Calculus of the embedding functor and spaces of knots

We associate a Taylor tower supplied by 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.

We determine the rational homology of the space of long knots in R^d for $d\geq4$. Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the E^1 page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with a bracket of degree d-1, which can be obtained as the homology of an explicit graph complex and is in theory completely computable. Our proof ...

We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on ...

We show that the invariants $ev_n$ of long knots in a $3$-manifold, produced from embedding calculus, are surjective for all $n\geq1$. On one hand, this solves some of the remaining open cases of the connectivity estimates of Goodwillie and Klein, and on the other hand, it confirms one half of the conjecture by Budney, Conant, Scannell and Sinha that for classical knots $ev_n$ are universal additive Vassiliev invariants over the integers. We actually study long knots in any...

We show that the map on components from the space of classical long knots to the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n-1) knot invariant. We also compute the second page in total degree zero for the spectral sequence converging to the components of this tower as Z-modules of primitive chord diagrams,...

This paper is a little more detailed version of math-QA/0010017 "Sur l'homologie des espaces de n\oe uds non-compacts", where the first term of the Vassiliev spectral sequence (computing the homology of the space of long knots in ${\mathbb R}^d$, $d\ge 3$) was described in terms of the Hochschild homology of the Poisson algebras operad for $d$ odd, and of the Gerstenhaber algebras operad for $d$ even. In particular, the bialgebra of chord diagrams arises as some subspace of t...

We study high-dimensional analogues of spaces of long knots. These are spaces of compactly-supported embeddings (modulo immersions) of $\mathbb{R}^m$ into $\mathbb{R}^n$. We view the space of embeddings as the value of a certain functor at $\mathbb{R}^m$, and we apply manifold calculus to this functor. Our first result says that the Taylor tower of this functor can be expressed as the space of maps between infinitesimal bimodules over the little disks operad. We then show tha...

Consider the space of `long knots' in R^n, K_{n,1}. This is the space of knots as studied by V. Vassiliev. Based on previous work of the authors, it follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson algebra. A partial description of a basis is given here. In addition, the mod-p homology of this space is a `free, restricted Gerstenhaber-Poisson algebra'. Recursive application of this theorem allows us to deduce that there is p-torsion of all orders in ...

Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the rational homology of Ebar(M,V). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main theorem states that if the dimension of V is more than twice the embedding dimension of M, the Taylor tower in the sense of orthogonal calculus (henceforward called ``the orth...

We study the spaces of embeddings $S^m\hookrightarrow R^n$ and those of long embeddings $R^m\hookrightarrow R^n$, i.e. embeddings of a fixed behavior outside a compact set. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. We find a natural fiber sequence relating these spaces. We also compare the $L_\infty$-algebras of diagrams that encode their rational homotopy type, when the codimension $n-m\geq 3$.