Topology of knot spaces in dimension 3

This paper is a computation of the homotopy type of K , the space of long knots in R the same space of knots studied by Vassiliev via singularity theory. Each component of K corresponds to an isotopy class of long knot, and we ‘enumerate’ the components via the companionship trees associated to the knot. The knots with the simplest companionship trees are: the unknot, torus knots, and hyperbolic knots. The homotopy-type of these components of K were computed by Hatcher. In the case the companionship tree has height, we give a fibre-bundle description of those components of K , recursively, in terms of the homotopy types of ‘simpler’ components of K , in the sense that they correspond to knots with shorter companionship trees. The primary case studied in this paper is the case of a knot which has a hyperbolic manifold contained in the JSJ-decomposition of its complement. AMS Classification numbers Primary: 57R40 Secondary: 57M25, 57M50, 57P48, 57R50