Fields
Undergraduate Network:
Workshop on Knot Theory September 24, 2011
Description:
Knot theory has its roots in Gauss’ investigations
of surfaces and a program of physics conceived by Lord Kelvin
to classify atoms as knots and links. This workshop aims
to highlight some of the more modern developments of the
theory, in particular where knot theory interacts with physics
and other parts of mathematics.
Schedule:
10:00
a.m. – Alison Henrich, Seattle University
11:00
a.m. – Coffee Break
11:15 a.m. – Iain Moffatt, University of South Alabama
12:15 p.m. – Lunch
1:15 p.m. – Panel Discussion
1:45 p.m. – Coffee Break
2:00 p.m. – Louis H. Kauffman, University of Illinois
at Chicago
Talks:
Dr. Louis H. Kauffman, University of Illinois at Chicago
Knots and Physics
This talk is a self-contained introduction to relationships
of knot theory and physics. We begin with the bracket model
for the Jones polynomial and show how it is related to the
Potts model in statistical mechanics. We then discuss how
knot invariants are related to quantum field theory via
Witten's functional integral and how this is related to
the theory of loop quantum gravity.
Dr.
Alison Henrich, Seattle University
The Link Smoothing Game: A Tale of Knots & Links,
Games and Graphs
Recently, the concept of combinatorial games on knots was
introduced. The classic game begins with the shadow of a
knot, and players take turns choosing which strand goes
over and which strand goes under at crossings in the diagram.
The goal of one player is to unknot the knot while the other
player wants to make something non-trivial. This game inspired
my collaborator Inga Johnson and I to invent a new game,
called the Link Smoothing Game. We begin our game in much
the same way with the shadow of a knot or a link, but in
our game the players proceed to smooth at the crossings.
One player hopes that the final result is a multi-component
link, while the other player wants to create something with
a single component. We have translated this game into a
game on graphs, which we were able to classify almost entirely
according to which player has a winning strategy.
Dr.
Iain Moffatt, University of South Alabama
How to get to the Jones polynomial via linear algebra
The Jones polynomial is an invariant of knots and links
that is a polynomial. If you take a course on knot theory,
the chances are that this will be one of the first two knot
polynomials you will meet (the other being the Alexander
polynomial). The chances also are that you will see exactly
two constructions of the Jones polynomial: one through a
skein relation and one through the Kauffman bracket. Both
of these constructions use combinatorial operations to `unknot
a knot'. In this talk I will describe a third, less often
seen, approach to the Jones polynomial that uses familiar
linear algebra. Starting with the simple, but inspired,
idea of cutting a knot diagram into basic pieces and associating
a linear map to these pieces, we'll build up a family of
knot polynomials that has the Jones polynomial as one if
its most basic members. In fact, by using only the mathematics
that we all saw in our first courses in linear algebra (OK,
perhaps a tiny bit more), we'll obtain a far-reaching theory
with connections to statistical mechanics and quantum physics.
Sponsors:
Pure Math, Applied Math, and Combinatorics and Optimization
Club (Waterloo)
Mathematics Society (Waterloo)
Waterloo Mathematics
Ontario Government
NSERC
Fields Institute
