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The Unreasonable Affinity of Knot Theory and the Algebraic Sciences

Bar Ilan University, March 20, 2002


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 with a vague 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. I'll move on to a quick description of the theory of finite type knot invariants and their concrete and extensive relationship with Lie algebras. I'll conclude (again at a somewhat vague level) with a certain finite presentation of knot theory that highlights its relationship with 6j-symbols and with Drinfel'd's theory of associators and quasi-Hopf algebras. I will say nothing about homological algebra, and that's a shame, because Khovanov's recent work on the subject is quite exciting.

I will come out of my talk just as puzzled as I will enter it, and you, I hope, a little more.

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

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