De-looping embedding spaces.

This talk will focus on the homotopy-type of the space of smooth
embeddings of S^1 in S^3, or equivalently on the homotopy-type of the
space of `long knots' in R^3, this is the space of embeddings of R in R^3
which agree with a standard linear embedding outside of a fixed ball. The
first result is that this long knot space is similar to a 2-fold
loop-space, in that it has an action of the operad of 2-cubes, extending
the connect-sum operation. The second result is that this action is free
in the sense of May, with free generating space being the subspace of the
space of long knots consisting of the components corresponding to prime
long knots. This means the homotopy-type of the space of long knots
in R^3 is a functor in the homotopy-type of the space of prime long knots.
The 3rd result is the determination of the homotopy-type of the space of
prime long knots.  The end result describes the homotopy type of every
component of the space of long knots in R^3 as an iterated fibre bundle of
various K(pi,1) spaces.  Some of the monodromy is very `standard'
while some involve the hyperbolic geometry of certain `Brunnian-type'
links in the 3-sphere.