The topology of spaces of knots in dimension 3

Let K denote the topological space of C^∞ smooth embeddings of ℝ in ℝ³ that restrict to a (fixed) linear inclusion outside of some (fixed) ball. We call K the space of long knots in ℝ³. I will give a recursive description of the homotopy type of K, component-by-component. The path-components of K are the isotopy classes of long knots. First I’ll describe an ”indexing” of these components in terms of finite, labelled, rooted-trees, based on the JSJ-decomposition of 3-manifolds. Via this indexing, the homotopy type of any path-component of K can be described in terms of iterating three elementary bundle operations related to such things as: free little 2-cubes objects from homotopy theory, and ‘wreath product constructions’ that use natural signed symmetric representations of the isometry groups of certain hyperbolic link complements of a ‘brunnian type’.