Little cubes and long knots

This paper gives a partial description of the homotopy type of KK, the space of long knots in R3R3. The primary result is the construction of a homotopy equivalence K≃C2(P⊔{∗})K≃C2(P⊔{∗}) where C2(P⊔{∗})C2(P⊔{∗}) is the free little 2-cubes object on the pointed space P⊔{∗}P⊔{∗}, where P⊂KP⊂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 KK. Beyond studying long knots in R3R3 we show that for any compact manifold MM the space of embeddings of Rn×MRn×M in Rn×MRn×M with support in In×M admits an action of the operad of little (n+1)(n+1)-cubes. If M=DkM=Dk this embedding space is the space of framed long nn-knots in Rn+kRn+k, and the action of the little cubes operad is an enrichment of the monoid structure given by the connected-sum operation.