Topology of knot spaces in dimension 3

Abstract

This paper gives a detailed description of the homotopy type of 𝒦, the space of long knots in ℝ3, 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 list the components via the companionship trees associated to knots. The knots with the shortest 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 where the companionship tree has more than one vertex, we give a fibre-bundle description of the corresponding components of 𝒦, recursively, in terms of the homotopy types of components of 𝒦 corresponding to knots with shorter companionship trees. The primary case studied in this paper is the case of a knot that has a hyperbolic manifold contained in the JSJ-decomposition of its complement. Moreover, the homotopy type of 𝒦 as an SO2-space is determined, which gives a detailed description of the homotopy type of the space of embeddings of S1 in S3.