June 17, 2024

Fields Undergraduate Network:
Workshop on Knot Theory September 24, 2011


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.


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

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.

Pure Math, Applied Math, and Combinatorics and Optimization Club (Waterloo)
Mathematics Society (Waterloo)
Waterloo Mathematics
Ontario Government
Fields Institute

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Note: If anyone attending has any dietary restrictions, please email us at uwaterloofun<at>gmail.com.