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)