The Second Miniconference on Knot Theory and its Ramifications
will be held on Saturday, March 30, 1996 at the George
Washington University.
10:30 - 11:30 Ken Millett, Physical Knots
Abstract. I would discuss results on polygonal knots, knot energies,
and thickness of knots.
12:00 - 1:00 Jeff Weeks, What is a hyperbolic knot?
Abstract.
The talk will give a gentle introduction to the theory
of hyperbolic knots. The computer program SnapPea will
illustrate the main ideas.
2:30 - 3:30 Wilbur Whitten, Knot inversion and the isometry
groups
of hyperbolic 3-manifolds
Abstract:
The problem of deciding whether a given knot K in the 3-sphere is
invertible is shown to be equivalent to deciding when two knot groups
are
isomorphic. These are the groups
of two satellite knots formed with K as their (only) maximal companion,
and
each of these satellite knots is itself noninvertible regardless of
the
invertibility status of K. This result shows clearly why the
invertibility
question of a knot is generally so difficult and it coincidently leads
to
three conjectures (too lengthy to include in this abstract)
concerning ths isometry groups of the hyperbolic 3-manifolds obtained
by surgery on hyperbolic links. If these conjectures are correct,
such
an isometry group would be one of a well-defined finite collection.
3:45 - 4:10 Yongwu Rong, Introduction to Higher Order Link
Polynomials
Abstract:
Two major link invariants are Homfly+PT polynomial
and the Vassiliev invariants. We introduce what we call ``higher
order link polynomials'' that combine ingredients from both.
We carry out a detailed study for the order one polynomials.
Various possible applications will be discussed.
4:20 - 4:45 Jozef H. Przytycki, What is new in skein modules?
Abstract:
Skein modules are the basic objects of algebraic topology based on
knots
(as homology and homotopy groups are the basic objects of a classical
algebraic topology). The last half a year brought big progress in the
theory (which is 9 years old). It will be illustrated by an example
of detecting torsion in the Kauffman bracket skein modules using
an SL(2,C) character variety and hyperbolic geometry.
4:55 - 5:20 Adam Sikora, Skein algebra of a handlebody from
the
point of view of algebraic geometry
Abstract:
This talk will be mainly concerned with a non-standard
'algebraic topology'
built on knots. In particular, we will define {\it the Kauffman bracket
skein
module} $S(M)$, a module associated to any 3-dimensional manifold $M$.
A particular version of this module has a structure of a commutative
algebra
and therefore it is called a {\it skein algebra}. The skein algebra
depends
only on the fundamental group of a manifold.
We are going to investigate skein algebras
using methods of algebraic
geometry. In particular, we will show how the skein algebra associated
to a
manifold $M$ is connected with a {\it character variety}, an algebraic
set
representing all traces of homomorphisms of $\pi_1(M)$ into $Sl_2(C)$.
This will give a very nice geometric interpretation of character varieties,
which essentially have been investigated since the time of Poincare,
Fricke
and Klein.
5:30 - 5:50 Michael McDaniel, Cablings of Vassiliev Invariants
Abstract:
We will study Vassiliev Invariants coming form cabling and
describe a practical way of calculating them.
Local Organizers: Jozef H. Przytycki e-mail: przytyck@math.gwu.edu
Yongwu Rong e-mail: rong@math.gwu.edu