups P_n(S^2)
admit a splitting as a direct sum.

Simon King

"Links and the geometry of triangulations
of S^3"

Let L be a link, formed by edges
of a triangulation T of S^3 with

n tetrahedra. By work of Armentrout
and Lickorish, the bridge number b(L)

is bounded from above by a linear
function of n, provided T or its dual is

shellable.

We prove by a series of examples
that in general there is no

subexponential upper bound for
b(L) in terms of L, if one drops the

shellability assumption. Thus,
there are triangulations of S^3 that are

"far" from being shellable. By
a study of the Rubinstein-Thompson

recognition algorithm of S^3, we
establish an upper bound for b(L) that is

exponential in n^2.

By the results of Armentrout and
Lickorish, the presence of links

in T^(1) with a big bridge number
implies the absence of geometric

properties of T. We prove a partial
converse: We define a numerical

invariant p(T), called "polytopality",
that can be seen as a bridge number

of the dual graph of T. We show
how to transform T into a polytopal

triangulation (i.e., a triangulation
that is isomorphic to the boundary of

a convex 4-dimensional polytope)
by inserting new vertices, whose number

is bounded from above by a quadratic
function of p(T). Thus, although

p(T) is defined in terms of topology,
it is a measure for the geometric

complexity of T.

Our results yield a conceptually
very simple recognition algorithm of S^3.

Toshitake
Kohno

"Surface braids and the moduli of
flat connections"

We construct a universal flat connection on the

configuration spaces of ordered points on a surface and

describe its holonomy representation with values in the

space of horizontal chord diagrams.

We also discuss an application to a deformation

quantization of the Poisson algebra of functions on the

moduli space of flat connections on surfaces.

Mustafa
Korkmaz

"The second homology groups of
mapping class groups of orientable surfaces"

(joint work with A. Stipsicz)

We first give an elementary computation
for the second homology

groups of mapping class groups
of closed orientable surfaces of genus at

least 4. This computation uses
only the presentation of the mapping class

group and the Hopf theorem which
gives the second homology of a group

from a given presentation. We then
use Harer's homology stability theorem

and the Hochshild-Serre spectral
sqeuence for group extensions to give a

new proof of Harer's theorem, by
extending to genus 4 case, on the second

homology groups of mapping class
groups.

Gregor
Masbaum

"Alexander-Conway polynomial, Milnor
numbers, and a new matrix-tree theorem"

(joint work with A. Vaintrob)

The lowest degree coefficient of
the Alexander-Conway

polynomial of an algebraically
split link can be expressed via

Milnor's triple linking numbers
in two different ways. One way is

via a determinantal expression
due to Levine. Using the

Alexander-Conway weight system,
we give another expression in

terms of spanning trees on a 3-graph.
The equivalence of the

two answers is explained by a new
matrix-tree theorem, relating

enumeration of spanning trees in
a 3-graph and the Pfaffian of a

certain skew-symmetric matrix associated
with it. Similar results

for the lowest degree coefficient
of the Alexander-Conway

polynomial exist if all Milnor
numbers up to a given order vanish.

Hugh Morton

"Choosing a basis for Homfly decorations"

(joint work with R. Hadji)

The Homfly polynomial of a knot
decorated by a diagram in the annulus,

such as the (2,1) cable or a Whitehead
double, depends on the diagram as

an element of the Homfly skein
of the annulus.

I shall present a basis for the
full skein of the annulus, including

reverse-string diagrams. The basis
elements Q_{\lambda,\mu} depend on

a pair of partitions \lambda and
\mu, and are eigenvectors of many

natural endomorphisms of the skein.

Specialisations of the Homfly polynomial
of a knot K decorated by

Q_{\lambda,\mu} can be identified
with quantum invariants of K coloured

by irreducible sl(N)_q modules.

Stefan
Papadima

"Braids and Koszulness"

(joint work with Alex Suciu)

We extend previously known relations,
between pure braid groups and various other

groups, associated to loop spaces
of higher configuration spaces.

Our generalization is based on
the Koszul property from homological algebra.

Luis Paris

"On a theorem of Artin"

(joint work with Arjeh M. Cohen)

Let B_n denote the braid group
on n strings, and let Sym_n denote the n-th

symmetric group. In 1947, Artin
proved that, with two exceptions for n=4, any

epimorphism of B_n onto Sym_n
is the standard epimorphism, up to

automorphism of Sym_n, and proved
that the pure braid group PB_n is a

characteristic subgroup of B_n.
The aim of this lecture is to explain Artin's

arguments, and to show how to extend
them to the other irreducible Artin

groups of spherical type

Bernard
Perron

"A new definition of the Casson
invariant"

Using a new definition of the Johnson-Morita
homomorphisms for the

mapping class group M(g,1) of a
surface of genus g with one boundary

component, we define a rational
invariant D(f) for any f in M(g,1).

For f in T(g,1) the Johnson
subgroup of M(g,1) (in fact a subgroup of the

Torelli group) ,we prove,using
the Casson surgery formula, that D(f) coincides

with the Casson invariant of the
homology sphereS(f) obtained by gluing

two handlebodies along f.

Reversing the point
of view, we can prove directly (i.e without reference to

Casson) thatD(f), for f in T(g,1),
depends only onS(f), by showing that D(f) is

invariant under Reidemeister- Singer
equivalence. The surgery formula, which

is a difficult poit in Casson version,
follows almost immediatly from the definition

D(f). This gives an independent
point of view of Casson invariant.

Claudio
Procesi

"Braid versus
symmetric group cohomology"

We discuss the problem of computing the genus of the covering of

polynomials with distinct roots by the roots, and describe a universal

obstruction and some explicit computations.

Kioji Saito

"The polyhedron
dual to the Weyl chamber system"

The polyhedron dual to the Weyl chamber system is a basic object in

a combinatorial and topological study of the braid groups. Namely,

it was used 1. to show the K(\pi,1) property of the configuration

space ('74 Deligne) and 2. to determine the braid relation of the

fundamental group of the configuration space ('70 Brieskorn).

In this talk, I shall show that the dual polyhedron is closely

related to the flat sturcure (Frobenius manifold) on the quotient

variety by the Weyl group action, and then, in application, give

a geometric construction of the generators of the fundamental group

satisfying the braid relation, which answers to a question of Deligne.

Mina
Teicher

"Computational problems related
to braid monodromy type"

Abstract

Miguel
Xicotencatl

"Orbit configuration spaces and
surface braid groups"

Given a n-manifold M, define its
pure braid group on k strands,

P_k(M), as \pi_1 F(M,k), where
F(M,k) is the configuration space

of k distinct points in M. It is
well known that the natural

homomorphism \varphi: P_k(M)
\to (\pi_1 M)^k

is an isomorphism for n>2 and epimorphism
for n=2.

In this talk we describe the kernel
of \varphi in the case M = S_g,

a compact orientable surface of
genus g>1. Namely, if S_g is the

quotient of the upper-half plane
H^2 by the subgroup \Gamma \leq SL(2,R),

then \ker \varphi \cong \pi_1 F_{\gamma}(H^2,k)
(the fundamental group

of an orbit configuration space).
As a consequence we obtain a spectral

sequence converging to the homology
of P_k(S_g).

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Page last updated on June 13, 2002.

For comments and suggestions please contact Carlo Petronio.