Goodwillie’s cosimplicial model for the space of long knots and its applications

We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of compactly generated weak Hausdorff spaces. Using this, we compute the first page of the integral Bousfield–Kan homotopy spectral sequence of the tower of fibrations, given by the Taylor tower of the embedding functor associated to the space of long knots. Based on the methods in Conant (Am J Math 130(2):341–357. https://doi.org/10.1353/ajm.2008.0020, 2008), we give a combinatorial interpretation of the differentials d1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^1$$\end{document} mapping into the diagonal terms, by introducing the notion of (i, n)-marked unitrivalent graphs.


Introduction
Manifold calculus is a theory introduced by Goodwillie and Weiss, cf.[Wei99] and [GKW01], that produces a sequence of functors approximating a given good functor (Definition 1.1.3)from the category of open subsets of a manifold to the category CGH of compactly generated weak Hausdorff spaces.Let M and N be smooth manifolds.We are interested in studying the space Emb ∂ (M, N, f ) of smooth embeddings that are germ equivalent on the boundary of M to a fixed smooth boundary-preserving embedding f : M → N .We topologise this space with the compact-open topology.Now we can apply manifold calculus to the embedding functor Emb ∂ (−, N, f ) : Open (M ) op ∂ → CGH, sending an open subset V ⊆ M with ∂M ⊆ V to the embedding space Emb ∂ (V, N, f ).In this way we obtain information about the space Emb ∂ (M, N, f ) by studying the embedding functor and its sequence of approximations.
In this paper, we focus on analysing the following embedding functor associated to the space K = Emb ∂ (I, R 2 × D 1 , c) of long knots where c is an embedding representing the trivial long knot.Our motivation for studying this embedding functor is its close relation to knot theory, in particular, the theory of Vassiliev invariants.In Section 1 we recall some background on manifold calculus and give a precise definition of Date: October 5, 2023.Email: y.shi@uu.nl.Affiliation: Utrecht University, Utrecht, the Netherlands.
the space K.At the end of this section, we summarise briefly the original construction of the Vassiliev invariants and present how the theory of these knot invariants and the theory manifold calculus relate.As mentioned above, manifold calculus associates to Emb(−) a sequence of polynomial functors T n Emb(−) (Definition 1.1.7)approximating the functor Emb(−), Emb(−) . . .
Our approach to understanding the maps π 0 (η n (I)) is to compute the Bousfield-Kan homotopy spectral sequence associated to the tower of fibrations −−→ T 0 Emb(I). (1) As a first step, we study in Section 2 the construction [GKW01, Contruction 5.1.1]that relates cosimplicial spaces with good functors from Open ∂ (I) to the category of spaces.In Theorem 2.12 we give a more precise formulation of this construction, summarised as the theorem below.
Theorem 2. There is an isomorphism between the homotopy class of augmented cosimplicial spaces and the homotopy class of good functors F : Open ∂ (I) op → CGH.
We were not able to find a proof of the above theorem or of [GKW01, Contruction 5.1.1]in the literature.Thus, we give a proof in Section 2, using some results of [AF15].Therefore, we can associate a cosimplicial space Emb • to the embedding functor Emb(−) via the above equivalence, which provides us with a cosimplicial model for the tower of fibrations (1), namely We find the construction of this cosimplicial model very natural, yet it is the least used one in the study of the space of long knots.There is another cosimplicial model C • of the tower of fibration (1) which is defined using the Fulton-MacPherson compactification, cf.[Sin06] and [BCKS17].In Remark 2.27 we briefly recall the construction and properties of C • .
The n-th level Emb n of the cosimplicial space Emb • is weakly homotopy equivalent to the cartesian product of the configuration space Conf n (R 2 × D 1 ) and n copies of the sphere S 2 .In Section 3.3, we compute the Bousfield-Kan homotopy spectral sequence {E r p,q } p,q≥0 with integral coefficients associated to Emb • , using a presentation of the homotopy groups of the configuration spaces via iterated Whitehead products.More specifically, we are able to give an explicit description of the abelian groups E 1 p,q (Proposition 3.3.6 and Remark 3.3.9).Thanks to this, we obtain a simplification of the calculation of the differentials mapping into the diagonal terms on the E 1 -page.
Theorem 3 (Theorem 3.3.14).For p ≥ 3,1 the differential d 1 : E 1 p−1,p → E 1 p,p can be expressed as an explicit sum of four iterated Whitehead products.
In Section 4, we give a combinatorial interpretation of the groups E 1 p,p and E 1 p−1,p , as well as the differentials d 1 : E 1 p,p → E 1 p−1,p , based on the methods of [Con08].The group E 1 p,p is isomorphic to the abelian group T p−1 of labelled unitrivalent trees of degree p − 1 with a total ordering on its leaves, modulo AS-and IHX-relations (Proposition 4.5).In Figure 1 we draw an example of a labelled unitrivalent tree of degree 4.
Figure 1.A labelled unitrivalent tree of degree 4. The arrow is not part of the tree.
In order to describe the groups E 1 p−1,p , we introduce the notion of (i, p)-marked unitrivalent graphs (Definition 4.10).In Figure 2 we draw an example of a marked unitrivalent graph of degree 8. Proposition 4 (Proposition 4.15).Let D p−1 be the abelian group generated by (i, p − 1)marked unitrivalent graphs with 1 ≤ i ≤ p, modulo AS-and IHX sep -relations.Then we can identify D p−1 with the torsion-free part of E 1 p−1,p .The differentials d 1 have the following interpretation using unitrivalent graphs.
Theorem 5 (Theorem 4.20).Let p ≥ 4.Under the identifications above, the differential d 1 : E p−1,p → E p,p maps a (k, p − 1)-marked unitrivalent graph Γ k,p−1 to the linear combination Γ 1 p − Γ2 p − (Γ 2 k − Γ 1 k ) of unitrivalent trees of degree p − 1, where i) the linear combination Γ 1 p − Γ 2 p is obtained by performing the STU-relation (Definition 4.17) on Γ k,p−1 at the edge connecting the leaf labelled by p − 1 and the marked node v p−1 , and ii) the linear combination Γ 2 k − Γ 1 k is obtained by performing the STU-relation on Γ k,p−1 at the edge connecting the leaf labelled by k and the marked node v p−1 .
See Figure 3 for a visualisation of the graphs occurring in the theorem.We call the equivalence relation on the abelian group T p−1 (labelled unitrivalent trees of degree p − 1) generated by the image of d 1 the STU 2 -equivalence relation (Definition 4.17).As a corollary we have the following proposition.Proposition 6 (Conant).A tree τ ∈ T p−1 is STU 2 -equivalent to 0 if and only if τ ∈ im(d 1 ) under the isomorphism E 1 p,p ∼ = T p−1 of Proposition 4.5.
We also obtain a combinatorial interpretation of E 2 p,p for p ≥ 1.
We conclude in Section 5 by illustrating the connection between the manifold calculus tower of Emb(−) and some geometrical and combinatorial aspects of Vassiliev knot invariants, and mention some future work.
Situation.We work with simplicial categories, i.e. categories enriched over the category of simplicial sets equipped with the Kan model structure 2 .Most of the categories we consider in this text are by nature enriched over the category of compactly generated weakly Hausdorff spaces.We regard them as simplicial categories by applying the singular complex functor to the mapping spaces.For simplicity, we call the mapping simplicial sets obtained as above again mapping spaces.In particular, every ordinary category is a topologically enriched category with discrete morphism spaces.Thus we also consider them as simplicial categories.For an introduction on (simplicial) enriched categories and enriched functors see [Lur09, A.1.4,A.3] and [Hir03,Chapter 9].
We denote by holim the homotpy limits in simplicial categories.We refer the reader to [Hir03, Chapter 18] for a detailed explanation of homotopy limits.For the calculation of homotopy limits in the category of compactly generated weak Hausdorff spaces see [Hir03,Chapter 18.2], [Rie14, Chapter 6] and [MV15, Section 8].
2 See [Lur09, Appendix A.2.7] for the Kan model structure on the category of simplicial sets.

Background and motivation
1.1.Manifold Calculus.In this section we are going to briefly introduce the basic building blocks of manifold calculus.The main references for this section are [BW13], [Wei99] and [GW99], which also contain further motivation for this subject.
is a weak homotopy equivalence.Definition 1.1.4.Let M and N be smooth manifolds.
i) We define Emb(M, N ) to be the space of smooth embeddings of M into N .
ii) When M and N are smooth manifolds with boundary, we define Emb ∂ (M, N ) to be the space of smooth embeddings F : M → N which preserve the boundary, i.e.F (∂M ) ⊆ ∂N .iii) We define Emb ∂ (M, N, f ) to be the space of smooth embeddings that coincide with a given smooth embedding f : M → N near the boundary that is transverse to ∂N , i.e.
We topologise Emb(M, N ) and Emb ∂ (M, N, f ) with the compact-open topology.
Theorem 1.1.5(Weiss).Let M and N be smooth manifolds with dim M ≤ dim N , and let f : M → N be a fixed smooth embedding with f ⋔ ∂N .Then the embedding functor Definition 1.1.7.For a manifold M without boundary, a good functor F is a polynomial functor of degree at most n if for every open subset U ∈ Open (M ) and A 0 , A 1 , . . ., A n pairwise disjoint closed subsets of M which lie in U , the (n + 1)-cube is homotopy cartesian, i.e. the natural map χ(∅) → holim S̸ =∅ χ(S) is a weak homotopy equivalence.In other words, Remark 1.1.8.One obtains the definition of polynomial functors of degree at most n for manifolds M with boundary by replacing Open (M ) with Open ∂ (M ) and requiring that each A i has empty intersection with ∂U so that The name 'polynomial functor' may come from the following criterium for polynomial functions.
Lemma 1.1.9.A smooth function p : R → R satisfies Definition 1.1.12.Let M be a manifold and let F be a good functor.The n-th Taylor approximation T n F of F is the homotopy right Kan extension op , together with the natural transformation η n : F → T n F coming from the universal property of the homotopy right Kan extension.Written as a homotopy limit, the functor T n F is Example 1.1.13([Wei99, Section 0]).Let M, N be smooth manifolds (without boundary).The first Taylor approximation T 1 Emb(V ) of the embedding functor Emb(−, N ) is weakly homotopy equivalent to the immersion functor Imm(−, N ), which associates to an open subset V ⊆ M the space of immersions Imm(V, N ).In other words, the "linear" approximation of Emb(M, N ) is the space Imm(M, N ) of "local" embeddings.
Proposition 1.1.14(Weiss).Using the notation from Definition 1.1.12,the functor T n F toghether with the natural transformation η n has the following properties.
i) The functor T n F is a polynomial functor of degree at most n, ii) For any V ∈ Open n ∂ (M ), the map η n (V ) is a weak homotopy equivalence.iii) If F is a polynomial functor of degree at most n, then η n is a weak equivalence, i.e.
η n (V ) is a weak homotopy equivalence for every V ∈ Open ∂ (M ).iv) If µ : F → G is a natural transformation where G is a polynomial functor of degree at most n, then the natural transformation µ factors through T n F , unique up to contractible choices.
Proof.See [Wei99, Theorem 3.9, Theorem 6.1].□ Remark 1.1.15.In other words, the natural transformation η n : F → T n F is the best approximation of F by a polynomial functors of degree at most n and T n F is unique up to weak homotopy equivalence.
Definition 1.1.16.Let M be a smooth manifold and let F : Open ∂ (M ) → CGH be a good functor.The Taylor tower of F is the tower of natural transformations r i : Definition 1.1.17.In the situation of Definition 1.1.16,The Taylor tower of F converges if for every V ∈ Open ∂ (M ), the canonical map η(V ) : The Taylor tower of a good functor does not converge in general.However, for some embedding functors, we have the following convergence criterium.
Theorem 1.1.18(Goodwillie-Weiss).Let M and N be two smooth manifolds.Then the Taylor tower of the embedding functor Emb(−, N ) converges, if dim N − dim M ≥ 3.
Proof.See [GW99, Corollary 2.5].□ Thus, in the codimension 2 case, where interesting knot theory is developed, this convergence theorem is not applicable.Nevertheless, manifold calculus allows us to study the space of knots from a homotopy theoretic viewpoint, as shown in the later sections.For this purpose, let us first recall the basic notions of knot theory.1.2.Long knots and Vassiliev's knot invariants.Classically, knot theory studies smooth embeddings of S 1 into S3 up to isotopy.A long knot is an embedding from I to R 2 × D 1 coinciding with a fixed linear embedding near the boundary.We consider long knots instead of knots in this paper because of technical convenience.For example, the space K of long knots has an E 1 -algebra 3 structure induced by concatenation, cf.[BCKS17, Section 4] and [Bud08].The one point compactification of each long knot induces an isomorphism between π 0 (K) and π 0 (Emb(S 1 , S 3 )).Thus for the study of knot invariants with values in an abelian group A, i.e. elements of H 0 (Emb(S 1 , S 3 ); A), it does no harm to use long knots instead of knots.However, note that Emb(S 1 , S 3 ) and the space of long knots K have different higher homotopy groups, cf.[Bud08, Theorem 2.1].
For the joy of the reader, see Figure 4 for an example of a long knot.Definition 1.2.2.Two long knots K 0 , K 1 ∈ K are called isotopic4 if there is a smooth map and F | I×{t} ∈ K for every t ∈ I.We call F an isotopy between K 0 and K 1 , and write Classifying knots up to isotopy has always been a central problem in knot theory, so finding computable knot invariants plays an important role.The knot invariants that we are interested in are the so called finite type invariants, which are a collection of knot invariants discovered by Vassiliev, see [Vas90] for Vassiliev's original approach and [Bar95] for an alternative explanation.There is a precise definition of Vassiliev invariants using combinatorics, which links the study of Vassiliev invariants to Chern-Simon theory and algebraic structures of Feynman diagrams, cf.[Bar95] and [Kon94].For motivation, we sketch Vassiliev's original approach here: Instead of focusing on one specific knot invariant, Vassiliev considered the whole set5 H 0 (K; A) of all knot invariants with values in a given abelian group A. The main steps of his computation are the following: i) Embed K in the space C ∞ ∂ (I, R 2 × D 1 , c) of all smooth maps from I into R 2 × D 1 which are germ equivalent with c on the boundary (Definition 1.2.1).ii) Compute the homology of the complement of K in C ∞ ∂ (I, R 2 × D 1 , c). iii) Use Alexander duality to obtain H • (K; A), and in particular H 0 (K; A).In order to perform step ii) and iii), Vassiliev finds a filtration by finite dimensional vector spaces {Γ i } i∈N , which approximate the space A) via a homological spectral sequence associated to this filtration.Furthermore, this filtration gives a filtration of the homology groups H In each of the finite dimensional vector spaces we can apply Alexander duality to obtain a filtration Finally, a Vassiliev invariant of degree at most n with values in A is defined to be an element of ) be a smooth map.We call a point p ∈ im(K) a singularity of K if K −1 (p) contains more than one element.The filtration ) \ K arises by distinguishing K by the type and the number of its singularities.Thus it appears natural to conjecture that the system of Vassiliev invariants classifies knots.On the other hand, it is still open whether Vassiliev invariants detect the unknot.
Notation 1.2.5.We abbreviate the embedding functor Emb ∂ (−, R 2 × D 1 , c) associated to the space K as Emb(−), since this is the only embedding space we are going to consider in the rest of the text.
The convergence for the Taylor tower of the embedding functor Emb(−) corresponding to the space K indeed fails, because the set π 0 (K) is countable, but the homotopy limit of the corresponding Taylor tower can be shown (formally) to be uncountable.However, the natural transformations η n in the Taylor tower of Emb(−) do provide us with a sequence of knot invariants, i.e. π 0 (η n (I)) : π 0 (K) → π 0 (T n Emb(I)) for every n ∈ N.These are actually finite type invariants: Theorem 1.2.7 (Budney-Connant-Koytcheff-Sinha).Let n ∈ N. The map π 0 (η(I)) is an additive finite type knot invariant of degree at most n − 1.
Proof.See [BCKS17, Theorem 6.5].□ Remark 1.2.8.Let A be an abelian group.An additive knot invariant is a abelian monoid homomorphism π 0 (K) → A, where the abelian6 monoid structure of π 0 (K) is induced by connected sum of knots.Thus, one non-trivial point in Theorem 1.2.7 is to give π 0 (T n Emb(I)) an abelian monoid structure such that it is compatible with the connected sum of knots.The authors of [BCKS17] solve this by defining compatible E 1 -algebra structures on the spaces K and T n Emb(I), cf.[BCKS17, Section 4].For more survey on the operadic structures on the space of long knots (not necessarily in codimension 2), cf.
We refer the readers to [Hab00], [CT04a], [CT04b] and [Sta00] for various geometric and combinatorial descriptions of universal additive Vassiliev invariants.If the conjecture is true, then we expect π 0 (T n Emb(I)) to be isomorphic to the abelian groups generated by certain combinatorial diagrams.The rest of the text aims at explaining one method of understanding the homotopy groups (at least π 0 ) of T n Emb(I), namely, the computation of a Bousfield-Kan homotopy spectral sequence of the following tower of fibration: In order to do this, we introduce in the next section a cosimplicial space associated to the embedding functor Emb(−).

A cosimplicial model constructed by Goodwillie
Goodwillie observed that a good functor on Open ∂ (I) corresponds to an augmented cosimplicial space, and vice versa, cf.[GKW01, Section 5].The cosimplicial spaces arise naturally in this way enjoy nice properties and can be considered as cosimplicial models for the Taylor tower of the corresponding good functors (Definition 2.15).Because certain details are left out in the paper [GKW01] for the construction of the cosimplicial spaces and their properties, we reformulate this correspondence in terms of equivalence of simplicial functor categories and give proofs in full detail, using some results from [AF15].This equivalence further facilitates the computations in Section 3.3.Definition 2.1.We define the following two categories.
i) The simplex category ∆ ∆ ∆ consists of the objects [n] = {0, 1, . . ., n} ⊆ N with n ≥ 0, and the morphisms are the order-preserving maps.ii) Denote by ∆ ∆ ∆ + the category of finite, totally ordered sets.The morphisms are order-preserving maps.
Remark 2.2.The simplex category ∆ ∆ ∆ is equivalent to the category of non-empty finite totally ordered sets, which we also denote, by abuse of notation, by ∆ ∆ ∆.
Notation 2.5.By restricting an augmented cosimplicial space Y • + to the subcategory ∆ ∆ ∆, we obtain the associated cosimplicial space, which we denote as Y • .
Convention 2.6.By the totalisation Tot X • of a cosimplicial space X • we always mean the totalisation Tot X • of a fibrant replacement X • of X • , with respect to the model structure introduced in [BK72, Section X.4.6].Similarly, by the n-th partial totalisation Tot n X • of X • we always mean Tot n X • .
Remark 2.7 ([BK72, Chapter XI.4.4]).We have natural weak homotopy equivalences In the following, we reformulate and prove the correspondence described in [GKW01, Construction 5.1.1]between augmented cosimplicial spaces and good functors from Open ∂ (I) op fin to the category of spaces.We need first few notation from the theory of simplicially enriched categories [Lur09, A.3].
Situation 2.11.Denote by Cat ∆ the category of simplicially enriched categories.In other words, objects of Cat ∆ are simplicially enriched categories and morphisms are simplicial functors.A functor In fact, the category Cat ∆ admits a left proper combinatorial model structure with this notion of weak equivalences [Lur09,Proposition A.3.2.4].Therefore, we can form the homotopy category h Cat ∆ of Cat ∆ .
Theorem 2.12.Let κ : is the subset of good functors.Remark 2.13.A more elegant way to formulate this is to use the language of ∞-categories: the functor κ induces an equivalence Here, Fun g denotes the ∞-category of good functors, N denotes the nerve functor, and Ho denotes the ∞-category of homotopy types which can be obtained from CGH by inverting all weak homotopy equivalences [Lur09, Remark 1.2.16.3] .In particular, here we have an "enriched"-version of the statement of the above theorem, instead of only talking in the homotopy category h Cat ∆ .We will see later in the proof that Theorem 2.12 is a consequence of the universal property of simplicial localisation, which can be in general only formulated in h Cat ∆ because the construction itself involves various cofibrant and fibrant replacements.However, we choose not to go further into the ∞-categorical setting, since it is not necessary for our later applications.
Corollary 2.14 ([GKW01, Remark 5.1.3]).Let F : Open ∂ (I) op fin → CGH be a good functor.Denote by F • + an augmented cosimplicial space such that κ • F • + ≃ F .Then the associated cosimplicial space F • has the following properties: Definition 2.15.Using the notations of the above corollary, we call a cosimplicial space F • satisfying the i) and ii) of Corollary 2.14 a cosimplicial space associated to the good functor F .
Because of F • enjoys the property of Corollary 2.14.ii), we call F • a cosimplicial model for the tower of fibrations obtained by evaluation of the Taylor tower of F on I.
Remark 2.16.Let n = #(π 0 (I \ V )) − 1. Denote by Emb • a cosimplicial space associated to Emb(−).Corollary 2.14.i) implies that Emb n is weakly homotopy equivalent to the Cartesian products of the configuration space of n-points in R 2 × D 1 (points of embeddings) and n-copies of S 2 (tangent vectors at the embedded points).We will use this relation to configuration spaces in Section 3. Now we work towards proofs of Theorem 2.12 and Corollary 2.14, which is, to the best of our knowledge, not available in the literature.Let us begin by introducing several categories.
For our application, we consider m = 1 and M = I.
Definition 2.22.Define the subcategory Disc ∂ 1/I of Disc 1/I whose objects are the embeddings such that the boundary ∂I of I is in the image.
In the same way, let Isot ∂ 1/I be the subcategory of Isot 1/I whose objects are the embeddings such that ∂I is in their images.Explicitly, objects of Disc ∂ 1/I are embeddings of the form Using the same proof strategy as Proposition 2.20, we have Corollary 2.23.The canonical functor Disc ∂ 1/I → Disc ∂ 1/I induces an equivalence of simplicial categories The functor is an equivalence of simplicial categories.ii) The functor Im : is an equivalence of ordinary categories.Let Isot(I) be the subcategory of Open ∂ (I) fin which has the same objects as Open ∂ (I) fin with only the morphisms that are isotopy equivalences.
is an equivalence of categories.Proof.i) We can prove that the functor is essentially surjective and fully faithful.This is true since the every connected component of each space of morphism of Disc ∂ 1/I is contractible.See also [AF15,Lemma 3.11].
ii) First Im is essentially surjective, because the boundary of I is in the image of i for any i ∈ Isot ∂ 1/I .For any two objects i 1 : V 1 → I and i 2 : V 2 → I in Disc ∂ 1/I , the morphism set Mor Disc ∂ 1/I (i 1 , i 2 ) is either empty or a one element set.Also the morphism set Mor Open ∂ (I) (im(i 1 ), im(i 2 )) is either empty or has one element.Thus Im is fully faithful.Similarly it follows that Im| Isot ∂ 1/I is an equivalence.□ Corollary 2.25.We have the following equivalences of simplicial categories Proof of Theorem 2.12.Recall the functor κ : Open ∂ (I) op fin → ∆ ∆ ∆ + with κ(V ) = π 0 (I \ V ) from Theorem 2.12.This functor fits into the equivalences of Corollary 2.25, i.e. we have the following commutative diagram where κ loc is the induced map of κ by the universal property of the localisation functor L, since κ sends an isotopy equivalence morphisms in Open ∂ (I) fin to isomorphisms in ∆ ∆ ∆ + .By the universal property (2.19) of localisation and isotopy invariance of good functors, precomposing with the localisation functor L induces an bijection of the set of homotopy classes of functors.By the equivalences of Corollary 2.25, the map κ loc in the diagram (2.26) also induces an equivalence of simplicial functor categories.Therefore, the map κ induces an bijection Proof of Corollary 2.14.Let F • + be an augmented cosimplicial space such that κ • F • + ≃ F .Then, part i) follows by composition of functors.
As for the proof of Corollary 2.14.ii),We have Remark 2.28.One way to generalise Theorem 2.12 in higher dimension, i.e. good functors from Open ∂ (M ) op with dim M > 1, is to consider configuration categories con(M ) associated to a smooth manifold M .See [BW18] for a detailed explanation on this subject.

An integral homotopy spectral sequence for Emb •
To a cosimplicial space Emb • (Definition 2.15) associated to the functor Emb(−), we can associate the Bousfield-Kan homotopy spectral sequence {E p,q } q≥p≥0 with integral coefficients, cf.[BK72, Chapter X].In this section, we will first briefly recall some properties of this spectral sequence (Section 3.1) and then give a concrete computation of the d 1 -differentials that map into the diagonal terms (Section 3.3).For the computation, we make use of the calculation of the homotopy groups for Conf n (R 2 × D 1 ), which we will recall in Section 3.2.

3.1.
A spectral sequence for cosimplicial spaces.We begin by introducing techniques that we need for the computation of Bousfield-Kan spectral sequences.The main reference for this section is [BK72, Chapter X].
Given a cosimplicial space X • : ∆ ∆ ∆ → CGH, there is a tower of fibrations (cf.[BK72, Chapter 6, Section 6.1]) Applying Bousfield-Kan homotopy spectral sequence, cf.[BK72, Section X.6], to the tower of fibrations (3.1.2),we obtain a spectral sequence approximating the homotopy groups of Tot X • , whose first page is given by E 1 p,q = π q−p (L p X • ), where q ≥ p ≥ 0, and E 1 p,q = 0 otherwise.With the help of the cosimplicial structure, we can calculate the homotopy groups of the spaces L p X • and the differentials in terms of the homotopy groups of X p .
Proposition 3.1.3(Bousfield-Kan).Given a cosimplicial space X • : ∆ ∆ ∆ → CGH, we have where the push-forward s i * : π q (X p ) → π q (X p−1 ) is induced by the codegeneracy maps s i .Proof.See [BK72, Section X.6.2].□ Proposition 3.1.4(Bousfield-Kan).Given a cosimplicial space X • , the first page of the Bousfield-Kan homotopy spectral sequence of X • is given by where q ≥ p ≥ 0, and the push-forward s i * : π q (X p ) → π q (X p−1 ) is induced by the codegeneracy maps s i .The differential d 1 : E 1 p,q → E 1 p+1,q on the first page is given by where the push-forward δ i * : π q (X p ) → π q (X p+1 ) is induced by the coface maps δ i on X p .Proof.See [BK72, Chapter X.7].□ 3.2.Homotopy groups of Conf n (R 2 × D 1 ).From Section 3.1 we see that we need to compute the homotopy groups of Emb n for n ≥ 0, in order to compute the Bousfield-Kan spectral sequence associated to the cosimplicial model Emb • .By Remark 2.16 we know that Emb n relates closely to the configuration spaces of n points in R 2 × D 1 .Therefore, let us gather some information about the homotopy groups of configurations spaces in this section.The main reference for this section is [FN62] and [FH01].
Definition 3.2.1.Let M be a smooth manifold (possibly with boundary).Define the configuration space Conf n (M ) of n ≥ 1 points on M as Now we focus on Conf n (R 2 × D 1 ) for n ≥ 0.
Proof.See [FN62, Theorem 2].□ Thus we can compute π * (Conf n (R 2 × D 1 )) inductively via the splitting long exact sequences for the fibre bundles pr k,n for 0 ≤ k ≤ n and n ≥ 2. And we can conclude the following corollary.
Corollary 3.2.5 (Fadell-Neuwirth).For n ≥ 2 and i ≥ 1, we have Now we are going to introduce a set of generators for π 2 (Conf n (R 2 × D 1 )), which we will use in the computations of Section 3.3.Definition 3.2.6.For 1 ≤ i ̸ = j ≤ n, define the map x ij as the composition of the following two maps x → (q i + x, q j+1 , . . ., q n ), and Proof.The image S ij := im(x ij ) of x ij is homeomorphic to a 2-sphere.For a fixed j with 1 ≤ j ≤ n, the space Note that for every i with 1 ≤ i < j, the map x ij is the positive generator of π 2 (S ij ).Thus, by Hurewicz isomorphism theorem, the maps . Now let j varies and apply Corollary 3.2.5, we have that the maps The case k = 0 works since we are looking at Euclidean spaces.
where for 1 ≤ i < j ≤ n the positive generator of S ij is x ij .Thus by the following theorem of Hilton about the homotopy groups of wedges of spheres, we reduce the computation of homotopy groups of Conf n (R 2 × D 1 ) to homotopy groups of spheres.
Definition 3.2.9([Hil55],[Whi78, Page 511-512]).Let and denote by ι i the positive generator of π r i +1 (S r i +1 ).Note that ι i can be considered as an element of π r i +1 (T ) via the canonical embedding S r i +1 → T .10i) The basic products of weight 1 are the elements ι 1 , ι 2 , • • • , ι k .We order the set of basic products of weight 1 by we have c ≤ a.We declare every basic product of weight ω to be greater than any basic product of smaller weight.We order the set of basic products of weight ω lexicographically, i.e. for two basic products [a, b] and ii) Thus a basic product p of weight ω is a suitably bracketed word in the symbols ι i for i = 1, . . ., k. Assume ι i appears w i times in p.We define the height h(p) of p as k i=1 r i w i .Theorem 3.2.10(Hilton).Using the notation of Definition 3.2.9,let P be the set of (formal) basic products of ι 1 , . . ., ι k .We have where the direct summand π * (S h(p)+1 ) is embedded in π * (T ) by composition with the basic product p ∈ π h(p)+1 (T ).

Proof. See [Whi78, Theorem 8.1]. □
The Whitehead products of x ij for 1 ≤ i, j ≤ n and i ̸ = j of π 2 (Conf n (R 2 × D 1 )) satisfy some relations, which we will use in the computation in the Section 3.3.Proposition 3.2.11(Hilton, Nakaoka-Toda, Massey-Uehara).Let X be a topological space.Then the Whitehead product [−, −] on π * (X) is bilinear, antisymmetric and satisfies the Jacobi identity, i.e. for α ∈ π a+1 (X), β ∈ π b+1 (X) and γ ∈ π c+1 (X) we have Proof.See [Hil61], [UM57] and [NT54].□ Proposition 3.2.12.Let 1 ≤ i, j ≤ n and i ̸ = j.The element Proof.See the proof of [FH01, Section II.3, Theorem 3.1].□ 3.3.A homotopy spectral sequence for the Taylor tower of Emb(−).In this section we perform some computation of the integral homotopy Bousfield-Kan spectral sequence of cosimplicial space Emb • (Definition 2.15) associated to the embedding functor Emb(−) (Notation 1.2.5).Recall that this spectral sequence aims at analysing the homotopy limit of the tower of fibrations according Corollary 2.14.By Corollary 2.14 we have the weakly homotopy equivalence For the computation of the homotopy spectral sequence associated to Emb • we need to compute the induced maps on homotopy groups of the coface and codegeneracy maps.From (3.3.1)we have By abuse of notation, we consider x ij (Proposition 3.2.7),for 1 ≤ i < j ≤ n, as elements of π * (Emb n ) under the natural inclusion.
Proposition 3.3.3.Let i, j, l, n ∈ N and 1 ≤ i < j ≤ n + 1 and 0 ≤ l ≤ n and n ≥ 2. i) We have ii) Denote by Z the set of basic products of the elements x i,j containing x u,l+1 or x l+1,v for 1 ≤ u ≤ l and l + 2 ≤ v ≤ n + 1.Under the isomorphism in Theorem 3.2.10, the kernel of the map s l * (c) is isomorphic to p∈Z π r (S h(p) + 1), for r ≥ 2. iii) For r ≥ 2 , the map s l * (t) is the canonical projection where forgetting the l-th component.Thus the kernel of s l * (t) is isomorphic to (0) l−1 × π r (S 2 ) × (0) n−l .Proof.i) and iii) follows from the description of s l right above the proposition.
For the proof of ii), let us abbreviate s l * (c) by s l * in this part of the proof.Note that for n ≥ 2, we have s l * (x uv ) = 0 if and only if u = l + 1 or v = l + 1.Therefore, for n ≥ 2 and z ∈ Z, we have s l * (z) = 0 by the naturality of the Whitehead product.Thus Similar analysis of the definition of the coface maps tells us that these maps δ l of Emb • corresponds to "breaking" the embeddings of the (l + 1)-th interval into the embedding of two subintervals.Therefore, one representative for the map δ l with 0 < l < n + 1 is the following: where the scalar ϵ ∈ R is so chosen that (x 1 , . . ., x l , x l + ϵv l , x l+1 , . . ., x n ) is a well-defined point in Conf n+1 (R 2 × D 1 ).For l = 0 and l = n + 1, we have where x −1 = (0, 0, −1) and x +1 = (0, 0, 1) and e = (0, 0, 1).
By explicit calculation we obtain the following Proposition 3.3.4.Let i, j, l, n ∈ N such that n ≥ 2 and 1 ≤ i < j ≤ n and 0 ≤ l ≤ n+1.i) For n ∈ N and n ≥ 2, we have ii) Denote by y k a generator for the k-th component π 2 (S 2 ) of (π 2 (S 2 )) n .We have that otherwise.

□
Now we can compute E 1 p−1,p and E 1 p,p in the homotopy spectral sequence associated to the cosimplicial space Emb Proof.We have that s l * = s l * (c) × s l * (t).□ Proposition 3.3.6.i) For p ≥ 3 and 1 ≤ i < p − 1, let T be the set of basic products of the elements x i,p−1 of height p − 2, such that each x i,p−1 appears exactly once.Let F be the set of basic products of elements x i,p−1 of height p − 1, such that one x k,p−1 appears exactly twice and all other x i,p−1 appear exactly once.Then we have where π p (S p−1 ) and π p (S p ) are embedded in π p (Conf p−1 (R 2 × D 1 )) by composition with the basic products in T and F respectively.ii) For p ≥ 2, let H be the set of basic products of height p − 1 of the elements in x i,p for 1 ≤ i ≤ p − 1 such that each x i,p appears exactly once.Then where the direct summands π p (S p ) are embedded in π p (Conf p (R 2 × D 1 )) by composition with the basic products in H.
Proof.i) Recall from Proposition 3.1.4that E 1 p−1,p ∼ = π p (Emb p−1 ) ∩ p−2 l=0 ker(s l * ).By Corollary 3.3.5 we only need to consider the π p (Conf p−1 (R 2 ×D 1 )) component of π p (Emb p−1 ).In other words, Recall from Corollary 3.2.5 that k } k∈N be the set of basic products of the elements x ij for i = 1, . . ., j − 1.Using Theorem 3.2.10,we have Next we need to examine which elements of π p (Conf p−1 (R 2 × D 1 )) lie in p−2 l=1 ker(s l * (c)).By Proposition 3.3.3.iii), it is sufficient to see which basic products lie in p−2 i=1 ker(s l * (c)).Let us consider the following cases: a) Suppose j ̸ = p − 1.In this case, we have s p−1 * (b In this case there exists at least one index 1 ≤ i < p − 1 such that x i,p−1 does not appear in b ii) Similar to i), we recall that E 1 p,p □ Remark 3.3.9.Using the same method as in the proof above, we can see that the abelian group E 1 p,q for 0 ≤ p ≤ q is isomorphic to the direct sum of the q-th homotopy groups of spheres of dimension at most q, indexed by the basic products where all elements {x i,p | 1 ≤ i ≤ p − 1} appear at least once11 .Remark 3.3.10.Budney-Conant-Koytcheff-Sinha gives the general formula for the abelian groups E 1 p,q , cf. [BCKS17, Proposition 7.2] by computing the Bousfield-Kan spectral sequence with integer coefficients associated to the cosimplicial model C • (Remark 2.27).The results in the above proposition agree with those from [BCKS17, Proposition 7.2] once one removes the homotopy groups of spheres that are 0. Our proof, using another cosimplicial model Emb • and doing the computation directly from definitions, provides an alternative approach of the computation of the spectral sequences, as well as more details for the arguments of [BCKS17, Proposition 7.2].

With the description of E 1
p−1,p and E 1 p,p in terms of elements x ij where 1 ≤ i ≤ j and j = p − 1 or j = p, we are going to give an explicit and simplified formula for the differential d 1 : E 1 p−1,p → E 1 p,p now.Proposition 3.3.11.
Proof.First we have with generator x 12 .Thus with generator y 1 for the component π 2 (S 2 ), and with generator x 12 for the component π 2 (Conf 2 (R 2 × D 1 )).Applying Proposition 3.1.4,we have Therefore, According to Proposition 3.3.6we have for p ≥ 4, Since E 1 p,p is torsion free, we see that d 1 is trivial on T π p (S p−1 ), and we conclude that we only need to consider the restriction of d 1 to F π p (S p ). Notation 3.3.12.We denote the torsion-free part of E 1 p−1,p by E 1 p−1,p /tors, i.e. the summand F π p (S p ) in Equation 3.3.7.Proposition 3.3.13.For p ≥ 4, denote by D sep p the set of iterated Whitehead products of the elements x i,p−1 for i = 1, . . ., p − 2 with the following properties: i) For every w ∈ D sep p , there exists one x k(w),p−1 that appears exactly twice and all other x i,p−1 with 1 ≤ i ≤ p − 2 and i ̸ = k(w) appear exactly once.ii) Every w ∈ D sep p is of the form w = [c 1 , c 2 ] where c 1 is an iterated Whitehead product of elements x i,p−1 with i ∈ I and c 2 is an iterated Whitehead product of elements x i,p−1 with j ∈ J such that I, J ⊆ {1, . . ., p − 2}, I ∩ J = {k(w)} and I ∪ J = {1, 2, . . ., p − 2}.
Then, E 1 p−1,p /tors is generated by D sep p .Proof.Denote by D p the set of iterated Whitehead products of the elements x i,p−1 with i = 1, . . ., p − 2 satisfying only condition i).Using the same argument as in the proof of Proposition 3.3.6,we see that In particular, the basic products in F , which are a basis of E 1 p−1,p /tors, are contained in D p .We have reduced the desired statement to the following claim which we prove by induction.
Claim.For p ≥ 4, any element of D p can be written as a linear combination of elements of D sep p using only the Jacobi identity and antisymmetry relations (Proposition 3.2.11).For p = 4, the claim follows by listing all the elements of D 4 and using the Jacobi identity of the Whitehead product.
Assume that the claim is true for all p ≤ n with n ≥ 4. Let p = n + 1 and consider w = [a 1 , a 2 ] ∈ D n+1 .Without loss of generality, we can assume that x 1,n is the repeated element in w.If the two copies of x 1,n appear in a 1 and a 2 separately, then w is already an element of D sep p .Otherwise, both copies of x 1,n appear in either a 1 or a 2 , say they appear in a 1 .By assumption a 1 is a Whitehead product of elements x m,n .Define the set M := {m ∈ N|x m,n appears in a 1 }.So we know #M ≤ n − 2 and 1 ∈ M .The element x m,n appears exactly once in a 1 , for by replacing each occurrence of x m,n in a 1 by r(x m,n ).By an inductive assumption, we can write a ′ 1 as a finite sum is a Whitehead product of the elements x m,n with m ∈ M , where x 1,n appears exactly twice and x m,n appears exactly once for m ̸ = 1.
Therefore w can be written as where ϵ 1 and ϵ 2 denote the signs which come from the Jacobi identity for the Whitehead product.For every i ∈ I, we have that The upshot is that it is sufficient, for the computation where Before we prove the theorem, let us take a look at an example of computation of d 1 .Now let us calculate d 1 (w) using the formulas in Theorem 3.3.14.We have and Proof of Theorem 3.3.14.First we proof the proposition for k ≤ p − 3.

Combinatorial interpretation
Since we computed the abelian groups E 1 p,p , E 1 p−1,p and the differential of the spectral sequence associated to the cosimplicial model Emb • in the previous section, we would also know E 2 p,p .In this section we give a combinatorial interpretation of E 1 p,p and E 1 p−1,p and the differentials d 1 , based on the calculation of Proposition 3.3.6 and Theorem 3.3.14.As a corollary, we obtain a graphic interpretation of the groups E 2 p,p .From these interpretations we will see that this spectral sequence relates closely to the theory of Vassiliev invariants.Definition 4.1.A unitrivalent graph Γ is a graph whose nodes have only degree 1 or 3, together with a cyclic order on the edges at each node.We call the nodes of degree 1 leaves and nodes of degree 3 trivalent nodes.When Γ has n leaves, we define a labelling (or total ordering) on Γ to be a bijection of the set {1, 2, . . ., n} to the set of leaves.Denote by UTG the collection of labelled unitrivalent graphs.We define the degree of Γ as the number of nodes divided by 2.
When we draw a labelled unitrivalent graph, we place the leaves on an oriented line, ordered according to the labelling.Unless explicitly mentioned, the cyclic orders of the trivalent nodes are given counterclockwise.See Figure 5 for an example of labelled unitrivalent graph.Definition 4.2.We define the following relations on Z[UTG]: i) Two labelled unitrivalent graphs Γ 1 and −Γ 2 are AS-related if Γ 1 and Γ 2 are the same up to the cyclic order at one node.This is depicted in Figure 6.ii) Let Γ be a labelled unitrivalent graph.Let e be an edge in Γ between two trivalent nodes v and w.Then Γ is IHX-related to the difference Γ ′ − Γ ′′ of the following two labelled unitrivalent graphs Γ ′ and Γ ′′ : Let {e, e ′ v e ′′ v } be the ordered set of edges at the node v, i.e. we have e < e ′ v < e ′′ v (cyclic order).In the same way, let {e, e ′ w , e ′′ w } be the ordered set of edges at the node w.The graph Γ ′ arises from Γ by deleting the edge e and the nodes v and w of Γ , and adding an edge e ′ and two trivalent nodes v ′ and w ′ such that the ordered set of edges at v ′ are {e ′ , e ′′ v , e ′ w } and the ordered set of edges at w ′ are {e ′ , e ′′ w , e ′ v }.The unitrivalent graph Γ ′′ is constructed in a similar fashion.For Γ ′′ the ordered set of edges at the nodes v ′ is {e ′ < e ′′ v < e ′′ w } and at the nodes w ′ is {e ′ < e ′ w < e ′ v }.This is depicted in Figure 7.
Construction 4.4.For p ≥ 2, denote by T p the set of iterated Whitehead products of the elements x ip with 1 ≤ i < p such that x ip appears at most once in a given iterated Whitehead product.We are going to construct a one-to-one correspondence labelled unitrivalent tree of degree at most p − 1 together with a monotone bijection of their labelling with a subset of {1, . . ., p} containing p .
Define the length of τ ∈ T p to be the total number of occurrences of x i,p with i = 1, . . ., p − 1 in τ .We will define Ψ T inductively on the length n of τ .Define Ψ T (x ip ) to be the degree 1 labelled unitrivalent tree consisting of two nodes labelled by i and p and an edge connecting them.Assume that for all τ k ∈ T p of length k with k ≤ n − 1 < p, we have that Ψ T (τ k ) is a degree k − 1 unitrivalent tree with labelling L τ k := {i ∈ N | x ip appears in τ k } ∪ {p}.For a tree τ n = [τ ′ , τ ′′ ] ∈ T p of length n, we know that τ ′ and τ ′′ are elements of T p and of length smaller than n.By induction hypothesis, both Γ ′ := Ψ T (τ ′ ) and Γ ′′ := Ψ T (τ ′′ ) have a leaf with label p.We define the labelled unitrivalent tree Ψ T (τ n ) to be the tree that arises by joining the tree Γ ′ and Γ ′′ at the respective leaves labelled by p, and joining to this joint point a new leaf labelled by p.The set of labels of Ψ T (τ n ) is L τ ′ ∪ L τ ′′ .This construction is depicted in Figure 8, where also the cyclic order at the joint node is indicated.Now we proceed to define the inverse map Φ T .For a labelled unitrivalent tree Γ 1 of degree 1 with the set of labellings {i, p}, set Φ T (Γ 1 ) = x ip .Assume that we have already defined Φ T for unitrivalent trees of degree smaller than n with n < p−1.Every unitrivalent tree Γ n of degree n can be depicted as in Figure 8. Then define where Γ ′ and Γ ′′ are the trees depicted in Figure 8.By construction, the two maps Ψ T and Φ T are inverse to each other.Lemma 4.6.Let J p ⊆ T p be the set of iterated Whitehead products of the elements x ip with i = 1, . . ., p − 1 such that each x ip appears exactly once in an iterated Whitehead product.Then we have where ∼ denotes the antisymmetry and Jacobi identity relations from Proposition 3.2.11.
Proof.Recall H from Proposition 3.3.6,and note that H ⊆ J p ⊆ E 1 p,p .Thus any element of J p \ H can be written as linear combination of elements of H. Furthermore, this linear combination is produced by applying the Jacobi identity and antisymmetry relation to the element, cf.[Hal50, Theorem 3.1].As a result, we obtain the desired a group isomorphism E 1 p,p Proof of Proposition 4.5.We are going to define group homomorphisms such that the induced map ψ T and ϕ T on the quotients E 1 p,p and T p−1 are inverse to each other.We will define our morphisms on generators and extend linearly to the whole group.
For p = 2, define ψ T (x 12 ) = Γ and ϕ T (Γ ) = x 12 , where Γ is the labelled unitrivalent tree of degree 1 and with labelling set {1, 2}.There is no relation to consider when passing to the quotients E 1 2,2 and T 1 .Thus ψT and φT are inverse to each other by definition.
where13 L i := {j ∈ N | x jp appears in v i } ∪ {p} and To see that the anti-symmetry of the Whitehead product corresponds to the AS-relation, we look at (4.7) Recall from Construction 4.4 that the only difference between the tree Ψ T ([v 1 , v 2 ]) and the tree Ψ T ([v 2 , v 1 ]) is the cyclic order at the trivalent node which is adjacent to the leaf with label p, i.e.
Note that the sum of signs in Equation 4.7 is Thus we obtain that the element in Formula 4.7 AS-related to 0.
As for the Jacobi identity, take where ϵ 1 , ϵ 2 and ϵ 3 are the suitable signs.Again by Construction 4.4, we have that is IHX-related to 0, and thus the element in Equation 4.8 is IHX-related to 0. Therefore, ψT is well-defined.
Let Γ p−1 be a labelled unitrivalent tree of degree p − 1, drawn as in Figure 8, define where L 1 and L 2 is the set of labels of Γ ′ and Γ ′′ respectively.Similar to the discussion of ψ T , one can show that ϕ T is well-defined.Therefore, the maps ψ T and ϕ T are inverses to each other by construction.□ Definition 4.10.Let i, j ∈ N with i ̸ = j.An (i, j)-marked unitrivalent graph Γ ij is a unitrivalent graph of degree j together with two marked nodes v i and v j that satisfy the following properties: i) The underlying graph of Γ is connected and has exactly one simple cycle14 .
ii) The two marked nodes v i and v j are adjacent to the leaf with label i and j, respectively, and lie on the simple cycle.Denote by UTG i,j the set of (i, j)-marked unitrivalent graphs.
In Figure 5, we can mark the nodes adjacent to the leaf with lable 3 and 8, respectively, and obtain a (3, 8)-marked unitrivalent graph, see Figure 9. Definition 4.11.For p ≥ 3, define D p to be the abelian group generated by the collection of (i, p)-marked unitrivalent graphs with 1 ≤ i < p, modulo AS-and IHX sep -relations, i.e.
, where the IHX sep -relation is the usual IHX-relation, except that the edge e, which appears in Definition 4.2.ii), is not allowed to be an edge adjacent to the marked nodes.
Similar as in the proof of Proposition 4.5 we can check by explicit computation that the induced maps ψD and φD on the quotients E  ii) Let Γ be a labelled unitrivalent graph.Then  Proof.This is because the STU 2 -relation does not change the connectivity and degree of the unitrivalent graphs, and it also does not add simple cycles to the graphs.□ Recall the computation of the differential d 1 : E 1 p−1,p → E 1 p,p from Proposition 3.3.14.Note that in the computation only d 1 | E 1 p−1,p /tors is relevant, and by abuse of notation we will write d 1 instead of d 1 | E 1 p−1,p /tors in the following.By Proposition 4.5 and Proposition 4.15 we can consider d 1 as a map between two groups of unitrivalent graphs as follows where Γ 1 , Γ 2 and Γ w are depicted in Figure 14.We will discuss the signs ϵ 1 , ϵ 2 and ϵ w at the end.Recall the formula for ∂ k (w) from Proposition 3.3.14.We write where c k 1 , c k+1 2 , c k+1 1 and c k 2 correspond exactly to the four Whitehead brackets in the formula of ∂ k (w).Thus we have 1 and Γ k 2 are the labelled unitrivalent trees depicted below Note that the underlying unlabelled trees of Γ k i and Γ k+1 i for i = 1, 2 are the same as the ones for Γ i respectively.Furthermore, we have where Γ 1 k and Γ 2 k are the labelled unitrivalent trees depicted in Figure 15.We perform the same steps for where Γ p 1 and Γ p 2 are the labelled unitrivalent trees depicted in Figure 16.
p .Now let us check the signs.For i = 1, 2, let L i be the set of labels of Γ i .Thus we have where ϵ 1 and ϵ 2 appeared right at the beginning the proof.Let L j i be the set of labels of Γ j i and L p i be the set of labels of Γ p i , for j ∈ {k, k + 1} and i ∈ {1, 2}.Then we have In conclusion, we are justified to write Corollary 4.22 (Conant).Let τ ∈ T p−1 , then τ ∈ im(d 1 ) if and only if τ is STU 2 -related to 0.
Proof."⇐" It follows from the formula for d 1 in the previous theorem that the image of the generators under d 1 are STU 2 -related to 0. Thus, any element in im(d 1 ) is also STU 2 -related to 0. "⇒" It is sufficient to prove that any linear combination of the form Γ 1 p − Γ 2 p − (Γ 2 k − Γ 1 k ) with 1 ≤ k ≤ p − 2, lies in the image of d 1 .Note that Γ 2 k − Γ 1 k is the result of performing a STU-relation at the trivalent node adjacent to the leaf with label k in Γ w = ψD (w), cf.Corollary 4.24.i) For p ≥ 4, the group E 2 p,p is isomorphic to the abelian group generated by unitrivalent trees of degree p − 1, modulo AS-, IHX-, and ST U 2 -relations.ii) For p = 3, we have E 2 3,3 iii) For p = 0, 1, 2, we have E 2 p,p = 0.
Proof.See [Con08,Theorem 3.3].□ Remark 4.26.We refer the readers to [Bar95], [CT04a], [CT04b] and [Hab00], for the close relation between Vassiliev knot invariants and unitrivalent graphs.As already points out by Bott in [Bot95], studying the groups E r n,n , and especially the passage from E 2 n,n to E ∞ n,n may be another approach to the theory of Vassiliev invariants.

Conclusion and further work
The connection between Vassiliev invariant and the embedding functor Emb(−) can be expressed in the following diagram with n ≥ 3 ii) The clasper surgery equivalence ∼ Cn is an equivalence relation on the set π 0 (K) of isotopy classes of knots defined by Habiro in [Hab00], and he proves that the canonical map π 0 (K) → π 0 (K)/∼ Cn is the universal additive Vassiliev invariant of degree n − 1. iii) The group G n−1 has a geometric interpretation using gropes.There is an isomorphism G n−1 ∼ = ker π 0 (K)/∼ Cn → π 0 (K)/∼ C n−1 A grope is an embedded (CW)-complex in R 3 , whose boundary components are knots.Conant and Teichner [CT04a;CT04b] explain the connection among gropes, clasper surgery and Vassiliev invariants.
In this text we showed the commutativity of the right most square in the above diagram, by giving combinatorial interpretations to our computations of the spectral sequence.The key ingredient for the calculations is the equivalence between the category of augmented cosimplicial spaces and the category of good functors from Open ∂ (I) op to spaces, which provides us with a cosimplicial space Emb • associated to the embedding functor Emb(−).
We find it interesting that the upper row of Diagram 5.1 is algebraic in nature whereas the lower row is geometric and combinatorial.If Conjecture 1.2.9 holds, the map ηn (I) would be an isomorphism of groups.In [Kos20] Kosanovic shows that the square marked with "⋆" commutes, thus making Diagram 5.1 commutative.A natural question that arises for us now is whether we can give a geometric or combinatorial interpretation of the Goodwillie-Weiss tower of knots, e.g. using gropes or clasper surgeries.In our forthcoming work [KST], we are working towards extending the map ηn (I) to a map of spaces.More specifically, we are constructing a space of gropes together with a continuous map to T n Emb(I) such that the induced map on path-connected components is ηn (I), for every n ∈ N and n ≥ 0.

Figure 2 .
Figure 2. A (3, 8)-marked unitrivalent graph.The black dots indicate the marked nodes.The arrow is not part of the graph.

Figure 3 .
Figure 3.An example of d 1 applied to a (k, p − 1)-marked unitrivalent graph.The triangles are placeholders for subgraphs, which stay unmodified.
is a good functor.Proof.See [Wei99, Proposition 1.4].□ Now we are going to introduce the approximation sequence for good functors produced by manifold calculus.Definition 1.1.6.Denote by [n] the set {0, 1, . . ., n}. i) Define the category Pow([n]) as the category whose objects are the subsets of [n] and the morphisms are inclusions of subsets.ii) Define the full subcategory Pow([n]) ̸ =∅ of Pow([n]), whose objects are the nonempty subsets of [n].
S⊆[n](−1) #S p i∈S x i = 0(1.1.10)forany collection of real numbers x 0 , x 1 , . . ., x n if and only if p is a polynomial of degree at most n.□ Definition 1.1.11.Let M be a smooth manifold and let n ∈ N. Define Open n ∂ (M ) to be the full subcategory of Open ∂ (M ) whose objects are the open subsets W of M that are diffeomorphic to N(∂M ) ⊔ ( k R m ) with 0 ≤ k ≤ n.Here N(∂M ) denotes a (non-fixed) tubular neighbourhood of ∂M in M .

Figure 4 .
Figure 4.An example of a long knot.
Definition 2.8.Let Open ∂ (I) fin be the full subcategory of Open ∂ (I) whose objects are the open subsets of I that contain ∂I and have only finitely many path connected components, i.e.Ob (Open ∂ (I) fin ) := {I} ∪ n≥0 Ob (Open n ∂ (I)) Proposition 2.9.A good functor on Open ∂ (I) op is determined up to weak homotopy equivalence by its restriction on Open ∂ (I) op fin .Proof.This follows by Definition 1.1.3.ii) of good functor.□ Remark 2.10.The restriction of a good functor on Open ∂ (I) op to Open ∂ (I) op fin is isotopy invariant in the sense of Definition 1.1.3.i).An isotopy invariant functor on Open ∂ (I) op fin fullfills Definition 1.1.3.ii)automatically.
Definition 2.17.i) Define the category Man m of smooth oriented m-dimensional manifolds.Objects of Man m are smooth oriented manifolds of dimension m, and the morphisms are the orientation-preserving smooth embeddings.ii) Define the simplicial category Man m of smooth oriented m-dimensional manifolds.This category has the same objects as Man m , and the morphisms are spaces of orientation-preserving smooth embeddings, equipped with the compact-open topology.iii) Define the full subcategory Disc m of Man m whose objects are finite disjoint unions of R m and R ≥0 × R m−1 .iv) Define the full simplicial subcategory Disc m of Man m that has the same objects as Disc m .v) Let M be a smooth oriented manifold of dimension m.Define the category Disc m/M := Disc m × Manm Man m/M , where Man m/M is the slice-category over M .Objects of Disc m/M are embeddings of finite disjoint unions of R m and R ≥0 × R m−1 into M .vi) Let M be a smooth oriented manifold of dimension m.Define the simplicial category Disc m/M := Disc m × Manm Man m/M , where Man m/M is the over category over M .vii) Let M be a smooth oriented manifold of dimension m.Define the subcategory Isot m/M of Disc m/M which has the same objects as Disc m/M , but only the morphisms that are isotopy equivalences.Situation 2.18.Let C and D be simplicially enriched categories and W be a set of morphisms in C. Denote by C[W −1 ] the simplicial localisation of C with respect to W .It is a simplicial category together with a functor L : C → C[W −1 ].There are several models [DK80b] 7 , [DK80a] and [Lur09, A.3.5] etc. for simplicial localisations, which are all equivalent after suitable fibrant or cofibrant replacements [Ste17].Simplicial localisations enjoys the following universal property [Lur09, Proposition A.3.5.5]: the functor L induces an injective map Hom h Cat ∆ C[W −1 ], D → Hom h Cat ∆ (C, D) (2.19) whose image is the subset of functors C → D that sends W to equivalences in D. Proposition 2.20 (Ayala-Francis).The canonical functor Disc m/M → Disc m/M induces an equivalence of simplicial categories We define basic products of weight bigger than 1 recursively.A basic product of weight ω is a Whitehead product [a, b], where a and b are both basic products of weights α < ω and β < ω respectively such that a) α + β = ω and a < b, and b) if b is defined as the Whitehead product [c, d] of basic products c and d, then • .Corollary 3.3.5.Let l, n, r ∈ N and n ≥ 2 and 0 ≤ l ≤ n and r ≥ 2, and recall the notations from Convention 3.3.2.For the degeneracy map Emb n+1 s l − → Emb n , we have ker s l * = ker s l * (c) × ker s l * (t) and * (c) × (0) n .
(j) k , and thus s i−1 * (b (j) k ) ̸ = 0. c) Suppose j = p − 1 and h(b (j) k ) = p − 2. In this case each x i,p−1 with 1 ≤ i ≤ p − 2 appears in b (j) k exactly once.Thus for all 0 ≤ l ≤ p − 2, we have s l * (b (j) k ) = 0, since s l * (x l+1,p−1 ) = 0 and x l+1,p−1 appears in b (j) k .d) Suppose j = p − 1 and h(b (j) k ) = p − 1.In this case there exists an index i k such that x i k ,p−1 appears exactly twice in b (j) k , and all other x i,p−1 with 1 ≤ i ≤ p − 2 and i ̸ = i k appear exactly once in b (j) k .As in c) we see s l * (b (j) k ) = 0 for 0 ≤ l ≤ p − 2. Thus p−2 i=1 ker(s i * (c)), or E 1 p−1,p , is generated by basic products of the form in c) and d), which yields (3.3.7).

Figure 5 .
Figure 5.A unitrivalent graph of degree 8.The arrow is not part of the graph.

Figure 6 .
Figure 6.A visualisation of the AS-relation.

Figure 9 .
Figure 9.A (3, 8)-marked unitrivalent graph.The blacks dots indicate the marked nodes.The arrow is not part of the graph.

Figure 11 .
Figure 11.A visualisation of the STU-relation.
obtained by performing the STU-relation at the leaf of Γ labelled by n, and Γ ′ 2 − Γ ′′ 2 is obtained by performing the STU-relation at the leaf of Γ labelled by m.The STU 2 -relation is depicted in Figure 12.

Figure 13 .
Figure 13.An example of d 1 applied to a (k, p − 1)-marked unitrivalent graph.The triangles are placeholders for subgraphs, which stay unmodified.

Figure 14 .
Similarly, Γ 1 p − Γ 2p is the result of performing a STU-relation at the trivalent node adjacent to the leaf with label p − 1 in Γ w .Therefore, we can obtain the linear combinationΓ 1 p − Γ 2 p − Γ 2 k + Γ 1 k via performing STU-relations on a (k, p−1)-marked unitrivalent graphs at its two marked trivalent nodes.By Proposition 4.15, the domain of the differential d 1 is exactly the set of (k, p − 1)-marked unitrivalent graphs.□ Remark 4.23.Our proofs of Theorem 4.20 and Corollary 4.22 supplement the proof of [Con08, Proposition 4.8] with more details.With the notion of marked unitrivalent graph, we are able to give the combinatorial interpretation of the map d 1 , instead of only its image.
π 0 (T n Emb(I)) explain and give references to some of the notations in the diagram: i) Recall that E i n,n with i ∈ {1, 2, ∞} denotes the diagonal terms of the homotopy Bousfield-Kan spectral sequence associated to the tower of fibrations• • • → T n Emb(I) → T n−1 Emb(I) → • • • → T 0 Emb(I) with E ∞ n,n∼ = ker π 0 (T n Emb(I)) → π 0 (T n−1 Emb(I)) Definition 1.1.1.i)Denote by CGH the simplical category of compactly generated weak Hausdorff spaces.We will call CGH the category of spaces for simplicity.ii)For a manifold M , denote by Open ∂ (M ) the category of open subsets of M which contain ∂M .Morphisms of Open ∂ (M ) are inclusions of these open subsets.For a manifold M ′ without boundary, we will simplify the notation as Open (M ′ ).Definition 1.1.2.A smooth codimension zero embedding i v : (V, ∂V ) → (W, ∂W ) between smooth manifolds V and W is an isotopy equivalence if there exists a smooth embedding i w : (W, ∂W ) → (V, ∂V ) such that i v • i w and i w • i v are isotopic to id (W,∂W ) and id (V,∂V ) respectively.
Definition 1.1.3.Let M be a smooth manifold of dimension m.A good functor on Open ∂ (M ) is a functor F : Open ∂ (M ) op → CGH of simplicial categories, which satisfies the following conditions: where L is the localisation functor in diagram (2.26) and F loc denotes the induced functor from Open ∂ (I) fin [Iso(I) −1 ] op to CGH by F .The first equality is by definition.The second weak homotopy equivalence comes from the equivalences of categories from Corollary 2.25.The third weak homotopy equivalence follows by observing that under the functor Fun Open ∂ (I) fin [Iso(I) −1 ] op , CGH L•− − − → Fun (Open ∂ (I) op fin , CGH) a homotopy limit cone over a functor Open ∂ (I) fin [Iso(I) −1 ] op → CGH restricts to a homotopy limit cone over a functor Open ∂ (I) op fin → CGH.□ Remark 2.27.There is another cosimplicial model C • for the above tower of fibrations, constructed via a compactification of Conf n (R 2 × D 1 ), cf.[BCKS17, Section 3] and [Sin09, Section 6].Compared with Emb • , the cosimplicial space C • has the advantage that it is geometric and various versions of C • have already been used in context concerning finite type invariants of 3-manifold, cf.[AS94] and [BT94].Using this cosimplicial space C • , the authors of [BCKS17] give the space Tot n C • ≃ T n Emb(I) an E 1 -algebra structure, which is used to define the abelian group structure on π 0 (T n Emb(I)) mentioned in Remark 1.2.8, cf.[BCKS17, Corollary 4.13, Section 5.7].We are exploring whether we can define similar multiplicative structure on the cosimplicial space Emb • .
4.14.Note that the usual Jacobi identity is not well-defined in Z[D sep p ]. Proposition 4.15.Recall the group E 1 p−1,p /tors from Proposition 3.3.13.For p ≥ 4, the construction above induces an isomorphism E 1 p−1,p /tors ∼ = D p−1 of groups.Proof of Proposition 4.15.Similar as in the proof of Proposition 4.5, we can define group homomorphisms such that the induced map ψ D and ϕ D are on the quotients are inverses to each other.Let w = [c 1 , c 2 ] ∈ D sep p , say with repeated occurrence of x k,p−1 .For i = 1, 2, define the set of labels L 1 p−1,p /tors and D p−1 are well-defined.More precisely, antisymmetry corresponds to AS-relation and the separated Jacobi identity corresponds to IHX sep -relation.By construction the maps ψ D and ϕ D are inverse to each other.□ Definition 4.17.Define the following relations on Z[UTG]: i) Let Γ be a labelled unitrivalent graph.Let e be the edge connecting the leaf labelled by n and the adjacent trivalent node v. Then Γ is ST U -related to the difference Γ ′ − Γ ′′ of the following two labelled unitrivalent graphs Γ ′ and Γ ′′ Denote by {e, e 1 , e 2 } the ordered set of edges at the node v, i.e. e < e 1 < e 2 (cyclic order).The graph Γ ′ arise from Γ the following steps: First, delete the edge e, the node v and the leaf labelled by n.Second, add two leaves, labelled by n and n + 1 with adjacent edges e 2 and e 1 respectively.Third, relabel the leaves labelled with m by m + 1 if m > n.The graph Γ ′′ is constructed similar to Γ ′ .For Γ ′′ the adjacent edges of the leaves labelled by n and n + 1 are e 1 and e 2 respectively.The STU-relation is depicted in Figure 11.
where Γ 1 p and Γ 2 p are depicted 15 in Figure16.Reviewing the leaves of Γ i k and Γ i p as placed on the oriented line,