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 enthrilling half the audience with the
definition of finite type (Vassiliev) invariants and their basic
theory, leading to their well known relationship with Lie
algebra^{2} (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
polynomial^{3,4}
and about L. Ng's new contact homology invariants^{5}.

^{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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