The Unreasonable Affinity of Knot Theory and the Algebraic Sciences

In an article bearing a similar title¹, Eugene Wigner once wondered about the numerous and unexpected ways in which mathematics comes into play in the natural sciences. While our topic and claims are much humbler, we find ourselves puzzled by the unreasonable affinity of knot theory and certain parts of algebra. It figures that knot theory is related to 3-dimensional topology. But why on earth should the mundane study of tangled shoelaces and unwieldy piles of seamen's rope be related to the elegance and sophistication of the likes of Lie algebras, 6j-symbols and homological algebra?

I'll tell you about the Jones polynomial and its relationship with Lie algebras and about Khovanov homology^2,3 for knots and its relationship with yet-to-be-explored parts of algebra. Everything will be very basic; fancy words like homology will be defined and a picture of Khovanov will be shown. The experts will be disapointed and everybody else will have fun.

1 Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications in Pure and Applied Mathematics 13-I (1960).

2 Mikhail Khovanov, A categorification of the Jones polynomial, arXiv:math.QA/9908171.

3 Dror Bar-Natan, Khovanov's Homology for Tangles and Cobordisms, arXiv:math.GT/0410495.

Slides:

1. WignersPaper.html.
2. KnotsStink.html.
3. TopInt.html.
4. GeomInt.html.
5. KnotsDull1.html.
6. KnotsDull2.html.
7. KauffmansBracket.html.
8. KauffmansCube.pdf.
9. DivideAndConquer.gif.
10. InvariantsOfSL2.gif.
11. FurtherAffinities1.html.
12. FigureEightSeifert.html.
13. BuiltInCancellations.gif.
14. KhovanovsCube.pdf.
15. WhoCares.html.
16. Kronecker.html.