What was the course about?
This course was an introduction to Knot Theory. We covered the following topics with varying degrees of depth:
- What is a knot?
- How do we distinguish between knots? Knot diagrams and Reidemeister moves.
- Tabulations of knots through history (we saw some of the famous mistakes in tabulations).
- Combinatorial Invariants: Tricolorability and Fox-n colourings.
- Polynomial Invariants: Conway Polynomial and Johns Polynomial.
- Properties undetectable from invariants (left and right trefoil, for example).
- Gauss Diagrams and the Casson Invariant.
- Prime knots and connected sums.
- How do we prove that a knot is prime? (We massively hand waved a proof 🙂 )
- Kauffman Bracket
- Little short story on Vassiliev Knot Invariants
What motivated me?
When I was preparing this course, I knew I was close to graduating from the Ph.D. After graduation I would not be able to participate as easily in teaching high school courses for a time due to restrictions on the work permits. Thus, I wanted to end with a course that was successful and satisfying for me.
In other courses I had always included knots as examples of the ideas being explored, but I have never taught about Knots themselves. I decided it was the time to do it! I have always loved knots and, at the time of this course, I have studied them for many years, even though they were not my main research area: I was ready for this!
It has been one of the most successful courses, with very good comments and reviews from the students, that I have ever given 😀
Exercise Sheets:
Other Resources:
Some of the material I used was based on the following online lectures, which I really like, and recommended to the students during the course (the one below is just the first video, there are many).
Malors 