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