Abstracts of talks delivered at the Third Conference on Knot Theory and

its Ramifications: KNOTS in WASHINGTON

held October 18-20, 1996 at the George Washington University.

Friday, October 18
1:00 - 2:00     Doug Bullock (Boise State Univ.), Skein Quantizations
  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)
     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
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
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
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

Saturday, October 19

10:30 - 11:00   Yongwu Rong (GWU), Derivatives of Link Polynomials
(joint work with W. B. R. Lickorish)
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
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
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.
   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
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

  1. ${\cal S}_{2,\infty}(F\times I;R,A)$ has no zero divisors, provided $R$ has no zero divisors.
  2. 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.
  3. ${\cal S}_{2,\infty}(T^2\times I;Z[A^{\pm 1}],R)$ is a unique factorization domain.
As a corollary we prove Bullock conjecture for surface groups that
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

Sunday,  October 20

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