I was a bit more sophisticated when I first heard of knot theory. My first thought was that it was either trivial or intractable, and most definitely, I wasn't going to learn it is interesting. But it is, and I was wrong, for the reader of knot theory is often lead to the most interesting and beautiful structures in topology, geometry, quantum field theory, and algebra.
Today I will talk about just one minor example, mostly having to do with the link to algebra: A straightforward proposal for a group-theoretic invariant of knots fails if one really means groups, but works once generalized to meta-groups (to be defined). We will construct one complicated but elementary meta-group as a meta-bicrossed-product (to be defined), and explain how the resulting invariant is a not-yet-understood yet potentially significant generalization of the Alexander polynomial, while at the same time being a specialization of a somewhat-understood "universal finite type invariant of w-knots" and of an elusive "universal finite type invariant of v-knots".
Handout and related links at http://www.math.toronto.edu/~drorbn/Talks/Regina-1206/
Let $G$ be a compact Lie group. The loop group $\Omega G$ is the set of maps from $S^1$ to $G$.
The based loop group (those loops which send the basepoint of $S^1$ to the identity element of $G$)
is an infinite-dimensional analogue of a coadjoint orbit of a compact Lie group. It is equipped with a natural $G$
action (pointwise conjugation) and a circle action (rotation of the loop); these two actions commute.
Its cohomology, K-theory and equivariant cohomology (under the $G$ action) have been studied since the work of
Bott in the 1940's, but its equivariant K-theory has not been studied until recently. I describe our recent results, which compute the equivariant $K$-theory of the based loop group of $SU(2)$ (both as a module and as an algebra).
(The talk is based on joint work with David Garcia-Perez, Carlos Palazuelos among others).