A couple of weeks ago Matt Daws and I gave a masterclass on knot theory as part of this year’s Leeds Festival of Science [pdf].
Many thanks go to Ruth Holland and Hazel Kendrick for organising us, and Dave Pauksztello for turning up and helping out.
First we spent some time playing with knotted electric wires, investigating when two knots are equivalent (i.e one can be pulled into the shape of the other, or more technically they are ambient isotopic), and when they’re not.
Here’s a motivating example, which begins to suggest that this is a tougher question than you might first imagine.
Then we cranked up the science, working through the writhe, Kauffman’s bracket polynomial, and ultimately the Jones polynomial for a selection of knots and links.
Matt created some hand-outs which he’s put online:
- Some knots: PDF file and LaTeX source.
- The writhe: PDF file and LaTeX source.
- Kauffman’s bracket polynomial: PDF file and LaTeX source.
- The Jones polynomial: PDF file and LaTeX source.
(As he says, we did tweak the definition of the Jones polynomial, omitting the final change of variable.)
Finally, here’s my lo-tech contribution which works through calculations of the bracket polynomial for some knots and links.
Simple City 