In an article bearing a similar title^{1}, 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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