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 basepoint. 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.

Let Emb(S^j,S^n) denote the space of C^infty-smooth embeddings of the j-sphere in the n-sphere. This paper considers homotopy-theoretic properties of the family of spaces Emb(S^j,S^n) for n >= j > 0. There is a homotopy-equivalence of Emb(S^j,S^n) with SO_{n+1} times_{SO_{n-j}} K_{n,j} where K_{n,j} is the space of embeddings of R^j in R^n which are standard outside of a ball. The main results of this paper are that K_{n,j} is (2n-3j-4)-connected, the computation of pi_{2n-3j-3} (K_{n,j}) together with a geometric interpretation of the generators. A graphing construction Omega K_{n-1,j-1} --> K_{n,j} is shown to induce an epimorphism on homotopy groups up to dimension 2n-2j-5. This gives a new proof of Haefliger's theorem that pi_0 (Emb(S^j,S^n)) is a group for n-j>2. The proof given is analogous to the proof that the braid group has inverses. Relationship between the graphing construction and actions of operads of cubes on embedding spaces are developed. The paper ends with a brief survey of what is known about the spaces K_{n,j}, focusing on issues related to iterated loop-space structures.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.1991 Mathematics Subject Classification. 57M27, 55R80, 57R40, 57M25, 55P99.

This paper gives a detailed description 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 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 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 which has a hyperbolic manifold contained in the JSJ-decomposition of its complement. Moreover, the homotopy type of K as an SO 2 -space is determined, which gives a detailed description of the homotopy-type of the space of embeddings of S 1 in S 3 .