A Geometric Homology Representative in the Space of Long Knots
AA GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OFLONG KNOTS
KRISTINE E. PELATT
Abstract.
We produce explicit geometric representatives of nontrivial homology classes inEmb( ˆ S , R d ), the space of long knots, when d is even. We generalize results of Cattaneo, Cotta-Ramusino and Longoni to define cycles which live off of the vanishing line of a homology spectralsequence due to Sinha. We use configuration space integrals to show our classes pair nontriviallywith cohomology classes due to Longoni. Introduction
Knot spaces have recently been the subject of much interest. Let Emb( ˆ S , R d ) be the space ofembeddings from S to R d with fixed initial point and initial tangent vector, which is homotopyequivalent to the space of long knots. Using Goodwillie-Weiss embedding calculus, Sinha [9]defines spectral sequences converging to the homology and cohomology of Emb( ˆ S , R d ) for d > E page. There is another spectral sequence, due to Vassiliev [11], which convergesto the homology of Emb( S , R d ). The E term of Vassiliev’s spectral sequence agrees with the E term of the embedding calculus spectral sequence by work of Turchin [10]. These approachesallow one to combinatorially understand the ranks of the homology groups of Emb( S , R d ), butdo not immediately give geometric understanding or representing cycles and cocycles in knotspaces. We present representing cycles and cocycles defined through techniques which apply toall classes in the spectral sequence.In [3] Cattaneo, Cotta-Ramusino and Longoni produce explicit, nontrivial, k ( d − − dimensionalcycles and cocycles. We give a brief summary of these results in Section 3. They define a chainmap from a graph complex to the de Rham complex of Emb( S , R d ), and produce cocycles asimages of graph cocycles consisting of trivalent graphs. To produce cycles, they use families ofresolutions of singular knots with k transverse double points. These cycles all live along the( − q, q ( d − − diagonal in the first page of the homology spectral sequence, which also servesas a vanishing line. To establish nontriviality, they show the pairing between certain cycles andcocycles is nonzero. For d odd, Sakai produces a (3 d − − dimensional cocycle in the space of longknots coming from a non-trivalent graph cocycle. To establish the nontriviality of this cocycle, heevaluates it on a cycle produced using the Browder bracket coming from the action of the littletwo-cubes operad on the space of framed knots.The main result of this paper is the explicit production of a nontrivial cycle which lives off ofthe vanishing line of the homology spectral sequence for d even, using techniques which shouldgeneralize. We define this cycle by generalizing the methods of Cattaneo, Cotta-Ramusino andLongoni to families of resolutions of singular knots with triple points. In particular, we first definea topological manifold M β and an embedding of M β into Emb( ˆ S , R d ), extending and correcting a r X i v : . [ m a t h . A T ] N ov KRISTINE E. PELATT the results in a preprint of Longoni [5]. Longoni also defines a cocycle which is the image of anon-trivalent graph when d is even. We show that the pairing between Longoni’s cocycle and ourcycle is nonzero and thus both are nontrivial.Our cycle generalizes, and our techniques are closely related to the spectral sequence combina-torics, giving possible recipes for representatives of all cycles in the embedding calculus spectralsequence. This is in contrast to Sakai’s approach, which would require new input for any Browder-primitive classes off of the ( − q, q ( d − − diagonal. These results will appear in future work,but we discuss them briefly at the end of this paper.2. Definition of the cycle
The idea at the heart of our method to produce homology classes in knot spaces goes backto Vassiliev’s seminal work [11]. In finite type knot theory, one defines the derivative of a knotinvariant by taking an immersion with transverse double-points and evaluating the knot invarianton the resolutions of that immersion. We require a generalization of such immersions.
Definition 2.1.
An immersion γ : S (cid:44) → R d has a transverse intersection r -singularity at ¯ t =( t , t , . . . , t r ) ∈ I × r with 0 < t < t < · · · < t r <
1, if all of the γ ( t i ) coincide and the derivatives γ (cid:48) ( t i ) are generic in the sense that any d or fewer of them are linearly independent.To connect with the language naturally produced by the embedding calculus spectral sequence,we use bracket expressions to encode singularity data. Sinha calculates in [9] that the subgroup of P ois d ( p ), the p − th entry of the Poisson operad (see [7]), generated by expressions with q bracketssuch that each x i appears inside a bracket pair and the multiplication “ · ” does not appearinside a bracket pair, is also a subgroup of E − p,q ( d − in the reduced homology spectral sequence.This is the full E − p,q ( d − in the spectral sequence converging to the homology of the space ofembeddings modulo immersions. On this subgroup, the differential d : E − p,q ( d − → E − p − ,q ( d − is d = (cid:80) pi =0 ( − i ( δ i ) ∗ , where ( δ ) ∗ is defined by adding x in front of the expression and replacingeach x j by x j +1 , ( δ p +1 ) ∗ is defined by adding x p +1 to the end, and for 1 ≤ i ≤ p , the map ( δ i ) ∗ is defined by replacing x i by x i · x i +1 and x j by x j +1 for j > i . In [10], Tourtchine does furthercalculations in this spectral sequence. Example 2.2.
The bracket expression β = β + β where β = [[ x , x ] , x ] · [ x , x ] and β =[ x , x ] · [[ x , x ] , x ] is a cycle in E − , d − . Definition 2.3.
A pair ( γ, ¯ t ) of an immersion and a sequence ¯ t = 0 < t < t < · · · < t p < β ∈ P ois d ( p ) if γ has a transverse r -singularity at the sequence0 < t i < . . . < t i r < x i , . . . , x i r appear inside of a bracket in β .For example, the knots K and K in Figure 1 respect β and β , respectively. A knot canrespect a bracket expression but have higher singularities; for example K also respects [ x , x ] · [ x , x ]. Definition 2.4.
We will denote the subspace of all pairs ( γ, ¯ t ) ∈ Imm( ˆ S , R d ) × I × r respectinga bracket expression by Imm ≥ β ( ˆ S , R d ), with the convention Imm φ ( ˆ S , R d ) = Imm( ˆ S , R d ). Thesubspace of Imm ≥ β ( ˆ S , R d ) consisting of immersions which do not have higher singularities willbe denoted by Imm = β ( ˆ S , R d ). GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OF LONG KNOTS 3
Figure 1.
The singular knots K and K .In the spectral sequence, bracket expressions of the form (cid:81) km =1 [ x i m , x j m ] are E -cycles. Sub-manifolds representing these cycles are well known and described in Section 2 of [3]. Briefly, westart with a singular knot K ⊂ R d with k double points which respects (cid:81) km =1 [ x i m , x j m ], andresolve each double point by moving one strand passing through the double point off of the other.For each vector in S d − we have a possible direction in which to move the strand, and thereforea possible way to resolve the double point. The subset of Emb( ˆ S , R d ) consisting of all such reso-lutions of K is a submanifold parameterized by (cid:81) m S d − , and its fundamental class correspondsto the cycle (cid:81) km =1 [ x i m , x j m ] of the spectral sequence.For higher singularities, we start with ideas of Longoni [5] and produce resolutions of transverseintersection singularities by moving one strand at a time off the intersection point. Assume therank of the singularity r is less than d , so the (tangent vectors of the) strands in question span aproper subspace. There are two cases - resolving a double point and resolving a higher singularity.If r ≥
3, we are moving a strand off the intersection point. The complementary subspace to the(tangent vector of the) strand has a unit sphere S d − which parametrizes the directions to moveone strand off the intersection point. If r = 2, we consider a unit sphere S d − in the complimentarysubspace which parametrizes the directions to move one strand off another.Resolutions of triple point singularities (and higher singularities) can produce further singu-larities (see Figure 4). By restricting away from neighborhoods of those “additional singularity”resolutions, we produce submanifolds with boundary which we show can be pieced together tobuild representatives of E -cycles in the spectral sequence. We formalize as follows. Definition 2.5. If β is a bracket expression, let β (ˆ i ) denote the bracket expression obtained from β by removing x i and the minimal set of other symbols as required to have a bracket expression,and replacing x k by x k − for all k > i .For example, with β = [[ x , x ] , x ] · [ x , x ], we have β (ˆ4) = [ x , x ][ x , x ]. For each strandthrough a transverse intersection r -singularity, we can define a resolution map which moves thatstrand off of the singularity. To accommodate the two cases, we let d ( r ) = (cid:26) d − r > d − r = 2 . By the rank of x i in a bracket expression β , we will mean the number of variables in β (counting x i ) which appear inside of common brackets with x i . In β , x has rank three and x has ranktwo. KRISTINE E. PELATT
Definition 2.6. If β is a bracket expression in which x i has rank r (with r >
0) define theresolution map ρ i : Imm ≥ β ( ˆ S , R d ) × S d ( r ) × I × I → Imm ≥ β (ˆ i ) ( ˆ S , R d )by ρ i ( γ, ¯ t, v, a, ε )( t ) = (cid:40) γ ( t ) + a · v exp (cid:16) t − t i ) − ε (cid:17) if t ∈ ( t i − ε, t i + ε ) γ ( t ) otherwise . We call the triple ( v, a, ε ) ∈ S d ( r ) × I × I the resolution data. We often fix a and ε so that theresolutions do not have unexpected singularities and by abuse denote the restriction by ρ i as well.The resolution map produces immersions in which the strand (between times t i − ε and t i + ε ) ismoved in the direction of v , as shown in Figure 2. Figure 2.
The resolution of a double point.
Definition 2.7.
Let S = { x i , x i , . . . , x i k } be an ordered subset of the variables in β . Define ρ β,S to be the composite ρ i k ◦ ( ρ i k − × id ) ◦ · · · ◦ ( ρ i × id ) : Imm ≥ β ( ˆ S , R d ) × (cid:89) m (cid:16) S d ( r m ) × I × I (cid:17) → Imm ≥ ∅ ( ˆ S , R d ) , where r m is the rank of x i m in β ( ˆ i , . . . , ˆ i m − ).The set S encodes which strands get moved in the resolution defined by ρ β,S .We now specialize. Let β = [[ x , x ] , x ] · [ x , x ], β = [ x , x ] · [[ x , x ] , x ] and choose theordered subset of variables for each to be S = { x , x , x } . We choose embeddings K and K of S in R (cid:44) → R d as shown in Figure 1, as well as a sequence 0 < t < t < · · · < t < K , ¯ t ) respects β and ( K , ¯ t ) respects β .We restrict the directions in which the singularities are resolved to ensure we produce notjust immersions but embeddings. We assume that in the disk of radius 1 /
10 centered at eachsingularity, both K and K consist of linear segments intersecting transversely, as shown inFigure 3. Fix ε > t i − ε, t i + ε ], i = 1 , , . . . ,
5, are disjoint and K ([ t i − ε, t i + ε ]) is contained in B ( K ( s i )) for i = 1 , , . . . ,
5. These intervals are the strands we willmove to resolve the singularities.Let w , . . . , w be the unit tangent vectors to each line segment at the singular points of K .Fix δ > { v ∈ S d − : (cid:107) v − w (cid:107) < δ } and { v ∈ S d − : (cid:107) v − w (cid:107) < δ } are disjoint. Asmentioned above, we avoid moving the third strand off of the triple point in these directions toprevent the introduction of a double point. GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OF LONG KNOTS 5
Figure 3. B ( K ( t )) ∩ K and B ( K ( t )) ∩ K .We produce a manifold M β as the image of a topological manifold M β embedded in Emb( ˆ S , R d )by resolving singular knots with triple and double points. The manifold M β decomposes as theunion (cid:83) i =1 M i , where each M i is the image in Emb( ˆ S , R d ) of a resolution map defined below.The domains of the resolution maps for the main pieces, M and M , are denoted M and M and are homeomorphic to (cid:0) S d − \ ∪ B δ (cid:1) × S d − × S d − . The domains of resolution maps definingthe remaining four families are denoted M i × I , where M i is homeomorphic to S d − × S d − × S d − for i = 3 , , , Definition 2.8.
For any triple ( ε , ε , ε ) with each ε i ≤ ε for ε as above, define M ( ε , ε , ε ) ⊂ Imm ≥ β ( ˆ S , R d ) × (cid:89) k =3 ( S d ( r k ) × I × I )as the subspace of all K × (cid:81) ( v i , a i , ε i ) , where a = , a = a = δ , and v is such that thedistances between v and the vectors ± w and ± w are all greater than or equal to δ . There areno restrictions on v , v ∈ S d − .We will suppress the dependence of M on the values of ε , ε , ε ≤ ε as well as δ except whenneeded. Lemma 2.9.
The restriction of ρ β ,S to M maps to Emb( ˆ S , R d ) ⊂ Imm ≥ φ ( ˆ S , R d ) . Choose the immersion K as shown in Figure 1, and assume that the constants δ > ε > K , to define M analogously. The restriction of ρ β ,S maps M to Emb( ˆ S , R d ) ⊂ Imm ≥ φ ( ˆ S , R d ). We denote the families of embeddings ρ β ,S ( M )and ρ β ,S ( M ) by M and M respectively, and connect the boundary components of M tothose of M to build a family without boundary.Each boundary component can also be described as the family of knots obtained by resolvinga singular knot with three double points. In fact, resolving the triple point in K by movingthe strand K ([ t − ε , t + ε ]) in the direction of ± w or ± w yields an immersion with threedouble points. The four boundary components of M are families of resolutions of these fourknots. KRISTINE E. PELATT
Definition 2.10.
Let K , K , K and K be the singular knots, each with three double points,defined below and shown in Figure 4. K = ρ (cid:0) K , w , , ε (cid:1) K = ρ (cid:0) K , − w , , ε (cid:1) K = ρ (cid:0) K , w , , ε (cid:1) K = ρ (cid:0) K , − w , , ε (cid:1) Figure 4.
Singular knots K , K , K , and K .We resolve these knots, restricting the directions so the resulting embeddings are those inthe boundary components of M . Initially, we focus on K . The double points corresponding to[ x , x ] and [ x , x ], labeled a and c , are resolved in the same way as the double points in K . Thedouble point corresponding to [ x , x ], labeled b , is resolved using only vectors in the direction v − w for some v such that (cid:107) v − w (cid:107) = δ . This guarantees that resolving this double point in K yields the (cid:107) v − w (cid:107) = δ boundary component of M . Definition 2.11.
Define M ( ε , ε , ε ) ⊂ Imm ≥ β ( ˆ S , R d ) × (cid:81) i =3 , , (cid:0) S d − × I × I (cid:1) where β =[ x , x ] · [ x , x ] · [ x , x ] as the subset of all K × (cid:81) i =3 , , (cid:0) u i , δ , ε i (cid:1) where u and u are unrestrictedand u satisfies (cid:107) w + δu (cid:107) = 1. Proposition 2.12.
Let S = { x , x , x } . The restriction of ρ β ,S maps M to Emb( ˆ S , R d ) ⊂ Imm ≥ φ ( ˆ S , R d ) , and ρ β ,S ( M ) is the (cid:107) v − w (cid:107) = δ boundary component of M .Proof. The resolution ρ β ,S ( K ) = ρ β (cid:0) ρ (cid:0) K , w , , ε (cid:1)(cid:1) using u as in the definition of M ( ε , ε , ε )is the same embedding as the resolution ρ β ( K ) using v = w + δu , since110 w exp (cid:18) t − t ) + ε (cid:19) + δ u exp (cid:18) t − t ) + ε (cid:19) = 110 v exp (cid:18) t − t ) + ε (cid:19) . (cid:3) GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OF LONG KNOTS 7
Similarly resolving the knots K , K , and K yields the boundary components of M corre-sponding to (cid:107) v + w (cid:107) = δ , (cid:107) v − w (cid:107) = δ , and (cid:107) v + w (cid:107) = δ respectively. This processcan also be applied to the boundary components of M . Let K , K , K , and K be the foursingular knots obtained from K by moving K ([ t − ε , t + ε ]) in the direction of the tangentvectors to the other two strands intersecting at the triple point, as shown in Figure 5. As with K , resolving these singular knots gives the four boundary components of M . Figure 5.
Singular knots K , K , K , and K .Since each of the four knots K , . . . , K has the same singularity data as one of K , . . . , K , wehave four pairs of knots which are isotopic in Imm = β i ( ˆ S , R ), and thus in Imm = β i ( ˆ S , R d ) with d ≥
4, where β , . . . , β each encodes singularity data for a knot with exactly three double points.If d >
4, we require that the isotopy be through knots in R ⊂ R d (with the standard embedding).If d = 4, we restrict the steps of the isotopy, as described in the Appendix, to simplify evaluationof Longoni cocycle on the cycle. Resolving each singular knot in these four isotopies yields fourfamilies, denoted M , M , M , and M , parametrized by S d − × S d − × S d − × I . Specifically,if h i : I → Imm ≥ β ( ˆ S , R d ) is an isotopy, then these M i are be the images of the composites(1) M i × I = S d − × S d − × S d − × I Id × h i −−−−→ S d − × S d − × S d − × Imm = β i ( S , R d ) ρ βi,Si −−−−→ Emb( ˆ S , R d ) . For i = 3 , , ,
6, the boundary of M i is the disjoint union of a boundary component of M anda boundary component of M , providing a way to glue the boundary of M to the boundary of M .The union of these six (3 d − S , R d ) gives a single family withoutboundary. Let M β = (cid:0) M (cid:116) M (cid:116) (cid:0) (cid:116) i =3 M i × I (cid:1)(cid:1) / ∼ where each boundary component of M , . . . , M is identified with a boundary component of M or M so as to be compatible with Proposition 2.12. Let M β be the image of the orientable KRISTINE E. PELATT topological manifold M β under the resolution map defined above. For d = 4, the resolution maptakes M β to Emb( S , R d ), as the isotopies we have chosen do not respect the fixed basepoint. Theorem 2.13. If d > is even then the fundamental class of M β is a non-trivial homology classin Emb( ˆ S , R d ) for any choice of isotopies h i through Imm = β i ( ˆ S , R d ) . For d = 4 the fundamentalclass of M β is a non-trivial homology class in Emb( S , R d ) if the isotopies h i satisfy a sequenceof specified steps. For more details on the case d = 4, see the Appendix and [6]. To prove [ M β ] is nontrivial, weevaluate a cocycle due to Longoni [5] on [ M β ] using configuration space integrals. This is themain result of Section 4. 3. The Longoni cocycle
In [3], Cattaneo, Cotta-Ramusino, and Longoni use configuration space integrals to define achain map I from a complex of decorated graphs to the de Rham complex of Emb( S , R d ). Thestarting point is the evaluation map ev : C q [ S ] × Emb( S , R d ) → C q [ R d ], where C q [ M ] is theFulton-MacPherson compactified configuration space. See [8] for more details. For some graphs G (namely those with no internal vertices), the image of the chain map I is defined by pullingback a form determined by G from C q [ R d ] to C q [ S ] × Emb( S , R d ) and then pushing forward toEmb( S , R d ).To understand the general case, let ev ∗ C q,r [ R d ] be the total space of the pull-back bundle shownbelow: ev ∗ C q,r [ R d ] ˆ ev (cid:47) (cid:47) (cid:15) (cid:15) C q + r [ R d ] (cid:15) (cid:15) C ordq [ S ] × Emb( S , R d ) ev (cid:47) (cid:47) C q [ R d ] , where C ordq [ S ] is the connected component of C q [ S ] in which the ordering on the points inthe configuration agrees with the ordering induced by the orientation of S . Fix an antipodallysymmetric volume form on S d − , denoted α . A choice of α determines tautological ( d − − formson ev ∗ C q,r [ R d ], defined by θ ij = ˆ ev ∗ φ ∗ ij ( α )where φ ij : C q ( R d ) → S d − sends a configuration to the unit vector from the i − th point to the j − th point in the configuration. We use integration over the fiber of the bundle ev ∗ C q,r [ R d ] → Emb( S , R d ), which is the composite of the projections ev ∗ C q,r [ R d ] → C ordq [ S ] × Emb( S , R d ) → Emb( S , R d ) , to push forward products of the tautological forms to forms on Emb( S , R d ). Which forms topush forward will be determined by graphs.Consider connected graphs which satisfy the following conditions. A decorated graph (of eventype) is a connected graph consisting of an oriented circle, vertices on the circle (called externalvertices ), vertices which are not on the circle (called internal vertices ), and edges. We requirethat all vertices are at least trivalent. The decoration consists of an enumeration of the edgesand an enumeration of the external vertices that is cyclic with respect to the orientation of thecircle. We will call the portion of the oriented circle between two external vertices an arc . GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OF LONG KNOTS 9
Definition 3.1.
Let D e be the vector space generated by decorated graphs of even type with thefollowing relations. We set G = 0 if there are two edges in G with the same endpoints, or if thereis an edge in G whose endpoints are the same internal vertex. The graphs G and G (cid:48) are equal ifthey are isomorphic as graphs and the enumerations of their edges differ by an even permutation.The vector space D e admits a bigrading as follows. Let v e and v i be the number of externaland internal vertices, respectively, and let e be the number edges. The order of a graph is givenby ord G = e − v i and the degree of a graph is defined bydeg G = 2 e − v i − v e . Let D k,me be the vector space of equivalence classes with order k and degree m . In [3], Cattaneo,Cotta-Ramusino and Longoni define a map from this vector space to the space of ( m + ( d − k ) − forms on Emb( S , R d ). Definition 3.2.
Define I ( α ) : D k,me → Ω m +( d − k (cid:0) Emb( S , R d ) (cid:1) as follows.(1) Choose an ordering on the internal vertices.(2) Associate each edge in G joining vertex i and vertex j to the tautological form θ ij .(3) Take the product of these tautological forms with the order of multiplication determinedby the enumeration of the edges, to define a form on ev ∗ C q,r [ R d ].(4) Integrate this form over the fiber to obtain a form on Emb( S , R d ).This integration over the fiber defines the pushforward and in this case is often called a con-figuration space integral. There is a coboundary map on D e which makes I ( α ) a cochain map. Definition 3.3.
Define a coboundary operator on D e by taking δG to be the signed sum of thedecorated graphs obtained from G by contracting, one at a time, the arcs of G and the edges of G which have at least one endpoint at an external vertex. After contracting, the edges and verticesare relabeled in the obvious way - if the edge (respectively vertex) labeled i is removed, we replacethe label j by j − j > i . When contracting an arc joining vertex i to i + 1, the sign isgiven by σ ( i, i + 1) = ( − i +1 , and when contracting the arc joining vertex j to vertex 1, the signis given by σ ( j,
1) = ( − j +1 . When contracting the edge l , the sign is given by σ ( l ) = l + 1 + v e ,where v e is the number of external vertices. Theorem 3.4. [3]
The map I ( α ) determines a cochain map and therefore induces a map oncohomology, which we denote I ( α ) : H k,m ( D e ) → H m +( d − k (Emb( ˆ S , R d )) . At the level of forms, I ( α ) depends on the choice of antipodally symmetric volume form α . Oncohomology, when d > α . Example 3.5.
From [3], we have the graph cocycle shown in Figure 6, originally investigated byBott and Taubes [1] for d = 3.This induces the cocycle14 (cid:90) ev ∗ C , [ R d ] θ θ − (cid:90) ev ∗ C , [ R d ] θ θ θ ∈ H d − (cid:16) Emb( ˆ S , R d ) (cid:17) . Figure 6.
Graph cocycle given by Cattaneo et al. in [3].In [3], Cattaneo et al. show that this cocycle evaluates non-trivially on ρ [ x ,x ] · [ x ,x ] (cid:0) K × S d − × S d − (cid:1) , where K is a singular knot with two double points respecting [ x , x ] · [ x , x ] (in this case, thecycle does not depend on the ordered subset S ⊆ { x , x , x , x } ). Example 3.6.
In [5], Longoni gives the example shown in Figure 7 of a graph cocycle G L in H , ( D e ) which uses nontrivalent graphs. There I ( α ) ( G L ) ∈ H d − (Emb( ˆ S , R d )) is the form ω = (cid:90) ev ∗ C , [ R d ] θ θ θ θ + 2 (cid:90) ev ∗ C , [ R d ] θ θ θ . We pair this cocycle with the cycle [ M β ] defined in Section 2 to see that both are nontrivial. Figure 7.
Graph cocycle given by Longoni in [5].4.
Nontriviality
Proposition 4.1.
Assume d > is even. Let [ M β ] ∈ H d − (Emb( ˆ S , R d )) be the cycledefined in Section 2, and let ω ∈ H d − (Emb( ˆ S , R d )) be the Longoni cocycle defined in thelast section. Then ω ([ M β ]) = ± . In particular, ω ([ M β ]) is nonzero, and therefore both ω and [ M β ] are non-trivial. In [10], Turchin calculates that E − , d − has rank one, so [ M β ] is a generator of this group.The proposition also holds for d = 4 if Emb( ˆ S , R d ) is replaced by Emb( S , R d ). Proof.
First we show that ω ([ M β ]) = ±
1. Let g : ev ∗ C , [ R d ] → S d − × S d − × S d − be themap shown in the diagram below, where ¯ ψ = φ × φ × φ . Then ω is the pushforward along π : ev ∗ C , [ R d ] → Emb( ˆ S , R d ) of g ∗ ( α ⊗ α ⊗ α ). GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OF LONG KNOTS 11 ev ∗ C , [ R d ] (cid:47) (cid:47) (cid:15) (cid:15) g (cid:41) (cid:41) π (cid:41) (cid:41) C [ R d ] ¯ ψ (cid:47) (cid:47) id (cid:15) (cid:15) S d − × S d − × S d − C ord [ S ] × Emb( ˆ S , R d ) (cid:47) (cid:47) (cid:15) (cid:15) C [ R d ] M β (cid:31) (cid:127) (cid:47) Emb( ˆ S , R d )By naturality of pushforwards, ω ([ M β ]) = g ∗ ( α ⊗ α ⊗ α )([ π − ( M β )]). The bundle π : ev ∗ C , [ R d ] → Emb( ˆ S , R d ) is trivial, so g ∗ ( α ⊗ α ⊗ α )([ π − ( M β )]) = (cid:82) C ord [ S ] ×M β g ∗ ( α ⊗ α ⊗ α ).To calculate (cid:82) C ord [ S ] ×M β g ∗ ( α ⊗ α ⊗ α ), we first partition C ord [ S ]. For i = 1 , . . . , N i = ( t i − ε, t i + ε ), where the t i are the times of singularity in K and K , and ε is as inSection 2. Define C ( i )5 = (cid:110) ¯ s ∈ C ord : s j (cid:54)∈ N i for j = 1 , . . . , s / ∈ C ( m )5 for m < i (cid:111) , and C c = C ord [ S ] \ (cid:16) ∪ i =1 C ( i )5 (cid:17) , so C c is the set of all ¯ s ∈ C ord [ S ] such that t i − ε < s i 5. Then C ord [ S ] decomposes as C ord [ S ] = C c (cid:116) C (1)5 (cid:116) · · · (cid:116) C (5)5 , andwe obtain a corresponding decomposition of (cid:82) C ord [ S ] ×M β g ∗ ( α ⊗ α ⊗ α ). We will show that (cid:82) C ( m )5 ×M β g ∗ ( α ⊗ α ⊗ α ) = 0 for m = 1 , . . . , 5, so calculating ω ([ M β ]) reduces to evaluating theintegrals (cid:90) C c ×M i g ∗ ( α ⊗ α ⊗ α ) . For m = 3 , , 5, we show (cid:82) C ( m )5 ×M β g ∗ ( α ⊗ α ⊗ α ) = 0 by showing (cid:82) C ( m )5 ×M i g ∗ ( α ⊗ α ⊗ α ) = 0for i = 1 , . . . , 6. Recall that manifolds have only trivial forms in degrees above their dimension,so a form pulled back through a smaller dimensional manifold is always zero. To prove that theintegrals (cid:82) C ( m )5 ×M i g ∗ ( α ⊗ α ⊗ α ) are zero, we show that the map g factors through spaces ofsmaller dimension when restricted to each of the subspaces C ( m )5 × M i .First, consider the case (cid:82) C (3)5 ×M g ∗ ( α ⊗ α ⊗ α ). Recall that M is ρ β ,S (cid:16) K × (cid:81) k =3 ( v k , a k , ε ) (cid:17) .If t / ∈ N and γ ∈ M , the point γ ( t ) does not depend on the value of v in the preimage of γ .This gives us the following factorization of g (cid:12)(cid:12) C (3)5 ×M : C (3)5 × M g (cid:47) (cid:47) (cid:40) (cid:40) (cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81) S d − × S d − × S d − C (3)5 × S d − × S d − (cid:53) (cid:53) (cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107)(cid:107) Since dim( C (3)5 × S d − × S d − ) = 2 d − S d − × S d − × S d − ) = 3 d − 3, wehave (cid:82) C (3)5 ×M g ∗ ( α ⊗ α ⊗ α ) = 0.Similarly, for m = 4 or m = 5, the restriction g (cid:12)(cid:12) C ( m )5 ×M factors through C ( m )5 × { v ∈ S d − : (cid:107) v ± w (cid:107) > δ and (cid:107) v ± w (cid:107) > δ } × S d − , so the corresponding integrals are zero. This argument also shows that (cid:82) C ( m )5 ×M g ∗ ( α ⊗ α ⊗ α ) = 0for m = 3 , , 5. For i = 3 , , , m = 3 , , 5, the restriction g (cid:12)(cid:12) C ( m )5 ×M i factors through S d − × S d − × I and therefore (cid:82) C ( m )5 ×M i g ∗ ( α ⊗ α ⊗ α ) is zero. We show (cid:82) C (1)5 ×M β g ∗ ( α ⊗ α ⊗ α ) = 0by replacing M β with the family of embeddings obtained by moving the first strand (instead ofthe fourth) off of the double point K i ( t ) = K i ( t ), over which g ∗ factors through a space oflower dimension. We replace M β in two steps - first with the family of embeddings in which bothstrands are moved off the double point, and then by the family in which only the first strand ismoved.Let M (cid:48) β be the piecewise smooth subspace of Emb( ˆ S , R d ) defined similarly to M β , but bychoosing the ordered subset of variables in β and β to be S = { x , x , x , x } , and fixing a = a and v = − v . In other words, M (cid:48) β is obtained from K , . . . , K by moving both strands off thedouble point K i ( t ) = K i ( t ) in antipodal directions.We define a cobordism W between M β and M (cid:48) β as the subspace of Emb( ˆ S , R d ) parametrizedby ( (cid:116) i M i ) × I , with the embedding corresponding to the parameter u ∈ I determined by a = ua (so the I parametrizes how far the strand with K i ( t ) is moved off the double point).By Stokes’ theorem, (cid:90) C (1)5 × W dg ∗ ( α ⊗ α ⊗ α ) = (cid:90) ∂ ( C (1)5 × W ) g ∗ ( α ⊗ α ⊗ α ) . Since dg ∗ ( α ⊗ α ⊗ α ) = g ∗ d ( α ⊗ α ⊗ α ) = 0, we have(2) 0 = (cid:90) ∂C (1)5 × W g ∗ ( α ⊗ α ⊗ α ) + (cid:90) C (1)5 ×M β g ∗ ( α ⊗ α ⊗ α ) − (cid:90) C (1)5 ×M (cid:48) β g ∗ ( α ⊗ α ⊗ α ) . The restriction g ∗ (cid:12)(cid:12) ∂C (1)5 × W factors through ∂C (1)5 × ( (cid:116) i M i ). If ¯ s ∈ ∂C (1)5 then the parameter, u ∈ I determining how far the first strand is moved does not affect g (¯ s, γ ) for γ ∈ W . Thus, (cid:82) ∂C (1)5 × W g ∗ ( α ⊗ α ⊗ α ) = 0 and (cid:90) C (1)5 ×M β g ∗ ( α ⊗ α ⊗ α ) = (cid:90) C (1)5 ×M (cid:48) β g ∗ ( α ⊗ α ⊗ α ) . Let M (cid:48)(cid:48) β be the piecewise smooth subspace of Emb( ˆ S , R d ) obtained by choosing the orderedsubset of variables in β and β to be S = { x , x , x } . In other words, M (cid:48)(cid:48) β is obtained from K , . . . , K by moving only the first strand off the double point K i ( t ) = K i ( t ). Let W ⊂ Emb( ˆ S , R d ) be parametrized by ( (cid:116) i M i ) × I , with the embedding corresponding to the parameter u ∈ I given by choosing a (cid:48)(cid:48) = ua (so the interval parametrizes how far the strand with K i ( t )is moved off the double point). Then W gives a cobordism between M (cid:48) β and M (cid:48)(cid:48) β , as ∂W = GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OF LONG KNOTS 13 M β (cid:116) ( −M (cid:48) β ). Using Stokes’ Theorem and naturality again, we have(3) 0 = (cid:90) ∂C (1)5 × W g ∗ ( α ⊗ α ⊗ α ) + (cid:90) C (1)5 ×M (cid:48) β g ∗ ( α ⊗ α ⊗ α ) − (cid:90) C (1)5 ×M (cid:48)(cid:48) β g ∗ ( α ⊗ α ⊗ α ) . The restriction g ∗ (cid:12)(cid:12) ∂C (1)5 × W does not factor through ∂C (1)5 × ( (cid:116) i M i ). To show the first integralin (3) is zero, we consider W as a subspace of Imm ≤ [ x ,x ] ,t ,t ( ˆ S , R d ), the subset of Imm( ˆ S , R d )consisting of all immersions γ with at most one singularity - a double point γ ( t ) = γ ( t ).Since a configuration in ∂C (1)5 does not contain the point t , the map g is well-defined on ∂C (1)5 × Imm ≤ [ x ,x ] ,t ,t ( ˆ S , R d ). Letting the dependance on the lengths of the strands be ap-parent, we now work with W = W ( ε , ε , ε ) as a subspace of Imm ≤ [ x ,x ] ,t ,t ( ˆ S , R d ). Inthis larger space, W ( ε , ε , ε ) is cobordant to W ( ε , , ε ). The cobordism is given by W ⊂ Imm [ x ,x ] ,t ,t ( ˆ S , R d ) parametrized by ( (cid:116) i M i ) × I × I where the second unit interval parametrizesthe length of the strand centered at t moved by the resolution map.By Stokes’ Theorem and naturality,0 = (cid:90) ∂C (1)5 × W dg ∗ ( α ⊗ α ⊗ α ) = (cid:90) ∂ (cid:16) ∂C (1)5 × W (cid:17) g ∗ ( α ⊗ α ⊗ α ) , and thus,(4)0 = (cid:90) ∂ ( ∂C (1)5 ) × W g ∗ ( α ⊗ α ⊗ α ) + (cid:90) ∂C (1)5 × W ( ε ,ε ,ε ) g ∗ ( α ⊗ α ⊗ α ) − (cid:90) ∂C (1)5 × W ( ε , ,ε ) g ∗ ( α ⊗ α ⊗ α )= (cid:90) ∂C (1)5 × W ( ε ,ε ,ε ) g ∗ ( α ⊗ α ⊗ α ) . The second equality holds because ∂ ( ∂C (1)5 ) = ∅ and the dimension of W ( ε , , ε ) is 2 d − (cid:82) C (5)5 ×M β g ∗ ( α ⊗ α ⊗ α ) = 0. Calculating (cid:82) C ×M β g ∗ ( α ⊗ α ⊗ α ) thusreduces to calculating (cid:82) C c ×M i g ∗ ( α ⊗ α ⊗ α ) for i = 1 , . . . , α , to be concentrated near the points ¯ x =(0 , . . . , , ∈ S d − and ¯ x = (0 , . . . , , − ∈ S d − . Let τ ¯ x and τ ¯ x be the Thom classes of thesepoints, as defined in Section 6 of [2], so α = ( τ ¯ x + τ ¯ x ). Let y be the arc in S d − connecting(0 , . . . , , 1) and (0 , . . . , , − y = (cid:110)(cid:16) , . . . , , (cid:112) − s , s (cid:17) ∈ S d − : − ≤ s ≤ (cid:111) . The Thom class τ y of y can be chosen so that dτ y = τ ¯ x − τ ¯ x = 2( τ ¯ x − α ). We have (cid:90) C c ×M i g ∗ ( α ⊗ α ⊗ α − τ ¯ x ⊗ α ⊗ α ) = (cid:90) C c ×M i g ∗ (cid:0) − dτ y ⊗ α ⊗ α (cid:1) = − (cid:90) C c ×M i dg ∗ ( τ y ⊗ α ⊗ α )= − (cid:90) ∂ ( C c ×M i ) g ∗ ( τ y ⊗ α ⊗ α ) . If ( v , v , v ) ∈ g ( ∂ ( C c × M i )) at least one of the first two coordinates of v is non-zero, butevery ¯ x ∈ y ⊂ S d − has x , x = 0. Thus, the sets y and g ( ∂ ( C c × M i )) are disjoint and (cid:82) C c ×M i g ∗ ( τ y ⊗ α ⊗ α ) = 0, which means (cid:82) C c ×M i g ∗ ( α ⊗ α ⊗ α ) = (cid:82) C c ×M i g ∗ ( τ ¯ x ⊗ α ⊗ α ). By asimilar argument, (cid:90) C c ×M i g ∗ ( α ⊗ α ⊗ α ) = (cid:90) C c ×M i g ∗ ( τ ¯ x ⊗ τ ¯ x ⊗ τ ¯ x ) . This integral can be calculated by counting the transverse intersections of g ( C c × M i ) and(¯ x , ¯ x , ¯ x ) in S d − × S d − × S d − .Recall that g (¯ s, γ ) = (cid:18) γ ( s ) − γ ( s ) (cid:107) γ ( s ) − γ ( s ) (cid:107) , γ ( s ) − γ ( s ) (cid:107) γ ( s ) − γ ( s ) (cid:107) , γ ( s ) − γ ( s ) (cid:107) γ ( s ) − γ ( s ) (cid:107) (cid:19) . Thus, we are counting the number of pairs (¯ s, γ ) ∈ C c × M i for which γ ( s ) − γ ( s ) (cid:107) γ ( s ) − γ ( s ) (cid:107) = γ ( s ) − γ ( s ) (cid:107) γ ( s ) − γ ( s ) (cid:107) = γ ( s ) − γ ( s ) (cid:107) γ ( s ) − γ ( s ) (cid:107) = (0 , . . . , , . For γ ∈ M β , this is only possible if s i = t i for i = 1 , . . . , γ ∈ M , then γ ( t ) − γ ( t ) (cid:107) γ ( t ) − γ ( t ) (cid:107) = γ ( t ) − γ ( t ) (cid:107) γ ( t ) − γ ( t ) (cid:107) = γ ( t ) − γ ( t ) (cid:107) γ ( t ) − γ ( t ) (cid:107) = (0 , . . . , , v = v = v = (0 , . . . , , 1) and so (cid:82) C c ×M g ∗ ( α ⊗ α ⊗ α ) = ± 1. If γ ∈ M i for i = 2 , . . . , 6, then γ ( t ) − γ ( t ) (cid:107) γ ( t ) − γ ( t ) (cid:107) (cid:54) = (0 , . . . , , , and (cid:82) C c ×M i g ∗ ( α ⊗ α ⊗ α ) = 0. Thus, ω ([ M β ]) = ± ω ([ M β ]) = 0. Let f : ev ∗ C , ( R d ) → S d − × S d − × S d − × S d − be themap shown in the diagram below, where ¯ ϕ = φ × φ × φ × φ . Then ω is the pushforwardof f ∗ ( α ⊗ α ⊗ α ⊗ α ) along p : ev ∗ C , ( R d ) → Emb( ˆ S , R d ). GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OF LONG KNOTS 15 ev ∗ C , ( R d ) (cid:47) (cid:47) p (cid:15) (cid:15) f (cid:41) (cid:41) p (cid:41) (cid:41) C [ R d ] ¯ ϕ (cid:47) (cid:47) (cid:15) (cid:15) S d − × S d − × S d − × S d − C ord [ S ] × Emb( ˆ S , R d ) (cid:47) (cid:47) p (cid:15) (cid:15) C [ R d ] M β (cid:31) (cid:127) (cid:47) Emb( ˆ S , R d )Since p − ( M β ) = p − ( C ord [ S ] ×M β ), we have ω ([ M β ]) = (cid:82) p − ( C ord [ S ] ×M β ) f ∗ ( α ⊗ α ⊗ α ⊗ α ).Following the calculation of ω ([ M β ]), define C ( i )4 = (cid:110) ¯ s ∈ C ord [ S ] : s j (cid:54)∈ N i for j = 1 , . . . , s / ∈ C ( m )4 for m < i (cid:111) . Each configuration in C ord [ S ] has four points, so C ord [ S ] = C (1)4 (cid:116) · · · (cid:116) C (5)4 . The argumentsused to prove that (cid:82) C ( m )5 ×M β g ∗ ( α ⊗ α ⊗ α ) = 0 also show (cid:82) p − ( C ( m )4 ×M β ) f ∗ ( α ⊗ α ⊗ α ⊗ α ) = 0for m = 1 , . . . , (cid:3) Future Work The resolution map in Definition 2.7 can be generalized to define a resolution map for knotsrespecting any bracket expression. Instead of choosing an ordered subset of the variables, werepeatedly choose the strands to move so as to resolve the singularity data for the brackets whichare not contained inside of any other brackets.For example, if ( K, ¯ t ) respects [[ x , x ] , [[ x , x ] , x ]] the point K ( t ) = K ( t ) is first movedaway from the point K ( t ) = K ( t ) = K ( t ), turning the original singularity into a double pointand a triple point. The double point is then resolved as before, and the triple point is resolvedby first moving the fifth strand off the singularity and then resolving the remaining double point.The description in Section 2 of the first differential of the embedding calculus homology spectralsequence is given in terms of “doubling” the point x i . In [6] we develop another description of thisdifferential, call it ˜ d , which encodes the singularity data that occurs when a knot respecting abracket expression is resolved as prescribed in the generalization of the resolution map, but withthe directions chosen in such a way as to introduce a new singularity. The boundary componentsof the family of resolutions of a knot ( K, ¯ t ) respecting a bracket expression under the generalizedresolution map are the same as the families of resolutions of knots respecting the terms in ˜ d ofthat bracket expression (with appropriate choices).Suppose β = (cid:80) mi =1 β i is a cycle on the first page of the spectral sequence (where each β i isa bracket expression with a single term) in which the Jacobi identity is not used to simplifythe differential. Knots ( K i , ¯ t ) respecting the β i can be chosen so that the boundaries of thefamilies of resolutions under the generalized resolution map can be connected by families ofembeddings given by an isotopy of underlying singular knots, as in the cycle [ M β ] defined here.Thus the process used in this paper can be generalized to more cycles on the first page of the spectral sequence. Because Turchin proved linear duality of the Cattaneo, Cotta-Ramusino, andLongoni graph complex and the E page of the embedding calculus spectral sequence, we alsohave configuration space integrals to evaluate on the families we produce. Together these couldgive not only a second proof of the collapse of the spectral sequence (Lambrechts, Turchin andVolic use closely related configuration space integrals in their proof of the collapse in [4]), butalso geometric representatives and a clear starting point for considering any torsion phenomena. Appendix A. Isotopies When d = 4 the value of ω ([ M β ]) depends on the isotopies chosen, as x = (0 , , , R except for near crossing changes. This forces thecounts used to calculate the integrals (cid:82) C c ×M i g ∗ ( τ ¯ x ⊗ τ ¯ x ⊗ τ ¯ x ) to be the same as in the higherdimensional cases. We give an example of such an isotopy from K to K below, by specifyingsteps the isotopy must satisfy. All four isotopies will appear in [6].By a slide isotopy we will mean an isotopy through singular knots in which a singular point ismoved along one of the strands through the singularity while the other strand moves along withthe singular point. By a planar isotopy we will mean an isotopy which can be represented byan isotopy of knot diagrams. Isotopies corresponding to the Reidemeister moves in classical knottheory generalize to singular knots in R d . In addition to the usual Reidemeister I and II moves,we use Reidemeister III moves to move a strand past a crossing (as in classical theory) or past asingularity, as shown in Figure 8. Figure 8. Reidemeister III move for singular knots.By a “rotate the disk isotopy,” we mean an isotopy in which the disk centered at a singularity isrotated by 180 ◦ about the axis perpendicular to a particular great circle. Specifically, we take twodistinct nested disks centered at the singular point with radii small enough that the intersectionof the knot with the disks is the two strands intersecting at the singular point. The smaller ofthe two disks is rotated by 180 ◦ without changing anything inside of this disk. The strands insideof the larger disk but outside of the smaller disk are stretched through a planar isotopy. Thisisotopy is shown in Figure 9 from the perspective of the north pole of the larger disk. The knotremains unchanged outside of the larger disk.A suitable type of isotopy from K to K is shown in Figure 10, and the steps are given below.Each step occurs in R ⊂ R except (4), (6) and (10), in which one strand of the knot brieflymoves into R .(1) Simplify the shape of the strand from b to c and perform a Reidemeister II move on thestrand from a to b to eliminate crossings.(2) Move the points a , b and c to a , b and c through a planar isotopy. GEOMETRIC HOMOLOGY REPRESENTATIVE IN THE SPACE OF LONG KNOTS 17 Figure 9. View of a rotate the disk isotopy from the north pole.(3) Rotate the disk centered at c by 180 ◦ about the axis perpendicular to the great circleshown.(4) The crossing is changed, briefly moving the strand from b to c in the direction of thefourth standard basis vector.(5) Perform a sequence of Redemeister I,II and III moves on the strand from b to c .(6) The crossing is changed, briefly moving the strand from c to a in the direction of thefourth standard basis vector.(7) Perform a sequence of Reidemeister I, II and III moves on the strand from c to a .(8) Rotate the disk centered at a by 180 ◦ about the axis perpendicular to the great circleshown.(9) Perform a sequence of Reidemeister I, II and III moves on the strand from c to a andthe strand from a to b .(10) The crossing is changed, briefly moving the strand from a to b in the direction of thefourth standard basis vector.(11) Perform a sequence of Reidemeister I, II and III moves on the strand from a to b .(12) Through a planar isotopy, the points a , b and c are moved to the positions of the doublepoints of K , denoted a , b and c and the strands are moved to give the knot the sameshape as K . References [1] Raoul Bott and Clifford Taubes. On the self-linking of knots. J. Math. Phys. , 35(10):5247–5287, 1994. Topologyand physics.[2] Raoul Bott and Loring W. Tu. Differential forms in algebraic topology , volume 82 of Graduate Texts in Math-ematics . Springer-Verlag, New York, 1982.[3] Alberto S. Cattaneo, Paolo Cotta-Ramusino, and Riccardo Longoni. Configuration spaces and Vassiliev classesin any dimension. Algebr. Geom. Topol. , 2:949–1000 (electronic), 2002.[4] Pascal Lambrechts, Victor Turchin, and Ismar Voli´c. The rational homology of spaces of long knots in codi-mension > Geom. Topol. , 14(4):2151–2187, 2010.[5] Riccardo Longoni. Nontrivial classes in H ∗ (Imb( S , R n )) from nontrivalent graph cocycles. Int. J. Geom.Methods Mod. Phys. , 1(5):639–650, 2004.[6] Kristine E. Pelatt. Geometric Representatives of Homology classes in the space of long knots . PhD thesis,University of Oregon, 2012.[7] Dev P. Sinha. The homology of the little disks operad. arXiv:math.AT/0610236.[8] Dev P. Sinha. Manifold-theoretic compactifications of configuration spaces. Selecta Math. (N.S.) , 10(3):391–428, 2004.[9] Dev P. Sinha. The topology of spaces of knots: cosimplicial models. Amer. J. Math. , 131(4):945–980, 2009. Figure 10. Isotopy from K to K . [10] V. Tourtchine. On the other side of the bialgebra of chord diagrams. J. Knot Theory Ramifications , 16(5):575–629, 2007.[11] V. A. Vassiliev. Complements of discriminants of smooth maps: topology and applications , volume 98 of