We determine the rational homology of the space of long knots in R for d 4 . Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with bracket of degree d 1 , which can be obtained as the homology of an explicit graph complex and is in theory completely computable.

We show that the Bousfield-Kan spectral sequence which computes the rational homotopy groups of the space of long knots in R-d for d >= 4 collapses at the E 2 page. The main ingredients in the proof are Sinha's cosimplicial model for the space of long knots and a coformality result for the little balls operad.

As in the case of the associahedron and cyclohedron, the permutohedron can also be defined as an appropriate compactification of a configuration space of points on an interval or on a circle. The construction of the compactification endows the permutohedron with a projection to the cyclohedron, and the cyclohedron with a projection to the associahedron. We show that the preimages of any point via these projections might not be homeomorphic to (a cell decomposition of) a disk, but are still contractible. We briefly explain an application of this result to the study of knot spaces from the point of view of the Goodwillie-Weiss manifold calculus.

AbstractLet M be a smooth manifold and V a Euclidean space. Let $ \overline{{{\text{Emb}}}} $(M,V) be the homotopy fiber of the map Emb(M,V) → Imm(M,V). This paper is about the rational homology of $ \overline{{{\text{Emb}}}} $(M,V). We study it by applying embedding calculus and orthogonal calculus to the bifunctor (M,V)↦ HQ ∧ $ \overline{{{\text{Emb}}}} $(M,V)+. Our main theorem states that if $$ \dim V \geqslant 2{\text{ED}}{\left( M \right)} + 1 $$(where ED(M) is the embedding dimension of M), the Taylor tower in the sense of orthogonal calculus (henceforward called “the orthogonal tower”) of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor $$ HQ \wedge \overline{{{\text{Emb}}}} {\left( {M,V} \right)}_{ + }. $$The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of $ \overline{{{\text{Emb}}}} $(–,V).

We give an overview of how calculus of the embedding functor can be used for the study of long knots and summarize various results connecting the calculus approach to the rational homotopy type of spaces of long knots, collapse of the Vassiliev spectral sequence, Hochschild homology of the Poisson operad, finite type knot invariants, etc. Some open questions and conjectures of interest are given throughout.

It is well-known that certain combinations of configuration space integrals defined by Bott and Taubes [11] produce cohomology classes of spaces of knots. The literature surrounding this important fact, however, is somewhat incomplete and lacking in detail. The aim of this paper is to fill in the gaps as well as summarize the importance of these integrals.

We associate a Taylor tower supplied by the calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral sequence collapses along this line and that the Taylor tower represents a universal finite type knot invariant.