Little cubes and long knots

This paper gives a partial description of the homotopy type of K , the space of long knots in R 3 . The primary result is the construction of a homotopy equivalence K ≃ C 2 ( P ⊔ { ∗ } ) where C 2 ( P ⊔ { ∗ } ) is the free little 2-cubes object on the pointed space P ⊔ { ∗ } , where P ⊂ K is the subspace of prime knots, and ∗ is a disjoint base-point. In proving the freeness result, a close correspondence is discovered between the Jaco–Shalen–Johannson decomposition of knot complements and the little cubes action on K . Beyond studying long knots in R 3 we show that for any compact manifold M the space of embeddings of R n × M in R n × M with support in I n × M admits an action of the operad of little ( n + 1 ) -cubes. If M = D k this embedding space is the space of framed long n -knots in R n + k , and the action of the little cubes operad is an enrichment of the monoid structure given by the connected-sum operation.