  We consider the space of all smooth knots in the 3-sphere isotopic to a given
knot, with the aim of finding a small subspace onto which this large space
deformation retracts. For torus knots and many hyperbolic knots we show the
subspace can be taken to be the orbit of a single maximally symmetric placement
of the knot under the action of SO(4) by rotations of the ambient 3-sphere.
This would hold for all hyperbolic knots if it were known that there are no
exotic free actions of a finite cyclic group on the 3-sphere. For satellite
knots the situation is more complicated but still describable in fairly simple
terms. (This preliminary version of the paper does not include details for the
case of satellite knots.)
