The Fifth Conference on Knot Theory and its Ramifications

"Knots in Washington" will be held Saturday, November 22, 1997 at

the University of Maryland (College Park).

This regional miniconference is held every semester, in various
locations

in the Washington area. (The Sixth "Knots in Washington" conference:

"Knot Theory Days," will take place Feb. 7-9, 1998 at U.S. Naval
Academy,

Annapolis; M.Kidwell, local organizer).

You are cordially invited to participate in this and future meetings.

All talks will be in the Colloquium Room in the Mathematics Department

of the University of Maryland (Room number 3206).

TENTATIVE SCHEDULE:

10:00 - 10:30 Refreshments

10:30 - 11:30 Charles Frohman, University of Iowa;

"Skeins and Characters"

Abstract:

The lecture will begin with
a general method for

producing 3-manifold invariants
from a compact group.

Out of this we will establish
integral formulas for

the Turaev-Viro invariant.

More provocatively, we will
show that

the Kauffman bracket skein
module at a root of

unity of a 3-manifold can
be viewed as functions

on a cartesian product of
copies of SU(2). The

mode of evaluation depends
on a Heegaard diagram

of the 3-manifold. This
gives rise to invariants

of Heegaard diagrams coming
from the Kauffman bracket

skein module. It also gives
a path to a rigorous

analysis of the asymptotic
behavior of Turaev-Viro

invariants in terms of the
representation theory

of the fundamental group
of the manifold.

11:45 - 12:05 Jozef Przytycki, GWU; "Torsion
in skein modules:

Theorems, Conjectures and Speculations."

Abstract.

We discuss torsion
in skein modules of 3-manifolds.

1. A nonseparating 2-sphere
or 2-torus in a manifold

yields
a torsion in most of the skein modules.

2. A separating incompressible
2-sphere or 2-torus is

often yielding a torsion
(e.g. for Kaufman bracket,

Homflypt and Kauffman skein
modules).

3. A nonseparating
surface (of any genus) is a cause

of torsion in the second
skein module (related to L_+

-q L_0 skein relation).

We discuss, with more details,
torsion in the Kauffman

bracket skein module. In
particular we show that

1. If M is a connected sum
of M_1 and M_2 then KBSM of

M has a torsion provided
that $M_1$ and $M_2$ have

first homology groups that
are not 2-torsion groups.

2. If M is the double of
a hyperbolic manifold with

boundary torus then the
Kauffman bracket skein module

of M has a torsion.

12:10 - 2:10 Lunch

2:15 - 3:15 Ted Stanford, Naval
Academy; "Vassiliev invariants and the

lower central series of the pure braid group."

Abstract:

Vassiliev defined a new
set of knot invariants around

1990 using singularity theory.
Birman and Lin showed

that the Jones polynomial
and its generalizations

can be reparametrized to
fit into the Vassiliev

framework. Finding
an interpretation of the Jones

polynomial and its generalizations
in terms of classical

topology has been notoriously
difficult, but Vassiliev

invariants have proved a
little more amenable in this

regard.

We will prove the following
theorem: Let K1 and K2

be knots. Then v(K1)
= v(K2) for every Vassiliev

invariant of order less
than n if and only if there

exists a positive integer
m and a braid b in Bm

and a pure braid p in the
nth group of the lower

central series of Pm, such
that K1 is the closure

of b and K2 is the closure
of pb. Bm is the

braid group on m strands,
and Pm is the pure braid

group on m strands.

Thus we obtain an interpretation
of what it means for

two knots to have matching
invariants up to order n

in terms of classical group
theory and topology.

The proof of the theorem
was inspired by a recent

result of Habiro, which
gives a characterization

of knots with matching invariants
up to order n

in terms of "claspers",
which are curves and

handlebodies in a knot complement
on which surgery

is performed to modify the
knot.

3:30 - 3:50 Adam Sikora, UMCP;

"A topological approach to Sl_n character

varieties."

Abstract:

We will present a theorem
proved jointly with C. Frohman

which gives a purely topological
description of the

$SL_n$-character variety
of any fundamental group of a

manifold. We will discuss
applications of this

theorem to the representation
theory of groups, the

theory of character varieties
and knot theory.

4:00 - 4:50 Open problem session

Local organizer: William Goldman (wmg@math.umd.edu)

Organizing Committee: John Millson (jjm@math.umd.edu)

Jozef Przytycki (przytyck@math.gwu.edu)

Yongwu Rong (rong@math.gwu.edu)

Sergey Novikov (novikov@ipst.umd.edu)

Adam Sikora (asikora@math.umd.edu)