Topology of spaces of knots in dimension 3

This paper is a computation of the homotopy type of K, the space of long knots in R^3, 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.