Friday, October 18

1:00 - 2:00 Doug Bullock (Boise State Univ.),
Skein Quantizations

Abstract.

That fact That the Kauffman bracket skein module of a surface
is a

quantum deformation has been bandied about the knot theory community

for a decade or more. In this talk we will see, mostly be example,

what the words "quantum" and "deformation" actually mean, what the

undeformed object looks like, why it should be of interest, and how

the skein module unifies various points of view. In the process,
we

will touch lightly upon topology, representation theory,

non-commutative algebra, and mathematical physics.

2:30 - 3:00 Michael McDaniel (GWU), An Introduction
to Cabling

of Chord Diagrams

(joint work with Yongwu Rong)

Abstract.

This talk outlines some interesting results
about the subspaces of

Vassiliev Invariants generated by the Homflypt and Kauffman weight
systems

of connected cablings of knots. Using certain eigenvectors of cabling,

a deframing projector and careful attention to framing, it can be seen
that

dimensional bounds on these subspaces are quadratic in m (their order).

3:10 - 3:40 Mark Kidwell (Naval Academy and
GWU),

Fibered Knots from Signed Graphs

Abstract.

We recall how to span a surface in a link by the

checkerboard method and by Seifert's algorithm. When these two

methods coincide, we use graph theory to study the push-off maps

from the surface into its complement. We give a graphical

criterion for the link to be fibered and use Eulerian circuits to

construct graphs that meet the criterion.

3:50 - 4:20 Brian Mangum (Columbia Univ.), Three-dimensional

Representations of Punctured Torus Bundles

Abstract.

We construct a curve of irreducible SL$_3 \Bbb C$ representations of
the

fundamental grop of any orientable once punctured torus bundle over
the

circle. Moreover, infinitely many of these representations are
conjugate

to SU(3) representations. The construction employs classical
matrix group

representations of the four-strand braid group.

4:30 - 5.30 William M. Goldman (UMD), The topology
of relative

character varieties of surfaces

Abstract.

In this talk I will describe the geometry and topology of cubic surfaces

arising as sets of R-points of relative character varieties of surfaces

of low genus. In particular I will discuss some experimental results

begun in an REU activity with Robert Benedetto in 1993 on the quadruply

punctured sphere. A compact component of the relative character variety

which corresponds to SL(2,R)-representations (not SU(2)-representations)

will be demonstrated, and relations between the relative character

varieties of quadruply punctured sphere and punctured torus will be

given.

Saturday, October 19

10:30 - 11:00 Yongwu Rong (GWU), Derivatives of Link Polynomials

(joint work with W. B. R. Lickorish)

Abstract.

In this talk, we study the higher order link polynomials -- a class

of link invariants related to both Homfly polynomial and Vassiliev

invariants. We show that a $d$th partial derivative of
the Homfly

polynomial is a $d$th order Homfly polynomial. For $d=1$, we
show

that these derivatives span all the first order Homfly polynomials.

We also make similar constructions for other link polynomials.

In the case of the Conway polynomial, we note that its derivative does
not

span the space of first order Conway polynomials. Questions related

to Vassiliev invariants as well as computational complexities of

link polynomials will be raised.

11:15 - 12:15 Louis H. Kauffman (Univ. Illinois at Chicago),

Virtual Knot Theory

Abstract.

Knot and link diagrams are usually drawn on the plane or on the two-sphere.

Diagrams, taken up to equivalence by local Reidemeister moves, drawn
on a

surface F_{g} of genus g, classify links in F_{g} x I where I is the
unit

interval. Such a diagram, when projected back into the plane
will have

{\em virtual} crossings that are not regarded as crossings in F_{g}.
A {\em

virtual link diagram} is a link diagram in the plane in which a subset
of

the crossings are designated as virtual. Such a diagram can be regarded
as

representing a classical diagram on some F_{g}, but we define a theory
of

virtual knots and links (with an appropriate generalization of the

Reidemeister moves) that does not make reference to any specific genus
g.

Quantum link invariants,classical link invariants and Vassiliev invariants

all generalize to the category of virtual links. There are many surprising

phenomena. We give an example of a non-trivial virtual knot with trivial

Jones polynomial. (This is a real phenomenon in F_{g} X I for g=1).

The generalization of Vassiliev invariants is non-trivial with the weight

systems complicated by extra combinatorics of virtual vertices.

The topic of this talk is new, and it yields a chance to examine knot

theory from an unusual angle. I am following an analogy with virtual
knots

that is well known to graph theorists who study non-planar graphs that
are

uncolorable in order to understand coloring problems for planar graphs.

Here we regard combinatorial knot theory as a theory of planar knot
diagrams

and extend to non-planar knot diagrams whose abstract structure parallels

that of classical knot theory.

2:00 - 3:00 Joanna Kania-Bartoszynska (Boise
State Univ.),

Lattice gauge field theory

Abstract.

I will present a combinatorial setting in which the Kauffman bracket

skein module can be seen as a quantization of the $SL_2C gauge field
theory.

This is joint work with Doug Bullock and Charlie Frohman.

3:15 - 3:45 Maxim Sokolov (GWU), Turaev-Viro
invariants parameterized

by non-primitive roots of unity.

Abstract.

Every Turaev-Viro invariant, $TV(M)_q$, is a sum of three
non-trivial

summand-invariants: $TV_0(M)_q$, $TV_1(M)_q$, and $TV_2(M)_q$. Here
$q$

is a $2r$-th root of unity such that $q^2$ is a primitive root of unity

of degree $r$, and $r\ge 3$. We can prove that

$$

TV_N(M)_{-q} = (-1)^N TV_N(M)_q

$$

for any 3-manifold $M$ (even non-oriented) and $N\in \{ 0,1,2 \}$.

From the conditions on $q$ it follows that if $q$ is not
a primitive

$2r$-th root of unity then $-q$ is. Hence, by our formula, we can express

any Turaev-Viro invariant parameterized by a non-primitive $q$ via
the

summand-invariants parameterized by the primitive root $-q$.

It worth noting that the above mentioned formula allows
us to

express the numbers $TV_1(M)_q$ and $(TV_0(M)_q+TV_2(M)_q)$ via

Turaev-Viro invariants. Moreover, if $q=exp(\pi i/r)$ and $r$ is odd,

then sometimes we can get $TV_0(M)_q$ from the following analog of

the well-known Kirby-Melvin formula:

$$

TV_0(M)_q TV(M)_{exp(\pi i/3)} = TV(M)_q.

$$

4:00 - 4:30 Jozef H. Przytycki (GWU), Kauffman
bracket skein algebra

of a product of a surface and interval is an integral domain

Abstract.

We consider the Kauffman bracket skein module, ${\cal S}_{2,\infty}(M;R,A)$,

of an oriented 3-manifold $M$, that is the quotient of the module

of formal linear combinations of unoriented framed links in $M$ with

coefficients in a commutative ring with unit, $R$, by the submodule
generated

by the classical Kauffman bracket relations $L_+ - AL_0 -A^{-1}L_{\infty}$

and $L\sqcup O + (A^2 + A^{-2})L$, where $A$ is a fixed invertible

element in $R$. If $M= F \times I$, for an oriented surface

$F$ or $A=-1$ then ${\cal S}_{2,\infty}(M;R,A)$ is an algebra.

We show that

- ${\cal S}_{2,\infty}(F\times I;R,A)$ has no zero divisors, provided $R$ has no zero divisors.
- If $M$ is a twisted $I$ bundle over unoriented surface $F$ then ${\cal S}_{2,\infty}(M;R,A)$ has no zero divisors, provided $R$ has no zero divisors.
- ${\cal S}_{2,\infty}(T^2\times I;Z[A^{\pm 1}],R)$ is a unique factorization domain.

the skein algebra of a fundamental group of an oriented compact surface,

(with coefficients in $C$), is isomorphic to the coordinate ring of

the $SL(2,C)$ character variety of the group.

4:45 - 5:15 Adam S. Sikora (GWU), Skein algebras
and quaternion

representations

Abstract.

Sunday, October 20

Workshops

On Thursday (Oct. 17; 3.30p.m.) Charles Frohman (Univ. of Iowa) will
give

a talk at the Topology Seminar of UMD ("Quantizing the

SL(2,C)-characters of a surface group").

Local Organizers: Jozef H. Przytycki e-mail: przytyck@math.gwu.edu

Yongwu Rong e-mail: rong@math.gwu.edu