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The Unreasonable Affinity of Knot Theory and the Algebraic Sciences

Cornell Topology Festival, May 1, 2003


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 will start my talk enthrilling half the audience with the definition of finite type (Vassiliev) invariants and their basic theory, leading to their well known relationship with Lie algebra2 (the other half of the audience will be dead bored). I'll move on to a vague and less well known discussion of the relationship between knots, quandles and Lie algebras, which in my opinion is one of the quickest ways to notice that there's something algebraically deep about knots. Finally I will say a few words about Khovanov's categorification of the Jones polynomial3,4 and about L. Ng's new contact homology invariants5.

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

2 Dror Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423-472.

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

4 Dror Bar-Natan, On Khovanov's categorification of the Jones polynomial, arXiv:math.GT/0201043.

5 Lenhard Ng, Knot and braid invariants from contact homology I and II, arXiv:math.GT/0302099 and arXiv:math.GT/0303343.

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