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
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