September 4 (Tue):
Thang Le (State University of New York)
Homotopy quantum field theory and quantum groups
Abstract:
It is known that quantum groups at roots of unity give rise to
modular categories, which, in turn, define TQFT and invariants of
3-manifolds.
Turaev introduced "Homotopy quantum field theory" and showed that
a modular G-category (an enhanced version of a modular category) gives
rise to a HQFT. We will briefly explain the theory of HQFT in the case
of abelian groups and show examples of such theory using quantum groups
at roots of unity. This is a joint work with V. Turaev.
Kazuo Habiro (RIMS, Kyoto University)
On the colored Jones polynomial of links
Abstract:
The (normalized) $n$th colored Jones polynomial $J_K(n)$
(associated with the $n$ dimensional representation of $U_q(sl_2)$)
of a knot $K$ in a 3-sphere is defined to be
a Laurent polynomial in an indeterminate $q$ over the integers.
I will explain a result that says $J_K(n)$ modulo
some ideal in $Z[q,q^-1]$ is determined by the $J_K(i)$ with $1 \le i < n$.
I also extend this result to colored Jones polynomial of algebraically
split links.
Justin Roberts (University of California, San Diego)
6j-symbols and asymptotics
Abstract:
Recent interest in the Kashaev-Murakami-Murakami volume conjecture has
made it important to be able to understand the asymptotic behaviour of
certain special functions arising from representation theory, the
quantum $6j$-symbols being an important example.
In 1998 I worked out the asymptotic behaviour of the {\em classical}
$6j$-symbols (for $SU(2)$), proving a formula conjectured by Ponzano
and Regge in 1968 involving the geometry of a Euclidean tetrahedron. I
will explain the methods and philosophy behind this calculation, and
speculate on how similar techniques might perhaps be useful in
studying the quantum case or the volume conjecture.
September 6 (Thu):
Andrew Kricker (University of Sydney)
On some "rationality" properties of the Kontsevich integral
Abstract:
In some papers in the late '90s, Lev Rozansky showed
us how to expand the coloured Jones polynomial of a
knot (for instance) into a series whose coefficients
are rational functions with denominator some power
of the Alexander-Conway polynomial of the knot. This
led him to conjecture a similar structure for the
(entire) Kontsevich integral of the knot. The first
aim of this talk is to go through a nuts-and-bolts
demonstration of such a property, by applying a
liberal dose of "wheels magic" to the LMO invariant.
Stavros Garoufalidis and the author showed that this
construction in fact presented a knot invariant, and
indeed could be generalised to a "rational"
invariant of boundary links. This talk will continue
with a brief survey of this development, emphasising
open issues in the theory.
The theory will be illustrated with a generalisation
of a surgery formula of Casson's to this setting.
Thomas Kerler (Ohio State University)
Cut numbers and Reidemeister Torsion of 3-manifolds
from the (q-1)-expansions of the RT TQFT's
Abstract:
The $SO(3)$-Reshetikhin-Turaev invariant at a $p$-th root of unity,
with $p\geq 5$ a prime takes values in the cyclotomic integers $Z[\zeta_p]$
and extends to a TQFT over $Z[\zeta_p]$ at least for connected surfaces.
(Murakami, Masbaum, Roberts, Gilmer). Expansions in $(\zeta_p-1)$ of
the invariants for the closed 3-manifolds yield Ohtsuki's invariants.
In this talk we will address the structure of the underlying
$(\zeta_p-1)$-expansions of the TQFT's.
Particularly, we construct a family of irreducible TQFT's over $Z/pZ$
from the Lefschetz components of TQFT's obtained by Frohman-Nicas
via the intersection homology of Jacobians.
The latter describe the Alexander Polynomial (or Reidemeister Torsion)
of a 3-manifold. Using results of Kleshchev and Sheth on the theory of
modular representations of symmetric groups
we construct specific resolutions of these irreducible
TQFT's that allow us to derive trace formulae for the coefficients
of the image of the Alexander polynomial in $Z/pZ[\zeta_p]$.
We further construct extensions of the $Sp(2g,Z)$ representations to
the Johnson-Morita quotient of the mapping class group by the
group of bounding cycles.
For $p=5$ the RT-TQFT (Frobenius TQFT) the $Z/5Z$ reduction is shown to be
identical with with one of the so constructed JM-extension of
5-modular irreducible subquotients of FN-TQFT's. As a result the
image of the Alexander Polynomial with respect to a cocycle
$\chi:H_1(M) \to Z$
in $Z/5Z[\zeta_5]$ is equal to the sum of the order of $\mbox{ker}(\chi)$
and the RT-invariant of $M$ (which is independent of $\chi$!).
We discuss further
formulae for the Casson-invariant resulting by combination with Murakami's
formula, compare them to Morita's original formulae,
as well as speculate on the first order structure of the RT-TQFT's
for larger $p$.
The Frobenius TQFT over $Z[\zeta_5]$ also proves to be
a "half-projective" TQFT
with parameter $x=\zeta_5-1$. In joint work with Gilmer, this results in an
estimate of $cut(M)$ by the "quantum order" of $M$, where the ``cut number''
$cut(M)$ is defined as the maximal number of surfaces that can be removed from
$M$ without disconnecting it, or, equivalently,
the maximal rank of a free group
onto which $\pi_1(M)$ can be mapped. This used for specific computations
of $cut(M)$ and compared to the estimate by Melvin and Cochran
in which we have $\beta_1(M)/3$ instead of $cut(M)$.
Adam Sikora (University of Montreal)
Analogies between number theory and 3-dimensional topology
Abstract:
Arithmetic topology concerns some surprising similarities between
3-dimensional topology and algebraic number theory, which can be
summarized by saying that knots are like prime numbers.
We prove new results both in topology and in
number theory which further extend these analogies.
Furthermore, if time permits, we present a new approach to knots and
$3$-manifolds inspired by the method of ideles and adeles developed
in number theory by Chevalley, Weil, and Tate.
September 11 (Tue):
Dylan Thurston (Harvard University)
The Casson invariant via configuration spaces
Abstract:
Finite type invariants of integer homology spheres can be written as
elementary integrals over configuration spaces. The simplest case is
the Casson invariant of a homology sphere $M$, which can be written as
the integral of the cube of a certain 2-form on $M \times M$. We prove that
the invariant is finite-type using elementary cut and paste topology.
Joint work with G. Kuperberg.
Jozef Przytycki (George Washington University)
Skein modules with a cubic skein relation: properties and speculations
Abstract:
The Kauffman bracket skein module ${\cal S}_{2,\infty}(M;R)$
is based on a skein relation involving unoriented links
$L_1,L_0$ and $L_{\infty}$. It contains information
on the space of Fox $3$-colorings. The Kauffman skein
module ${\cal S}_{2,\infty}(M;R)$ is based on a skein relation
involving unoriented links $L_2,L_1,L_0$ and $L_{\infty}$.
It contains information on the space of Fox $5$-colorings.
We will discuss, in our talk, the skein module
${\cal S}_{4,\infty}(M;R)$ satisfying the skein
relation $b_0L_0 + b_1L_1 + b_2L_2 + b_3L_3 + b_{\infty}L_{\infty} = 0$
and the framing relation $L^{(1)} = a L$ where $L^{(1)}$
denote a link obtained from $L$ by twisting the framing of $L$ once
in the positive direction. We discuss the question
for which parameters trivial links (or only the trivial knot)
are linearly independent. We also discuss the conjecture
that the skein module of $S^3$ is generated by trivial links.
We address the question whether ${\cal S}_{4,\infty}(S^3;R)$
contains information on the space of Fox $7$-colorings.
We use the fact that a rational $\frac{7}{2}$-move
preserves the space of $7$-colorings.
Xiao-Song Lin (University of California, Riverside)
The Bennequin number of n-trivial closed n-braids is negative
Abstract:
A famous result of Bennequin states that for any braid
representative of the unknot the Bennequin number is negative. We will
extend this result to all n-trivial closed n-braids. This is a class of
infinitely many knots closed under taking mirror images. Our proof relies
on a non-standard parametrization of the Homfly polynomial.
September 12 (Wed):
Justin Sawon (University of Oxford)
Introduction to Rozansky-Witten invariants from a hyperkahler perspective
Abstract:
In 1996 Rozansky and Witten constructed a new invariant of three-manifolds
as the partition function of a 3-dimensional topological sigma model with
target space a hyperk{\"a}hler manifold X. A Feynman diagram expansion
shows that the invariant is essentially the LMO invariant, with a new
system of coefficients (a weight system on trivalent graphs) constructed
from X. In this talk I will give an introduction to this weight system:
how it is defined, how it may be calculated, some of its properties, some
examples, and what it tells us about hyperk{\"a}hler manifolds.
Ted Stanford (New Mexico State University)
Free Newton expansions and two-strand string links
Abstract: We discuss an algebraic construction that generalizes both
the Magnus expansion in a free group and the Alexander modules of
a finitely-presented group.
As an application, we obtain matrix representations
of the group of string links up to concordance. The Gassner representation
for string links, due to Le Dimet, is one example of such a representation.
Another example shows that the two-strand string link concordance group
is not nilpotent, in particular not abelian.
Christine Lescop (University of Grenoble)
On the Bott and Taubes anomaly in the Chern-Simons theory for links
Abstract:
The anomaly is a still unknown element of the space of Feynman diagrams
that is defined from a series of configuration space integrals.
In 1999, Sylvain Poirier proved that if the anomaly is zero in degree
greater than one, then the Kontsevich integral coincides with
the perturbative expansion of the Chern-Simons theory for knots
(that is a universal Vassiliev invariant defined from configuration
space integrals).
After introducing all the configuration space integrals mentioned above,
we will present the known properties of the anomaly and we will describe
some methods to compute its low degree terms.
September 13 (Thu):
Vladimir Turaev (University of Strasbourg)
Homotopy quantum field theory
Abstract:
We apply the idea of a topological quantum field theory (TQFT)
to maps from manifolds into topological spaces.
This leads to a notion of a $(d+1)$-dimensional homotopy quantum field
theory (HQFT) which may be described as a TQFT for closed $d$-dimensional
manifolds and $(d+1)$-dimensional cobordisms
endowed with homotopy classes of maps into a given space.
For a group $\pi$, we introduce cohomological HQFT's with target $K(\pi,1)$
derived from cohomology classes of $\pi$ and its subgroups of finite index.
The main body of the talk is concerned with $(1+1)$-dimensional HQFT's.
We classify them in terms of so called crossed group-algebras.
In particular, the cohomological $(1+1)$-dimensional HQFT's
over a field of characteristic 0
are classified by simple crossed group-algebras.
We introduce state sum models for $(1+1)$-dimensional HQFT's.
We shortly discuss the 3-dimensional HQFT's.
Tomoyoshi Yoshida (Tokyo Institute of Technology)
An abelianization of SU(2) CFT and Witten invariant
Abstract:
We carry out an abelianization of $SU(2)$ Conformal Field Theory.
We give an explicit representation of a base
of the conformal block of level $k$ of
$SU(2)$ CFT in terms of the classical Riemann
theta functions of degree $k+2$ on Prym
varieties of 2-fold branched covering
Riemann surfaces with coefficients of automorphic forms.
As corollaries, we prove the Verlinde formula
of the dimension of the conformal blocks,and
we construct the projectively flat connection
on the bundle of the conformal blocks over Teichm\"uller space.
Also we construct an hermitian product preserved by the
connectionon on the bundle.
Finally we give a definition of Witten invariant by pairings
of Riemann theta functions.
Dror Bar-Natan (Hebrew University)
Knotted trivalent graphs, tetrahedra and associators
Abstract:
If knot theory was finitely presented, one could
define knot invariants by assigning values to the generators so that
the relations are satisfied. Well, some mild generalization of knot
theory, the theory of knotted trivalent graphs, is finitely
presented, as we will see in this talk. We will also see that the
resulting theory is essentialy equivalent, though much more symmetrical
and elegant and topological, to the Drinfel'd theory of associators.
The talk will follow the handout at HUJI-001116.
(joint with Dylan Thurston).
Workshop:
September 17 (Mon):
Hitoshi Murakami (Tokyo Institute of Technology)
Good news and bad news about the volume conjecture
Abstract:
I will give a talk about some good news and bad news
about the volume conjecture,
a generalization of Kashaev's conjecture.
Good news: Supporting evidence for knots, links, and three-manifolds
with/without boundary. It also indicates a complexification of the volume
conjecture, which would give a topological Chern-Simons invariant.
Bad news: No rigorous proof is given except for the following cases: the torus
knots (links), the figure-eight knot, and the Borromean rings. I will explain
how hard it would be.
Riccardo Benedetti (University of Pisa)
On quantum hyperbolic invariants of 3-manifolds
Abstract:
Let $W$ be a compact closed oriented $3$-manifold, $L\subset W$ be a link,
$\rho$ be a principal flat $B$-bundle on $W$; $B$ is the upper
triangular Borel sub-group of $SL(2, C)$. One decsribes
a family of ``quantum hyperbolic state sum invariants''
$K_N(W,L,\rho)\in C$, $N=2p+1 \in N$. Their definition
consists of two main points: (1) the proof of
the existence of {\it distinguished and decorated triangulations}
${\cal T}=(T,H,{\cal D})$, for every triple $(W,L,\rho)$;
(2) the proof that a
suitable state-sum $K_N({\cal T})$ does not depend on the particular $\cal T$
so that an invariant $K_N(W,L,\rho)$ is actually defined.
The existence of such elaborated triangulations is not
evident because of non trivial global constraints. The state sum
is based on the quantum-dilogarithm $6j$-symbols at $\omega_N =
\exp(2\pi i/N)$. Kashaev had early proposed conjectural, purely
topological invariants $K_N(W,L)$ and one recognizes that they
correspond to the special case of the trivial flat bundle on
$W$. The algebraic properties of the $6j$-symbols ensure the
invariance of $K_N({\cal T})$ up to a suitable {\it decoration transit}.
It is not evident how to deduce the full invariance from the transit one.
Finally one would develop some consideration
about the so called ``Volume Conjecture''.
This talk arise from a joint work with Sthephane Baseilhac.
Justin Roberts (University of California, San Diego)
Rozansky-Witten invariants and derived categories
Abstract:
In 1996 Rozansky and Witten described a new family of
$(2+1)$-dimensional topological quantum field theories, quite
different from the now familiar Chern-Simons theories. Instead of
starting from a compact Lie group, one starts with a hyperk\"ahler
manifold $X^{4n}$; the partition function (a topological invariant)
for a closed $3$-manifold $M$ is then expressed as an integral over
the space of all maps from $M$ to $X$. Further analysis shows that
these invariants amount to evaluations of the universal finite-type
invariant of Le, Murakami and Ohtsuki, using weight systems derived
purely from the hyperk\"ahler manifold $X$.
I will give a brief explanation of the geometrical origin of these
weight systems and then describe (joint work with Simon Willerton) a
precise analogy between hyperk\"ahler manifolds and Lie algebras, the
connections with Vassiliev theory. The flavour of the theory is
appealingly algebro-geometrical: whereas constructions of Chern-Simons
theory start from the category of representations of a quantum group,
Rozansky-Witten theory turns out to be based on the derived category
of coherent sheaves on $X$. Don't panic! I'll try to make this seem
more reasonable than it sounds.
Justin Sawon (University of Oxford)
Rozansky-Witten invariants and TQFTs
Abstract:
The Rozansky-Witten invariant of a three-manifold is defined as a Feynman
path integral, and hence we can expect there to be an associated
(2+1)-dimensional TQFT. Rozansky and Witten only made a suggestion for
what the vector space associated to a surface should be.
In this talk I will describe joint work with Justin Roberts and Simon
Willerton. We define a (1+1+1)-dimensional TQFT, ie. a map which
associates categories to 1-manifolds, functors to 2-bordisms, and natural
transformations to 3-bordisms. This extended TQFT is built from the
derived category of a hyperk{\"a}hler manifold, and turns out to be a
legitimate candidate for the Rozansky-Witten TQFT.
Nobuya Sato (Osaka Prefecture University)
Turaev-Viro-Ocneanu invariants of 3-manifolds from subfactors
Abstract:
In this talk, I will explain an approach to topological invariants of
3-manifolds from subfactors of von Neumann algebras.
More specifically, starting with a subfactor of finite index and finite depth,
we construct a Turaev-Viro-Ocneanu topological quantum field theory
in three dimensions.
This theory turns out to
have enough symmetry to allow so-called Verlinde formula. A computational
method of a Turaev-Viro-Ocneanu invariant will be discussed.
This is a joint work with M. Wakui (Osaka University).
Soeren Hansen (University of Strasbourg) and
Toshie Takata (Niigata University)
Quantum invariants of Seifert manifolds for simply laced Lie algebras and
their asymptotic expansion
Abstract:
According to the asymptotic expansion conjecture (AEC) due to J. E. Andersen
the full asymptotic expansion of the quantum $G$--invariant of a closed
oriented $3$--manifold $M$ should be expressible by a sum over the set of
values of the Chern-Simons functional of flat $G$ connections on $M$, $G$
being a simple and simply connected compact Lie group.
If the AEC is true the asymptotic formula contains a lot of
geometrical/topological invariants and the AEC connects
the quantum invariants of $X$ to the fundamental group of $X$.
We start by explaining the AEC and give a short history of its origin and what
is known today. We give formulas of the Reshetikhin-Turaev invariants of
all Seifert manifolds in the general framework of a modular category.
We elaborate on these formulas in case the modular category is
induced by the representation theory of a quantum deformation of a simply
laced Lie algebra. This leads to explicit formulas which can be used to
calculate the asymptotic expansions of the invariants. We explicitly confirm
the existence part of the AEC for all lens spaces for the simply laced Lie
algebras.
September 18 (Tue):
Room 420 at RIMS, Chairperson: ???
Hugh Morton (University of Liverpool)
The Homfly polynomial for decorated Hopf links
Abstract:
The goal is to find the Homfly polynomial of a link formed by
decorating each component of the Hopf link with the closure of some
directly oriented tangle. Because these decorations are spanned in the
Homfly skein of the annulus by certain elements $Q_\lambda$ indexed by
partitions $\lambda$ it is enough to find the invariants
$$ where the components of the Hopf link are decorated
by $Q_\lambda$ and $Q_\mu$. I show how the 1-variable $sl(N)_q$
specialisation of $$ can be expressed in terms of an
$N\times N$ minor of the Vandermonde matrix $(q^{ij})$, and how to find
the 2-variable invariant from the Schur symmetric function $s_\mu$ of an
explicit power series depending on $\lambda$.
Vladimir Turaev (University of Strasbourg)
Abelian torsion of 3-manifolds
Abstract:
We discuss algebraic properties of the abelian torsions of 3-manifolds.
We also give a surgery formula for the torsions and Seiberg-Witten invariants
associated with $Spin^c$-structures on 3-manifolds.
Jozef Przytycki (George Washington University)
Symplectic structure on colorings, Lagrangian tangles and its applications
Abstract:
We show how to define a symplectic form on the space of
Fox $p$-colorings of the boundary of
n-tangles so that every tangle corresponds to a Lagrangian of
the symplectic structure (that is, a subspace of a maximal
dimension on which the form vanishes).
Inversely, for a field $R=Z_p$ for $p>2$, every Lagrangian can
be realized by a tangle. It does not hold for $Z_2$ and $n>3$.
For $R=Z$ we show that tangles yield virtual Lagrangians in a
$Z$-symplectic space of Fox colorings of the boundary.
We discuss applications of these unexpected connections to 3-manifold topology.
In particular we show that our symplectic space is related (via double
branched cover) to the symplectic structure on the first homology of a surface
(with the symplectic form given by the intersection number). It relates
our results to a known fact that 3-manifolds yield Lagrangians
in $H_1(\partial M;Q)$.
As an application we use Lagrangians
to find obstructions for embedding n-tangles into links.\\
Rotation of a tangle yields an isometry of our symplectic space, and we
analyze invariant subspaces of the map, in particular we look for
invariant Lagrangians of the rotation by $2\pi/n$ (along $z$-axis).
We use our analysis to answer, partially the question whether
rotation of a link preserves the homology
of the double branch cover of $S^3$ with the link as branching set.
This is related to the recent result of Traczyk that rotation
preserves the Alexander polynomial.
I would like to thank T.Januszkiewicz, J.Dymara, A.Sikora and
D.Silver for enlightening discussions.
Adam Sikora (University of Montreal)
On Kerler's conjecture on cut numbers of 3-manifolds
Abstract:
A cut number of a 3-manifold M is the maximal number
of disjoint embedded surfaces in M which do not disconnect M.
Equivalently it is the maximal rank of a free group being an
epimorphic image of the fundamental group of M.
One has cut_number(M)<= b_1(M).
We will present our work inspired by the following
conjecture: 1/3*b_1(M)<= cut_number(M).
Stephen Bigelow (University of Melbourne)
Homology and the Jones polynomial
Abstract:
The Jones polynomial of a knot is defined using representations of the
braid group which factor through the Hecke algebra. Lawrence has shown
that these representations can be defined more topologically using the
action of the braid group on configuration spaces of points in the disk
with $n$ punctures. I will explain how to do this using a natural and
explicit construction. I will discuss possible applications of this
topological perspective on the Jones polynomial and the Hecke algebra.
Gregor Masbaum (Institut de Mathematiques de Jussieu)
Alexander-Conway polynomial, Milnor numbers, and a new matrix-tree theorem
Abstract: 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.
(Joint work with A. Vaintrob)
September 19 (Wed):
Room 420 at RIMS, Chairperson: ???
Toshitake Kohno (University of Tokyo)
Loop spaces of orbit configuration spaces and finite type invariants
Abstract:
This is a joint work in progress with F. Cohen. The homology of
the loop space of a configuration space has a structure of a Hopf
algebra with relations described by generalized Yang-Baxter
equations. We establish a relation between the homology of the
loop spaces of orbit configuration spaces for a Fuchsian group
acting on the upper half plane freely and the algebra of chord
diagrams on a surface.
Dror Bar-Natan (Hebrew University)
Knot invariants, associators and a strange breed of planar algebras.
Abstract:
If knot theory was finitely presented,
one could define knot invariants by assigning values to the generators
so that the relations are satisfied. Well, knot theory is
finitely presented, at least as a Vaughan Jones-style
"planar algebra". We define a strange breed of planar algebras that can
serve as the target space for an invariant defined along lines as
above. Our objects appear to be simpler than the objects that appear in
Drinfel'd theory of associators - our fundamental entity is the
crossing rather than the re-association, our fundamental relation is
the third Reidemeister move instead of the pentagon, and our "relations
between relations" are simpler to digest than the Stasheff polyhedra. Yet
our end product remains closely linked with Drinfel'd's theory of
associators and possibly equivalent to it. The talk will follow the
slides at Fields-010111;
see also MSRI-001206. (joint with Dylan Thurston).
Xiao-Song Lin (University of California, Riverside)
Link-homotopy invariants of finite type
Abstract:
An explicit polynomial in the linking numbers $l_{ij}$ and
Milnor's triple linking numbers $\mu(rst)$ on six component links is shown
to be a well-defined finite type link-homotopy invariant. This solves a
problem raised by B. Mellor and D. Thurston. An extension of our
construction also produces a finite type link invariant which detects the
invertibility for some links.
Michael Polyak (Tel-Aviv University)
Binary rooted trees and Milnor's mu-invariants
Abstract:
We provide an explicit combinatorial formula for Milnor's
mu-invariants of string links by counting descending tree paths in a
diagram. In terms of Gauss diagrams this corresponds to counting
planar binary trees. We discuss some related cutting/stacking
operations on string links.
Chan-Young Park and Myeong-Ju Jeong (Kyungpook National University)
Polynomial invariants and Vassiliev invariants
Abstract:
In 1993, J. S. Birman and X.-S. Lin showed that after a suitable
change of variables, each coefficient of the Maclaurin series of
the Jones, HOMFLY and Kauffman polynomial is a Vassiliev
invariant. So some infinite linear combinations of the
coefficients these polynomials give Vassiliev invariants. But the
coefficients of the Jones, HOMFLY, and Kauffman polynomial are not
Vassiliev invariants.
In 2001, we defined a sequence of knots or links induced from a
double dating tangle and showed that any Vassiliev invariant has a
polynomial growth on this sequence. Based on this observation, we
gave a criterion to detect whether the derivatives of knot
polynomials at a point are Vassiliev invariants or not. As an
application we showed that for each nonnegative integer $n$,
$J_K^{(n)}(a)$ is a Vassiliev invariant if and only if $a=1$,
where $J_K^{(n)}(a)$ is the $n$-th derivative of the Jones
polynomial $J_K(t)$ of a knot $K$ at $t=a$. Similar results were
obtained for the Conway, Alexander, $Q$-, HOMFLY and Kauffman
polynomial. Also we constructed two polynomial invariants
$\bar{v}, {v}^*$ in one and two variables, from a numerical
Vassiliev invariant $v$ of degree $n$, by using sequences of knots
induced from double dating tangles. These new polynomial
invariants are Vassiliev invariants of degree $\leq n$ and their
values on a knot are also polynomials of degree $\leq n$.
In this talk, we generalize the polynomial constructions to a
polynomial invariant in $m$ variables, which is a Vassiliev
invariant with degree $\leq n$ and give some questions on
Vassiliev invariants and the new Vassiliev polynomial invariants.
Seiichi Kamada (Osaka City University)
Knot invariants derived from quandles and racks
Abstract:
A lot of invariants of knots are defined and calculated via their diagrams.
Fox's 3-colorability is an elementary example. This idea is generalized
into quandle (or rack) colorings. Quandles, racks and keis are algebraic
objects deeply related with knot theory.
We introduce how to obtain invariants
of knots/links (or framed links) associated with quandles (or racks).
September 20 (Thu):
Room 420 at RIMS, Chairperson: ???
Greg Kuperberg (University of California)
Perturbative 3-manifold invariants are finite type
Abstract:
In a recent paper [math.QA/9912167], Dylan Thurston and I presented
Kontsevich's configuration space integral 3-manifold invariants. I will review
this work, concentrating on the second half in which we show that the
invariants are non-trivially finite type. We conjecture that the invariants
coincide with the LMO invariant, having shown that their last non-vanishing
finite differences agree.
Dylan Thurston (Harvard University)
Knotted trivalent graphs and their algebra
Abstract:
We can represent knots using the algebra of knotted trivalent graphs
with the tetrahedron and Mobius strip as generators. This
simultaneously generalizes planar knot diagrams, non-associative
tangles, and several other combinatorial representations of knots. This
representation is equivalent to a version of Turaev's shadow world, in
which knots are represented by certain 2-complexes with integer or
half-integer decorations ("gleams") on the 2-cells.
Joint work with D. Bar-Natan.
(problem session, short excursion, dinner)
September 21 (Fri):
Room 420 at RIMS, Chairperson: ???
Thang Le (State University of New York)
On the integrality of perturbative invariants of homology 3-spheres
Abstract:
For any simple Lie algebra, quantum invariants of homology 3-spheres
can be expanded into power series which generalizes Ohtsuki's expansion
in the $sl_2$ case.
We show that the coefficients of this series are integers.
Kazuo Habiro (RIMS, Kyoto University)
Cyclotomic expansion of the Witten-Reshetikhin-Turaev invariant
Abstract:
The Witten-Reshetikhin-Turaev (WRT) invariant
(associated with the quantum group $U_q(sl_2)$)
of a closed 3-manifold is defined for each root of unity.
The Ohtsuki series of a rational homology sphere is known to
unify in some sense the WRT invariants at roots of unity of prime orders.
I will introduce an invariant $I(M)$ of an integral
homology sphere $M$ with values in a certain completion R of the Laurent
polynomial ring $Z[q,q^-1]$ such that the WRT invariant at each root of
unity, $z$, recovers $from I(M)$ by substituting $z$ for $q$.
Andrew Kricker (University of Sydney)
The LMO invariant and covering spaces
Abstract:
This talk will sketch the proof of a certain
formula which calculates the LMO invariant of the
p-fold branched cyclic cover of a knot (in the case
that it is a QHS^3) in terms of a certain "rational"
lift of the Kontsevich integral of the knot, together
with the knot's total p-signature.
In degree 1 this resolves a long-standing question:
the knot invariant that is the Casson-Walker invariant
evaluated on a brached cyclic cover can indeed be
determined from "known" knot invariants. Of more
interest is a surprising correspondence between the
"loop" expansion of the Kontsevich integral, and the
"degree" expansion of the LMO invariant, in terms of
which this formula is expressed. This is joint work
with S. Garoufalidis.
Christine Lescop (University of Grenoble)
Linking the configuration space integrals of the Chern-Simons theory
to the Kontsevich integral
Abstract:
We discuss the known common features of the Kontsevich integral and of the
perturbative expansion of the Chern-Simons theory for knots and we
give an algebraic characterization of
the knot invariants that share these common features.
Ted Stanford (New Mexico State University)
Computational results on mod 2 finite-type invariants of
knots and string links
Abstract:
We present methods and results of some mod 2 computations of finite-type
invariants. For knots, we show that there are some mod 2 congruences between
integer-valued invariants of even and odd orders. For two-strand string links,
we show that there is no mod 2 version of the Kontsevich integral.
Yoshiyuki Yokota (Tokyo Metropolitan University)
On the potential functions for the hyperbolic structures of a knot complement
Abstract:
Let $M$ be the complement of a hyperbolic knot in $S^3$.
Through the study of Kashaev's conjecture, we have found a complex function
which gives the volume and the Chern-Simons invariant of the complete
structure of $M$
at the critical point corresponding to the solution to the hyperbolicity
equations for $M$.
The purpose of this talk is to construct such functions for the
non-complete structures of $M$.
Seminar:
September 25 (Tue), Goussarov day:
Michael Polyak (Tel-Aviv University)
Goussarov's point of view on finite type invariants
Abstract:
During several discussions with Goussarov I heard some of his
ideas and opinions about finite type invariants. I'll try to
summarise what I learned from these meetings, e.g.: the cubic
spaces, n-equivalence, partially defined invariants, etc.
Kazuo Habiro (RIMS, Kyoto University)
Goussarov's n-equivalence and tree clasper surgery
Abstract:
Goussarov's $n$-equivalence surgery on a 3-manifold $M$ is surgery along $n$
disjoint Y-graphs $Y_1$, ..., $Y_n$ contained in a handlebody $V$ in $M$ such
that, for any proper subset $S$ of $\{Y_1, ..., Y_n\}$, surgery on $V$ along
the elements in $S$ yields a 3-manifold homeomorphic to $V$ relative to
the boundary. Surgery on a tree clasper of degree $n$ can be thought of
as a special kind of n-equivalence surgery in which some of their
leaves are paired in Hopf links.
I would like to explain the result due to Goussarov that
$n$-equivalence surgery and tree clasper surgery of degree $n$ generates
the same equivalence relations on 3-manifolds.
Yoshiyuki Ohyama (Nagoya Institute of Technology)
$C_n$-moves and their application
Abstract: M. N. Goussarov and K. Habiro showed independently that
two knots have the same Vassiliev invariants of order less
than $n$ if and only if they can be transformed into each
other by a finite sequence of $C_{n}$-moves.
We will talk the results for Vassiliev invariants and for
a local move on a link diagram. The key to them is the
property of $C_{n}$-moves.
Ted Stanford (New Mexico State University)
Some groups of knots
Abstract: It is now well-known, but one of the
most remarkable and surprising early theorems
about finite-type invariants was Gusarov's result
that knots modulo invariants of a fixed order form
a group under connected sum. This result may
be formulated and proved using braids and
closed braids. The braid approach gives rise
to various generalizations of Gusarov's groups.
Stavros Garoufalidis (Georgia Institute of Technology)
An introduction to the loop filtration
Abstract:
Y-graphs or clovers offer a good way to describe
surgery on 3-dimensional space and to study finite type invariants of
knotted objects in 3-dimensional space.
As an example of this principle, we will introduce a 'loop move' on the
set of pairs of knots in integer homology spheres, and show that a
suitable rational lift of the Kontsevich integral is a universal
invariant. The loop move is closely related to the notion of
S-equivalence, and to the grading of unitrivalent graphs given by the
number of internal trivalent vertices.
Dror Bar-Natan (Hebrew University)
Bracelets and the Goussarov filtration on the space of knots
Abstract:
Following Goussarov's paper "Interdependent Modifications of Links
and Invariants of Finite Degree" (Topology 37-3 (1998)) I will
describe an alternative finite type theory of knots. While (as shown
by Goussarov) the alternative theory turns out to be equivalent to
the standard one, it nevertheless has its own share of intrinsic
beauty.
September 26 (Wed):
Stavros Garoufalidis (Georgia Institute of Technology)
Automata, graphs and boundary links
Abstract:
Dror Bar-Natan (Hebrew University)
Khovanov's categorification of the Jones polynomial
Abstract:
In two recent and very novel papers,
arXiv:math.QA/9908171
and arXiv:math.QA/0103190,
Khovanov finds a graded chain complex whose graded Euler characteristic
is is the Jones polynomial, and proves that each individual homology
group of this complex is a link invariant. His construction is very
simple and elegant, and yet orthogonal to everything else we know about
knot theory and hence extremely interesting. I plan to explain
Khovanov's construction in about 2/3 of the time of the talk, and leave
the rest for discussion. There will be a handout; for now, see
Calgary-010824.
September 27 (Thu):
Hugh Morton (University of Liverpool)
Murphy operators and general Hopf link invariants
Abstract:
I discuss skein theoretic versions of the Murphy operators in
the Hecke algebras of type A, and their relation to the skein of the
annulus and Hopf link invariants. The skein theory view includes the use
of reverse string decorations, and leads to an extension of the Murphy
operators to algebras of tangles with mixed orientations.
Stephen Bigelow (University of Melbourne)
The Burau representation for $n=4$
Abstract:
The Burau representation of $B_n$ is known to be faithful when $n < 4$ and
unfaithful when $n > 4$. The case $n = 4$ is the last remaining open case.
This case also has special interest because a non-trivial braid in the
kernel would almost certainly give rise to a non-trivial knot with Jones
polynomial equal to one. I will explain the mathematics behind my computer
search for an such a braid.
Gregor Masbaum (Institut de Mathematiques de Jussieu)
Unimodular representations of mapping class groups in TQFT
Abstract:
Quantum invariants of $3$-manifolds give rise to
finite-dimensional representations of (central extensions) of mapping
class groups. These representations can be defined over number fields, and
preserve a non-degenerate Hermitian form. For the SO(3)-theories at odd
primes and for surfaces of genus one and two, we describe an explicit
mapping class group invariant full lattice defined over the corresponding
ring of integers. The mapping class group acting on this lattice
preserves a related Hermitian form which is unimodular over the
ring of integers.
(Joint work with P. Gilmer and P. van Wamelen.)
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