EElliptic Associators and the LMO Functor
Ronen KatzOctober 8, 2018
Abstract
The elliptic associator of Enriquez can be used to define an invariant of tangles embeddedin the thickened torus, which extends the Kontsevich integral. This construction by Humbertuses the formulation of categories with elliptic structures.In this work we show that an extension of the LMO functor also leads to an ellipticstructure on the category of Jacobi diagrams which is used by the Kontsevich integral, andfind the relation between the two structures. We use this relation to give an alternative prooffor the properties of the elliptic associator of Enriquez. Those results can lead the way tofinding associators for higher genra. a r X i v : . [ m a t h . G T ] O c t ontents A . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Unordered Elliptic Jacobi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Unordered Elliptic Jacobi Diagrams with no Struts . . . . . . . . . . . . . . . . . . 143.5 Ordered Elliptic Jacobi Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Categories of Pattern-Connected Diagrams . . . . . . . . . . . . . . . . . . . . . . 18 A
The Kontsevich integral was originally defined as an invariant of knots which returns values in aspace of Jacobi diagrams ([17]). Its importance comes mainly from the fact that it is universal withrespect to Vassiliev invariants (see also [1]). A new, combinatorial, formulation of the Kontsevichinvariant for framed knots and links was given in [18]. The idea of this definition is to break thelink to elementary tangles and asign values to each of them separately.The paper [18] already contains the extension of the Konstevich integral to tangles, which isdefined more explicitly in [19] and [3]. In this context it is convenient to consider all tangles asmorphisms in the category q ˜ T of non-associative framed tangles. This category has the structureof a ribbon category - a concept which encapsulates the above elementary tangles and the relationsbetween them. The Kontsevich integral becomes a functor to a category of Jacobi diagrams A ∂ ,which also gets a structure of a ribbon category.In [16], the Kontsevich integral is further extended to an invariant of tangles embedded in athickened torus T × I . The category of those tangles is denoted q ˜ T , and is an extension of theribbon category q ˜ T . The new ingredients in this category are two elementary tangles which goaround the generators of π ( T ). Those tangles are called beaks. The beaks, together with therelations between them and the other elementary tangles, is encapsulated in the concept of anelliptic structure.In order to define the extension of the Kontsevich integral to q ˜ T , one needs to find a categoryof Jacobi diagrams extending A ∂ such that this extension has an elliptic structure. This categoryis defined in [16] and denoted A - the category of elliptic Jacobi diagrams. To give this extensionan elliptic structure, [16] uses the concept of an elliptic associator. An elliptic associator is a pairof elements in the exponent of ˆ f ( A, B ) - the completed free Lie algebra generated by A and B ,satisfying several identities. This associator, when mapped into A , determines the value of theKontsevich invariant on the beaks. A specific elliptic associator e ( φ ) is introduced in [8] and [13].In a different direction, the Kontsevich integral was also used to define the LMO invariant,which is an invariant of closed 3-manifolds which returns values in some spaces of Jacobi diagrams([20]). It was extended to a TQFT first in [21] and, several years later, in [9]. A functorial variantof this construction was given in [10], called the LMO functor.In this work we extend the LMO functor to an invariant of 3-cobordisms with embeddedtangles. Restricted to tangles embedded in the thickened torus, this invariant can be comparedto the invariant of [16]. It turns out that those invariants are not equal. However, their valueson the beaks are equivalent modulo a certain relation called the homotopy relation. We use thisequivalence to give an alternative, more intuitive, proof that e ( φ ) is indeed an elliptic associator.This work is divided into the following sections:In section 2 we review the definitions of ribbon categories and elliptic structures. As we ex-plained, those concepts encapsulate the strucure of the categories of tangles q ˜ T and q ˜ T , and giveus a tool to define invariants of tangles.In section 3 we review several variants of categories of Jacobi diagrams which will be used later,and explain the relations between them.In section 4 we define our extension of the LMO functor to the category of embedded tangles in3-cobordisms. For that purpose we explain how to represent a tangle embedded in a 3-cobordismusing a representing tangle in D × I . We also give an explicit description of the beaks using thoserepresenting tangles, and verify their properties.In section 5 we introduce the Lie algebras t ,n , and explain how their universal envelopingalgebras are mapped into the spaces A
In this section we define the concepts of ribbon categories and elliptic structures, and introducethe categories of tangles which are the universal examples for those concepts.
In this subsection we recall the definition of a ribbon category, and give the main example - thecategory of tangles in D × I . The definitions are all taken from [16].Let ( C , ⊗ , , a ) be a non-associative monoidal category. Assume for simplicity that ⊗ U = U ⊗ = U for any object U ∈ C . For shortness we will denote U ⊗ V by U V . a is a familyof natural isomorphisms a X,Y,Z : ( XY ) Z → X ( Y Z ) for any objects
X, Y, Z ∈ C , satisfying thepentagon relation for any objects
X, Y, Z, W ∈ C : a W,X,Y Z a W X,Y,Z = ( id W ⊗ a X,Y,Z ) a W,XY,Z ( a W,X,Y ⊗ id Z ) A duality on C is a rule that associates to each object V an object V ∗ and 2 morphisms b V : → V ⊗ V ∗ and d V : V ∗ ⊗ V → , satisfying:( id V ⊗ d V ) a V,V ∗ ,V ( b V ⊗ id V ) = id V ( d V ⊗ id V ∗ ) a − V ∗ ,V,V ∗ ( id V ∗ ⊗ b V ) = id V ∗ A braiding on C is a family of natural isomorphisms c U,V : U V → V U for any 2 objects
U, V ∈ C , satisfying for any objects
U, V, W ∈ C : c UV,W = a W,U,V ( c U,W ⊗ id V ) a − U,W,V ( id U ⊗ c V,W ) a U,V,W c U,V W = a − V,W,U ( id V ⊗ c U,W ) a V,U,W ( c U,V ⊗ id W ) a − U,V,W
By convension we denote c − U,V := ( c V,U ) − . A twist on a monoidal category with braiding is a family of natural isomorphisms θ V : V → V for any object V ∈ C , satisfying: θ UV = c V,U c U,V ( θ U ⊗ θ V ) Definition 2.1. A ribbon category is a monoidal category ( C , ⊗ , , a ) with duality, braidingand twist as above, satisfying, for any object V ∈ C :( θ V ⊗ id V ∗ ) b V = ( id V ⊗ θ V ∗ ) b V The main example for a ribbon category is the category of framed oriented tangles, which wewill now describe.Let D ⊂ R be the unit disk. Denote by b n a sequence of some fixed n points in the interior of D (say, the n points uniformly distributed along the segment ( − ,
1) of the x axis). Let m, n ≥
05e integers, and ω s , ω t non-associative words in the symbols { + , −} of lengths m, n , respectively.A tangle in D × I of type ( ω s , ω t ) is an oriented 1-manifold γ embedded in D × I , such thatits only boundary points are γ ∩ ( D × { } ) = b m × { } and γ ∩ ( D × { } ) = b n × { } , andsuch that the orientations of the tangle around the points of b m × { } and b n × { } correspond tothe symbols of ω s and ω t (a “+” symbol corresponds to a strand going “up”, i.e. in the positivedirection of I , and a “ − ” symbol corresponds to a strand going “down”). The embedding of γ should be piece-wise smooth and transverse to the horizontal surfaces D × { t } at all but a finitenumber of points. We also require the tangle to be vertical (i.e. of the form { z } × I ) near theboundary points.A framing on a tangle γ is the homotopy class relative to the boundary of a non-zero normalvector field on the smooth points of γ , such that the limit of this vector field at the non-smoothpoints is the same from both sides. The vectors based on the boundary points should all beparametrized as (0 , − , Definition 2.2.
The category q ˜ T is the category whose objects are non-associative words in { + , −} , and whose sets of morphisms q ˜ T ( ω s , ω t ) are the sets of ambient isotopy classes of orientedframed tangles of type ( ω s , ω t ).The concept of a ribbon category is designed to encapsulate the structure of q ˜ T . More specifi-cally, we have the following proposition, which is easy to verify: Proposition 2.1. q ˜ T is a ribbon category, where we define the dual of ω = ( ω , ..., ω n ) to be ω ∗ = ( − ω n , ..., − ω ) , and: a ω ,ω ,ω = ω ( ω ω )( ω ω ) ω b ω = ω ω ∗ d ω = ω ω ∗ c ω ,ω = ω ω ω ω θ ω = ωω In fact, the category q ˜ T is the universal ribbon category, in the sense that there is a unique func-tor from it to any other ribbon category, preserving most of its properties. A precise formulationand proof of this theorem can be found in [22]. 6 .2 Elliptic Structure We will now define the concept of elliptic structure. The definitions in this section are also takenfrom [16].Let C be a ribbon category, C any category, and {·} : C → C a functor. Definition 2.3. An elliptic structure relative to ( C → C ) is a pair ( X, Y ) of natural automor-phisms of the functor {· ⊗ ·} : C × C → C (i.e. the composition of the tensor product of C with thegiven functor {·} ), satisfying the following identities for any objects U, V, W ∈ C (where for Z = X or Z = Y we denote Z (cid:48) U,V,W := { a − U,V,W } Z U,V W { a U,V,W } ): X UV,W = X (cid:48) U,V,W { c V,U ⊗ id W } X (cid:48) V,U,W { c U,V ⊗ id W } (2.1) Y UV,W = Y (cid:48) U,V,W { c − V,U ⊗ id W } Y (cid:48) V,U,W { c − U,V ⊗ id W } (2.2) Y U,V X U,V Y − U,V X − U,V = { c V,U c U,V } (2.3) Y (cid:48) U,V,W { c U,V ⊗ id W } X (cid:48) V,U,W { c U,V ⊗ id W } = { c V,U ⊗ id W } X (cid:48) V,U,W { c − U,V ⊗ id W } Y (cid:48) U,V,W (2.4)We will now describe the main example for a category with an elliptic structure, which is thecategory of framed oriented tangles in the thickened torus.Let T := S × S be the torus. We fix an embedding D ⊂ T . This embedding also gives us anembedding of all the sets of points b n into T . The torus T (minus an open neighborhood of a pointat infinity) is depicted in figure 1. This figure also shows the embedded disk D , and 2 generators x and y of π ( T ). xy D Figure 1: The torus T with generators of π Definition 2.4.
The category q ˜ T of framed tangles in the thickened torus is defined as follows:The objects of q ˜ T are non-associative words in { + , −} . For two such words ω s , ω t the morphismsset q ˜ T ( ω s , ω t ) is the set of ambient isotopy classes of oriented framed tangles in T × I of type( ω s , ω t ). As in the definition of q ˜ T , the tangles are piece-wise smooth, transverse to the planes T × { t } at all but a finite number of points, and vertical near the boundary points.7here is an obvious functor q ˜ T → q ˜ T , induced by the embedding D ⊂ T . We want to describean elliptic structure relative to this functor. In this context, the natural automorphisms X ω ,ω and Y ω ,ω act by composition with some invertible tangles in q ˜ T (( ω )( ω ) , ( ω )( ω )). We willnow describe those tangles.The tangent space at any point p = ( u, t ) ∈ T × I can be decomposed as T T u × R . If the point u is in D ⊂ T , a tangent vector at p can be parametrized by ( x, y, t ).Let γ : [0 , → T be a smooth simple closed path, with a nowhere vanishing derivative. Assumethat γ (0) = γ (1) is the first (left) point of b ⊂ T , and that γ (cid:48) (0) = c ( − ,
0) and γ (cid:48) (1) = c (1 , c , c (we will assume that the loops representing x and y from figure 1 havethis property). We define the tangle ˜ γ of type (++ , ++) as follows:The right strand is a constant strand at the second (right) point of b , with a framing parametrizedconstantly by (0 , − , γ ( t ) , t ) ∈ T × I , and framed by the unique framingwhich has the following two properties: (A) it is parametrized by (0 , ,
0) at ( γ (0) ,
0) (thus it isactually not in q ˜ T (++ , ++) as defined above), and (B) its parametrization has t = 0 at all points(this parametrization is unique due to the nowhere vanishing derivative assumption).By following closely the loops of figure 1 it can be seen that at ˜ γ (1) this framing is parametrizedby (0 , − , γ = x and γ = y .Let pt (=positive twist) and nt (=negative twist) be the following tangles of type (++ , ++):The right strand is constant, as in the definition of ˜ γ . The left strand is also a constant strand,but its framing makes a half twist from (0 , − ,
0) to (0 , , pt this twist is in the positivedirection, and in nt the twist is in the negative direction.For any 2 words ω , ω and any tangle u ∈ q ˜ T (++ , ++), define the cabling ∆ ++ ω ,ω ( u ) ∈ q ˜ T (( ω )( ω ) , ( ω )( ω )) to be the tangle obtained from u by duplicating the left strand | ω | timesalong its framing and giving the strands orientations according to the symbols of ω , and similarlyfor the right strand with ω . Proposition 2.2.
There is an elliptic structure relative to ( q ˜ T → q ˜ T ) with: X ω ,ω = ∆ ++ ω ,ω (˜ x · pt ) Y ω ,ω = ∆ ++ ω ,ω (˜ y · nt ) where x, y are the representatives of the elements of π ( T ) depicted in figure 1. The compositions ˜ x · pt and ˜ y · nt which appear in this proposition are defined by simply puttingthe first tangle on top of the second. Note that the framings of ˜ x and ˜ y at the bottom correspond tothe framings of pt and nt at the top, and after the compositions we get elements in q ˜ T (++ , ++).This proposition is proved in [16] as a special case of the general concept of genus g structures.Furthermore, it is proved there that ( q ˜ T → q ˜ T ) is universal, in the sense that there is a unique pairof functors from it to any other pair with an elliptic structure, preserving most of its properties.It is clear that the main ingredients of this elliptic structure are the tangles ˜ x · pt and ˜ y · nt .In [16], these are the tangles which are represented by “beaks” in beak diagrams. In section 4 we8ill give a different description of those tangles, and use this description to give another, pictorial,proof of proposition 2.2. 9 Categories of Jacobi Diagrams
In this section we review the definition of several categories of Jacobi diagrams, and the relationsbetween them. Most of those categories are obvious extensions of spaces which appear, in one wayor another, in [10], [15] and [16]. In the following sections we will see how those categories are usedto define invariants of tangles.
Definition 3.1. A pattern P is a compact (but not necassarily closed) oriented 1-manifold whoseboundary ∂P is divided into 2 ordered sets ∂ s P and ∂ t P . To each pattern P we can associate 2words ω s ( P ) and ω t ( P ) in the symbols { + , −} . The words ω s ( P ) and ω t ( P ) encode the orientationsof P around the ordered sets ∂ s P and ∂ t P : A + ( − ) in the i -th place of ω s ( P ) means that theorientation of P near the i -th point of ∂ s P is going away from (towards) this point. A + ( − ) inthe i -th place of ω t ( P ) means that the orientation of P near the i -th point of ∂ t P is going towards(away from) this point.The category of patterns P is the category whose objects are finite words in the symbols { + , −} ,and for any 2 such words ω s , ω t the morphisms set P ( ω s , ω t ) is the set of patterns P such that ω s ( P ) = ω s and ω t ( P ) = ω t . The composition P · P of 2 patterns is defined by attaching ∂ t P to ∂ s P . Graphically, we usually draw ∂ s P at the bottom and ∂ t P at the top. The composition isthen obtained by putting P above P . Note:
There are obvious functors q ˜ T → P and q ˜ T → P which map an object ω to itself (forget-ting the non-associative structure), and a tangle u to its skeleton. The split ∂P = ∂ s P ∪ ∂ t P isdetermined by whether the boundary point is in D × { } or in D × { } . A Let P be a pattern, and S a set. A Jacobi diagram D over P and S is a uni-trivalent graphwhose univalent vertices are either connected to a point in P or labeled by an element of S . Thetrivalent vertices are cyclically oriented. When we draw a Jacobi diagram in a 2-dimensional plane,we assume the orientations of the trivalent vertices are always counter-clockwise. An example for aJacobi diagram is given in figure 2. The pattern P is given by the solid lines, and the uni-trivalentdiagram is given by the dashed lines.The degree of a Jacobi diagram D is defined to be the number of trivalent vertices + thenumber of univalent vertices on P .The set of all Jacobi diagrams over P and S is denoted D ( P, S ). Notation:
Let A be a set with a degree map A → N , and F a field ( F will usually be a field ofcharacteristic 0, such as R or C ). Then S F ( A ) denotes the degree completion of the vector spacespanned by A over the field F . 10 bb Figure 2: An example for a Jacobi diagramLet A ( P, S ) be the quotient of S F ( D ( P, S )) by the relations
ST U , IHX and AS , which aredefined by: ST U : = − IHX : = − AS : = − At this point we are specifically interested in the spaces A ( P, ∅ ), i.e. spaces of Jacobi diagramswith no labeled vertices. We will denote those spaces by A ∂ ( P ). In these spaces the degree of alldiagrams is even. It is common to define the degree in those spaces as half the number of vertices,but we will continue to use the above definition of degree, in order to be compatible with otherspaces of Jacobi diagrams.Let A ∂ be the category defined as follows: The objects of A ∂ are words in the symbols { + , −} . For 2 such words ω s , ω t , the morphisms set A ∂ ( ω s , ω t ) is defined by: A ∂ ( ω s , ω t ) := (cid:83) P ∈ P ( ω s ,ω t ) A ∂ ( P ). For a ∈ A ∂ ( ω s , ω t ) and b ∈ A ∂ ( ω t , ω u ), the composition b · a is obtained byputting b above a and composing the underlying patterns.Recall the box notation, which is defined in figure 3. In this figure the vertical lines goingthrough the box can be either solid or dashed. The sign ε i is − i -th line is a solid line withorientation opposite to the orientation of the box (i.e. the direction which the arrow in the boxpoints to), and +1 otherwise. 11 = (cid:88) i ε i D i Figure 3: The box notation D Figure 4: An I relationIn each space A ∂ ( P ), we define I ( P ) to be the subspace generated by all sums of the type givenin figure 4. In this figure we assume that the lines which come out of the upper side of the box areexactly all the ends of the solid lines which lead to the points of ∂ t ( P ). We call each such sum an I relation.It can be verified that the union of all I ( P ) is a two-sided ideal of A ∂ (see [16], Lemma 1.4.4).Therefore we can define the quotient category A ∂ / I . We denote this category by A . Remark:
As we will see in the following sections, the categories of Jacobi diagrams decorated by ∂ are usually the targets of invariants of tangles in cobordisms whose boundary surfaces have oneboundary component. The corresponding quotient categories with no decoration are used whenthe boundary surfaces have no boundary. Specifically, A ∂ is used to defined invariants of tanglesin D × I , and A is used to define invariants of tangles in S × I . In the following sections we willbe more interested in invariants of tangles in T × I . The categories A ∂ and A will only be usedfor technical reasons in the way to define those invariants.We end this section by recalling a known notation (see [10], notation 3.13): Notation:
Let ω be a word of length n in the symbols { + , −} , and ω , ..., ω n other words inthose symbols. The map ∆ ωω ,...,ω n : A ∂ ( ω, ω ) → A ∂ ( ω · ... · ω n , ω · ... · ω n ) is the map obtainedby applying, for each 1 ≤ i ≤ n , the doubling map ∆ : A ∂ ( ↑ , ↑ ) iterated | ω i | − i -thstrand, and by applying the orientation-reversal map S to each new strand whose correspondingsymbol in ω i does not agree with the i -th symbol in ω . This map also induces a map on the12uotient categories: ∆ ωω ,...,ω n : A ( ω, ω ) → A ( ω · ... · ω n , ω · ... · ω n ) Let D ( P ) denote the set D ( P, H ( T )), i.e. the set of Jacobi diagrams over the pattern P withlabels coming from the first homology of the torus. Let A ∂ ( P ) be the quotient of S F ( D ( P )) bythe relations: ST U , IHX , AS and multilinearity. The multilinearity relation is defined as follows:( au + bv ) D = a (cid:32) u D (cid:33) + b (cid:32) v D (cid:33) ∀ a, b ∈ F u, v ∈ H ( T )The loops x, y from figure 1 induce generators of H ( T ), which we also denote by x, y . Giventhis basis (or any other basis), it is easy to see that A ∂ ( P ) is isomorphic to A ∂ ( P, { x, y } ), and themultilinearity relation is no longer needed. In general it is better to work with the definition whichallows all the elements of H ( T ) as labels, because this definition does not depend on a choice of abasis, and also because A ∂ ( P ) defined this way has a natural action of the symplectic group (see[14]). However, for our needs it would be more convenient to consider only diagrams with x and y labels, and forget about multilinearity. A strut in an elliptic Jacobi diagram D ∈ D ( P, { x, y } ) is a component in the diagram of theform uv with u, v ∈ { x, y } . D is called top-substantial if it has no struts labeled by y on bothvertices. Let ts A ∂ ( P ) ⊂ A ∂ ( P ) be the subspace generated by all the top-substantial diagrams.This restriction will allow us to define a composition of elliptic Jacobi diagrams (see also [10],section 3.1).For 2 elements D ∈ ts A ∂ ( P ) and D ∈ ts A ∂ ( P ) such that P and P are composable, wedefine the composition D · D to be the sum of all diagrams obtained by putting D on top of D and attaching all the y -labeled vertices of D to all the x -labeled vertices of D (thus, if thenumber of y -labeled vertices of D is not equal to the number of x -labeled vertices of D , this sumis empty). This composition is extended linearly to a map A ∂ ( P ) × A ∂ ( P ) → A ∂ ( P · P ).The category ts A ∂ is now defined as the category whose objects are words in the symbols { + , −} ,and for any 2 such words ω s and ω t , ts A ∂ ( ω s , ω t ) := (cid:83) P ∈ P ( ω s ,ω t ) ts A ∂ ( P ). The composition isdefined as above.The category ts A ∂ is a monoidal category. The tensor product of two objects ω and ω isthe concatenation ω ω , and the tensor product of 2 diagrams D and D is obtained by putting D alongside of D . In ts A ∂ ( ∅ ) this tensor product induces a multiplication. With respect to thismultiplication we can define an exponent. In particular we have the identity elements: id ω = exp yx ⊗ ωω
13 Let I ( P ) be the subspace of ts A ∂ ( P ) generated by sums of the type shown in figure 5. In thisfigure we assume that the lines which come out of the upper side of the box are exactly all theends of the solid lines which lead to the points of ∂ t ( P ), and all the ends of dashed lines endingwith the label y . We call each such sum an I relation. D y y Figure 5: An I relationIt may be verified that the union of all I ( P ) is a two-sided ideal of ts A ∂ (by a similar argumentto the one we mentioned in the previous section), which we denote by I . The quotient category ts A ∂ / I is denoted A . Denote the projection by π : ts A ∂ → A . Let D y ( P ) ⊂ D ( P ) be the subset of all diagrams D such that each component of D has at leastone trivalent vertex or one vertex on P . In other words, D has no struts. Since all the relations of ts A ∂ ( P ) preserve this subspace, we get a subspace A ∂y ( P ) of ts A ∂ ( P ).We use the spaces A ∂y ( P ) to define a category A ∂y similarly to the way we defined A ∂ but witha different composition. For 2 elements D ∈ A ∂y ( P ) and D ∈ A ∂y ( P ) such that P and P arecomposable, we define the composition D · D to be the sum of all diagrams obtained by putting D on top of D and attaching some of the y -labeled vertices of D to some of the x -labeledvertices of D . This composition is extended linearly to a map A ∂y ( P ) × A ∂y ( P ) → A ∂y ( P · P ).Define maps j ∂P : A ∂y ( P ) → ts A ∂ ( P ) by u (cid:55)→ exp yx ⊗ u . It is easy to see that j ∂P isinjective. Also, given u ∈ A ∂y ( P ) and u ∈ A ∂y ( P ) with P and P composable, it may beverified that j ∂P · P ( u · u ) = j ∂P ( u ) · j ∂P ( u ). So j ∂P induce an injective functor j ∂ : A ∂y → ts A ∂ .Let I y ( P ) be the subspace of A ∂y ( P ) generated by sums of the type shown in figure 6. Again,we assume that the lines which come out of the upper side of the box are exactly all the ends ofthe solid lines which lead to the points of ∂ t ( P ), and all the ends of dashed lines ending with thelabel y . We call each such sum an I y relation. Proposition 3.1.
The subspaces I y ( P ) induce a two-sided ideal of A ∂y .Proof. Let the two diagrams of figure 6 be denoted by I xy and I box , respectively. Let D be anydiagram composable with I xy and I box . We need to show that D ◦ ( I xy + I box ) is in I y (the other14 y yxy + D y y Figure 6: An I y relationorder of composition is trivial).Denote by A x the set of all x -labeled vertices of D . Let U x be a subset of A x , U y a subset of the y labels of I box with the same size of U x , and p : U y ∼ = −→ U x a bijection. Denote by D ◦ p I box thediagram obtained by attatching the x -labels in U x to the y -labels in U y according to p . Similarlydefine D ◦ p I xy . Also, for any l ∈ A x \ U x , define D ◦ ( p,l ) I xy to be the diagram obtained byattatching the x -labels in U x to the y -labels in U y according to p , and also attaching l to theleft-most y label in I xy . We have: D ◦ ( I xy + I box ) = (cid:88) ( U y ,U x ,p ) D ◦ p I box + D ◦ p I xy + (cid:88) l ∈ A x \ U x D ◦ ( p,l ) I xy For a given triple ( U y , U x , p ), we will show that the corresponding summand is an I y relation.Indeed, by using IHX and STU relations we can replace the box from I box by a box over allthe top solid lines of D , all the edges leading to y labels of D and all the labels leading to theremaining x -labels of D . The summands of this box which are near x -labels get canceled by thesum (cid:80) l ∈ A x \ U x D ◦ ( p,l ) I xy , and we are left with a new I y relation, as required.We denote the quotient category A ∂y / I y by A y , and the projection by π y : A ∂y → A y .For any u ∈ I y ( P ) we have j ∂P ( u ) ∈ I ( P ), so the functor j ∂ induces a functor j : A y → A . In this subsection we define the categories A ∂< and A < , which are isomorphic to A ∂y and A y ,respectively. Their definition is, in a sense, more complicated - we take more Jacobi diagrams, andquotient them by more relations. On the other hand, the composition rule in these categories ismuch simpler. An ordered elliptic Jacobi diagram over a pattern P is an elliptic Jacobi diagram in D y ( P ),with the additional data of a linear order on the labeled vertices. Denote the set of all orderedelliptic Jacobi diagrams over P by D < ( P ). In figures we use the convention that a labeled vertexis bigger if it appears higher in the figure.Let A ∂< ( P ) be the quotient of S F ( D < ( P )) by the relations: STU, AS, IHX, multilinearity15although we will assume, as above, that the labels are only x, y and there is no need for multilin-earity), and STU-like. The STU-like relation is defined as follows: vw D − wv D = (cid:104) v, w (cid:105) D (cid:104)· , ·(cid:105) is the intersection form on H ( T ). Since we will only use the labels x, y , all we need to knowis that (cid:104) y, x (cid:105) = 1.The category A ∂< is defined in a way similar to the previous categories we defined in thissection. The composition of 2 diagrams D , D is obtained by simply putting D on top of D ,and declaring all the labeled vertices of D to be bigger than all the labeled vertices of D . Thisinduces the composition of A ∂< by linearity.Let k ∂ : D → D < be the map which sends a diagram D to itself and declares all the x -labeledvertices to be smaller than all the y -labeled vertices. It may be verified that k intertwines thecompositions of A ∂y and A ∂< , so it induces a functor k ∂ : A ∂y → A ∂< . Proposition 3.2. k ∂ : A ∂y → A ∂< is an isomorphism. A short proof of this proposition, using an explicit formula for the inverse of k ∂ , can be found in[10]. We will give here a different proof. The idea of the proof is, for any diagram D ∈ D < ( P ), toiteratively use the STU-like relation to reduce the number of pairs of labeled vertices with y < x ,until we get a representation of D as a linear combination of diagrams from D y ( P ). This simpleidea is formalized using the language of filtrations and direct limits. In section 5 we will use thistechnique several more times. Proof.
Since k ∂ is the identity on objects, it is enough to show that for any pattern P , k ∂ : A ∂y ( P ) → A ∂< ( P ) is an isomorphism. For that purpose we will construct an inverse map ϕ ∂ : A ∂< ( P ) → A ∂y ( P ).For any diagram D ∈ D < ( P ), define n y The LMO functor was defined by Cheptea, Habiro and Massuyeau ([10]). It is a functor from thecategory of Lagrangian cobordisms to a certain category of Jacobi diagrams. In this section weextend the LMO functor to the category of Lagrangian cobordisms with embedded tangles. Theextension is quite straight-forward, so most of this section may be seen as a review of CHM’s work,with a slight generalization.On the other hand, our construction will be more restricted than the construction of CHM.They deal with cobordisms between surfaces of any genus, with or without boundary. We willrestrict ourselves to closed surfaces of genus 1, which is all we need here. We made this choice forconvenience, to make the notation simpler, but the extension to any genus should be obvious.At the end of this section we show how this extended LMO functor gives rise to an ellipticstructure on the categories of Jacobi diagrams introduced in section 3. Recall that T is the torus S × S . Denote by T ∂ a torus with one boundary component. A cobordism of T ∂ is an oriented compact connected 3-manifold M with an isomorphism m : ∂ ( T ∂ × [0 , ∼ = → ∂M . Similarly, a cobordism of T is an oriented compact connected 3-manifold M with an isomorphism m : ∂ ( T × [0 , ∼ = → ∂M .Recall that for any n ≥ b n in T . We can define those setsof points also in T ∂ . Let ω s , ω t be non-associative words in the symbols { + , −} , with lengths | ω s | = m , | ω t | = n . A cobordism with an embedded tangle of type ( ω s , ω t ) (either of T ∂ or of T ) is a cobordism M with an embedded framed oriented tangle T ⊂ M satisfying ∂T = m (( b m × { } ) ∪ ( b n × { } )), such that the orientations of T around the boundary points correspondto the words ω s and ω t , and the framings around the boundary points are all parallel to theboundary surfaces and parametrized (via m ) as (0 , − , M , m , T ) and ( M , m , T ) are said to be equivalent if thereis a homeomorphism h : M → M such that h ◦ m = m and h ( T ) = T (including the framingand the orientation of the tangle).The categories CT ∂ and CT (Cobordisms with Tangles) are defined as follows: The objects arenon-associative words in { + , −} . For any two such words ω s , ω t , the set of morphisms CT ∂ ( ω s , ω t )( CT ( ω s , ω t )) is the set of all equivalence classes of cobordisms of T ∂ ( T ) with embedded tangles oftype ( ω s , ω t ). The composition is defined by simply putting one cobordism on top of the other.We now explain how to represent a cobordism with tangle by another tangle embedded in asimpler manifold. In D we have the sets of points b n , and we choose 2 more points p and q . Let ω s , ω t be non-associative words in { + , −} . A representing tangle of type ( ω s , ω t ) is a framedoriented tangle T embedded in D × I with the following properties: • The boundary of T is (( b | ω s | ∪ { p, q } ) × { } ) ∪ (( b | ω t | ∪ { p, q } ) × { } ). • There is a component whose boundary is { p, q }×{ } , denoted by x , and there is a componentwhose boundary is { p, q } × { } , denoted by y .20 The orientations of T around the boundary correspond to the words ( − +)( ω s ) and ( − +)( ω t ).(Note that the convention used in [10] is opposite to ours, so they represent the orientationsat p and q by (+ − ), instead of ( − +).) • We are given a subset S of the closed components of T , and say that those components aremarked for surgery.Two representing tangles T and T are said to be equivalent if they can be related by a sequenceof ambient isotopies and Kirby moves on the components marked for surgery. This means that wecan add to S a trivial closed component with ± S , andwe can slide any component over components of S .The category RT (Representing Tangles) is defined as follows: The objects are non-associativewords in { + , −} . For any two such words ω s and ω t , the morphisms set RT ( ω s , ω t ) is the set ofall equivalence classes of representing tangles of type ( ω s , ω t ).There is a simple operation ◦ : RT ( ω s , ω t ) × RT ( ω t , ω u ) → RT ( ω s , ω u ) which takes 2 composabletangles and simply puts them one on top of the other. But the defintion of the composition in RT is different. Let T ∈ RT ( ω s , ω t ), T ∈ RT ( ω t , ω u ) be representing tangles with subsets markedfor surgery S and S , respectively. We define the composition T · T to be T ◦ T i ( ω t ) ◦ T , where T i ( ω ) is the tangle shown in figure 8. p q b | ω | p q b | ω | Figure 8: The tangle T i ( ω )The set of components marked for surgery in T · T is defined to be the union of S , S andthe (now closed) y component of T and x component of T . The identity in RT ( ω, ω ) is: p q b | ω | p q b | ω | There is a functor rep : RT → CT ∂ , which is the identity on objects, and for a representingtangle T , rep ( T ) is the cobordism of T ∂ with embedded tangle obtained by removing a tubu-lar neighborhood of x, y from D × I , and performing surgery on the components in S . Theparametrization m : ∂ ( T ∂ × [0 , → ∂rep ( T ) is chosen in such a way that the generators x and y of T ∂ from figure 1 are mapped to the following elements on the boundary of rep ( T ):21 xqp D xy D p q The segments of the paths x and y which are parallel to the removed components of the tangleare determined by the framing of the tangle.The functor rep is very much related to the functor D from Theorem 2.10 of [10]. D is a functorfrom the category of bottom-top tangles in homology cubes to the category of cobordisms. Thecategory of bottom-top tangles in homology cubes, when restricted to tangles with one bottomcomponent and one top component, is isomorphic to the category RT via standard Kirby calculus.Therefore rep factors through D , which proves that it is indeed a functor and an isomorphism.In order to define the LMO functor we need to restrict to the subcategories LCT ∂ ⊂ CT ∂ and LCT ⊂ CT of Lagrangian cobordisms with embedded tangles. The exact definition of Lagrangiancobordisms is not important here, and can be found in [10] (Definition 2.4). For our purposes it isenough to say that the corresponding subcategory LRT of RT is the subcategory of all representingtangles in which the determinant of the linking matrix of S equals ± 1, and the framing of y , afterperforming surgery on S , is 0. Let T be a tangle in LRT ( ω s , ω t ). Denote by P ∈ P ( ω s , ω t ) the skeleton of T . It is decomposed as P = { x, y } ∪ S ∪ P (cid:48) .Recall that a Drinfel’d associator is an element φ ( A, B ) in the exponent of the completed freeLie algebra generated by A and B , which satisfies several identities (see, for example, [4], Definition3.1). We define Z to be a functor from the category of tangles to the category of Jacobi diagrams A ∂ . The tangle T will be mapped by Z to Z ( T ) ∈ A ∂ ( P ). Z is a variant of the Kontsevich integralof tangles, which is defined over elementary tangles as follows:22 u ( v w )( u v ) w = ∆ +++ uvw (cid:32) φ (cid:32) , (cid:33)(cid:33) Z (+ +)(+ +) = exp( 12 ) Z (+ +)(+ +) = exp( − 12 ) Z ( − +) = Z ( − +) = n where ν ∈ A ∂ ( ↑ ) ∼ = A ∂ ( ) is the Kontsevich integral of the unknot with 0 framing.Let Z ν,S ( T ) be the value obtained from Z ( T ) by taking the connected sum of each componentof S with ν .In [10] (after Lemma 4.9), an element T g ∈ A ( ∅ , { − , ..., g − , + , ..., g + } ) is defined. We willconsider T as an element of A ( ∅ , { x, y } ) via the labels change 1 − (cid:55)→ x and 1 + (cid:55)→ y . For a word ω in the symbols { + , −} , let id ω be the identity pattern in P ( ω, ω ), and let T ( ω ) ∈ ts A ∂ ( id ω ) bethe element obtained by putting T alongside the empty pattern id ω .The LMO functor LM O : LCT ∂ ∼ = LRT → ts A ∂ is defined as follows: A non-associative word ω is mapped to itself, forgetting the non-associative structure. A morphism T ∈ LRT ( ω s , ω t ) ismapped to: LM O ( T ) := T ( ω t ) · (cid:18) U − σ + ( S )+ U − σ − ( S ) − (cid:90) S χ − S ∪{ x,y } Z ν,S ( T ) (cid:19) where: • χ S (cid:48) : A ( P (cid:48) , S (cid:48) ) → A ∂ ( P (cid:48) ∪ S (cid:48) ) is the symmetrization map defined in [1] (section 5.2, in theproof of Theorem 8), applied to the components of S (cid:48) (which are first considered as labels).For a diagram D ∈ A ( P (cid:48) , S (cid:48) ), χ S (cid:48) ( D ) is the average of all the diagrams which are obtainedby putting all the s labeled vertices on the s component of the pattern S (cid:48) (for all s ∈ S (cid:48) ), inany possible order. • (cid:82) S (cid:48) : is the ˚Arhus integral on the labels of S (cid:48) , as defined in [6] (section 2.1, specificallyDefinition 2.11). • U ± := (cid:82) χ − (cid:0) ν Z ( ± ) (cid:1) with ± being an unknot with framing ± • σ ± ( S ) are the numbers of positive/negative eigenvalues of Lk ( S ).For a more detailed account of this construction, and a proof of invariance and functoriality, see[10]. For our purposes it is almost enough to use this definition as a “black box”. The only23mportant thing to notice is that the integral (cid:82) S commutes with χ − { x,y } , i.e. we have: (cid:90) S χ − S ∪{ x,y } Z ν,S ( T ) = χ − { x,y } (cid:90) S χ − S Z ν,s ( T )In order to define the LMO functor on cobordisms of T with embedded tangles, all we needto do is choose a representing tangle T , map it by LM O to ts A ∂ , and then map it to A by thequotient map. Thus we get a functor LM O : LCT → A . The fact that this functor is well definedis also proved in [10], Theorem 6.2. Let HCT be the subcategory of LCT containing only tangles in cobordisms which are homologycylinders. Homology cylinders are cobordisms which are homologically equivalent to the cylinder T × I (an exact definition can be found in [10] Definition 8.1). The LMO functor, when restricted tothis category, gives values of the form exp yx ⊗ a , where a is a combination of diagrams from D y (see [10] section 8.2 and [15] section 4.2). We can therefore compose the LMO functor with theinverse of the injective functor j : A y → A defined above to get a functor LM O y : HCT → A y .Furthermore, we can also define LM O < : HCT → A < by LM O < := k ◦ LM O y , k being theisomorphism of categories defined in section 3.5.If we further restrict HCT to include only tangles in the trivial cylinder T × I , we get thecategory q ˜ T of framed tangles in the thickened torus defined in Section 2.2. Restricting the abovevariants of the LMO functor to this subcategory we get LM O y : q ˜ T → A yp and LM O < : q ˜ T →A
LM O y and LM O < to X + , + and Y + , + we get elliptic structures relative to A ∂ → A yp and A ∂ → A t ij [ v i , t jk ] = 0[ x i , y i ] = − (cid:88) j (cid:54) = i t ij where 1 ≤ i, j, k ≤ n are distinct indices, v, w ∈ { x, y } , and < · , · > is the intersection form of H ( T ) (the symbols x and y are considered to be the generators of H ( T ) from figure 1). U ˆ t ,n isthe degree completion of the universal enveloping algebra of t ,n .Denote by ↑ n the pattern in P (+ + · · · + (cid:124) (cid:123)(cid:122) (cid:125) n times , + + · · · + (cid:124) (cid:123)(cid:122) (cid:125) n times ) composed of n up-going strands. There isa map u n : U ˆ t ,n → A
Let R D
Let D ∈ D yp ( ↑ n ) be a diagram, and let v be a vertex in D on the rightmoststrand. We call v a lonely vertex if it belongs to a component of D which does not have morevertices on the pattern. Otherwise we call it a non-lonely vertex .30et L D yp ( ↑ n ) ⊂ D yp ( ↑ n ) be the subset of diagrams in which all the vertices on the rightmoststrand are lonely vertices. Let L A yp ( ↑ n ) denote the quotient of S F ( L D yp ( ↑ n )) by all STU relationscontained in it (there are no I y relations in this subspace). Proposition 5.2. The obvious map l : L A yp ( ↑ n ) → A yp ( ↑ n ) is an isomorphism.Proof. Let D ∈ D yp ( ↑ n ). For any non-lonely vertex v of D , let n l ( v ) be the total degree of allcomponents with a lonely vertex higher than v . Let n l ( D ) := ( (cid:80) v non-lonely n l ( v ) ) + |{ v non-lonely }| .Let ( D yp ( ↑ n )) m := { D ∈ D yp ( ↑ n ) | n l ( D ) ≤ m } . We get a filtration of D yp ( ↑ n ):( D yp ( ↑ n )) ⊆ ( D yp ( ↑ n )) ⊆ ( D yp ( ↑ n )) ⊆ · · · which induces the sequence: S F (( D yp ( ↑ n )) ) ⊆ S F (( D yp ( ↑ n )) ) ⊆ S F (( D yp ( ↑ n )) ) ⊆ · · · Let L m be the quotient of S F (( D yp ( ↑ n )) m ) by all STU, IHX and I y relations contained in it. Weget a sequence: L l −→ L l −→ L l −→ · · · L is isomorphic to L A yp ( ↑ n ) (the IHX relations in this space are implied by STU, and there areno I y relations). The direct limit of this sequence is A yp ( ↑ n ), and the maps l m induce the map l at the limit. So, in order to complete the proof it is enough to define an inverse for each l m .Let η m : S F (( D yp ( ↑ n )) m ) → S F (( D yp ( ↑ n )) m − ) be defined as follows: For D with n l ( D ) < m , η m ( D ) = D . For D with n l ( D ) = m , we have 2 cases. If the highest vertex on the right strand isnon-lonely, define η m ( D ) by: D' y y η m (cid:55)−→ − D' y yyx − D' y y In these figures we assume that there are no more y labels inside D (cid:48) .If the highest vertex on the right strand in D is a lonely vertex, denote the highest non-lonelyvertex in D by v , and define η m ( D ) by: D' v η m (cid:55)−→ D' v + D' v We claim that η m induces a map η m : L m → L m − . Indeed, if u ∈ S F ( D yp ( ↑ n )) m ) is an I y relationthen η m ( u ) is either again an I y relation, or is equal to 0 by definition of η m . If u is an IHX31elation, η m ( u ) is a sum of IHX relations. Suppose now u is an STU relation: u = u − u − u = D' vw − D' vw − D' v (Note: the labels v i , w i etc. are not part of the diagrams. We write them only to help keep trackin the following computations.)If none of the vertices v i , w i are the highest non-lonely vertices in their respective diagramsor the vertices immediately above the highest non-lonely vertices, then η m ( u ) is a sum of ST U relations. Otherwise, we have to deal with several different cases (in all those cases, we assumethat at least one of u , u , u has n l ( u i ) = m , because otherwise η m ( u ) = u ):A. If v is the highest non-lonely vertex in u , and either w or w is a lonely vertex, then bydefinition η m ( u ) = 0.B. If v is the highest non-lonely vertex in u and w ,w are non-lonely (and therefore also v and v ), then again we have 2 cases:B1. If v is the highest vertex on the right strand, we have: η m ( u ) ≈ (cid:13) D'' y y v − (cid:13) D'' y y v − (cid:13) D'' y y ++ 4 (cid:13) D'' y y v yx − (cid:13) D'' y y v yx − (cid:13) D'' y y yx ≈ D'' y y − D'' y y yx + 3 D'' y y ++ 4 D'' y y yx − D'' y y − D'' y y yx −− D'' y y yx − D'' y y yxyx + 9 D'' y y yx ++ 10 D'' y y yx + 11 D'' y y yxyx − D'' y y yx ≈ (cid:13) with 1 and 2 , 2 (cid:13) with 3 and 4 ,4 (cid:13) with 6 , 7 and 8 , and 5 (cid:13) with 9 , 10 and 11 . The second equivalence followsbecause: 1 , 3 and 5 are a sum of STU and IHX relations, 2 cancels 9 , 4 cancels6 , 7 , 10 and 12 are an IHX relation, and 8 cancels 11 .B2. If v is not the highest vertex on the right strand, we have:33 = u − u − u = D'' vwz − D'' vwz − D'' vz and η m ( u ) ≈ D'' vw + D'' vwz − D'' vw −− D'' vwz − D'' v − D'' vz ≈≈ (cid:13) D'' vw + 2 (cid:13) D'' vw + 3 (cid:13) D'' vwz − (cid:13) D'' vw − (cid:13) D'' vw − (cid:13) D'' vwz −− (cid:13) D'' v − (cid:13) D'' v − (cid:13) D'' vz ≈ (cid:13) , 5 (cid:13) and 7 (cid:13) are an IHX relation, 2 (cid:13) , 4 (cid:13) and 8 (cid:13) arean IHX relation and 3 (cid:13) , 6 (cid:13) and 9 (cid:13) are an STU relation.C. If v is the lonely vertex immediately above the highest non-lonely vertex, then: u = u − u − u = D'' vwz − D'' vwz − D'' vz and η m ( u ) ≈ D'' wz + D'' vwz − D'' wz − D'' vwz − D'' z − D'' vz ≈≈ D'' z + 2 D'' wz + 3 D'' vz ++ 4 D'' vwz − D'' z − D'' wz −− D'' vz − D'' vwz − D'' z − D'' vz ≈ η m : L m → L m − is well defined. It is easy to verify that it isthe inverse of l m − , which completes the proof that l is an isomorphism.We now have a representation of A yp ( ↑ n ) as the quotient of the vector space S F ( L D yp ( ↑ n ))by STU relations alone. Recall the isomorphism k : A yp ( ↑ n ) → A
The obvious map s : S F ( SR D
Proof. For a diagram D ∈ R D
38e claim that γ m induces a map γ m : S m → S m − . Indeed, if u is an STU-like relation, then γ m ( u ) is a sum of STU-like relations.Let u = u + u + u be an STU relation, and suppose n triv ( u ) = m . If the STU relation u does not involve the vertices v ( u ) and w ( u ), then γ m ( u ) is a sum of STU relations. If the STUrelation involves both v ( u ) and w ( u ), then by definition γ m ( u ) = 0. And if the STU relationinvolves w ( u ) but not v ( u ), then γ m ( u ) is a 4T relation.Now let u = u + u + u + u be a 4T relation. If u does not involve, in any of its summands,the vertex v ( u i ), then γ m ( u ) is a sum of 4T relations. If the 4T relation u involves, in any of itssummands, the vertex v ( u i ), then γ m ( u ) is either equal to a sum of 4T relations, or equivalent toit via STU (depending on whether v ( u i ) is involved in the 4T relation in all the u i ’s or only in 2of them). This follows from the fact that the following sum: D' − D' − D' + D' ++ D' − D' − D' + D' can be written as a sum of 8 4T relations, with 24 of the 32 summands cancelling in pairs.It is easy to verify that γ m is the inverse of s m − , which completes the proof.The map u r : RU ˆ t ,n → R A
If a diagram a ∈ A
The map ˜ u s : RU ˆ t ,n → SR A
OSR D · z D'' ≈≈ vzw D'' + < z, v > · w D'' − wzv D'' −− < z, w > · v D'' − < v, w > · z D'' ≈ vwz (cid:13) D'' ++ < z, w > · v (cid:13) D'' + < z, v > · w (cid:13) D'' − · wvz (cid:13) D'' −− < z, v > · w (cid:13) D'' − < z, w > · v (cid:13) D'' −− < v, w > · z (cid:13) D'' ≈ (cid:13) cancels 6 (cid:13) , 3 (cid:13) cancels 5 (cid:13) , and 1 (cid:13) , 4 (cid:13) and 7 (cid:13) are STU-like.In all other cases, β m ( u ) is a sum of STU-like relations.42f u is an O4T relation: u = D e f − D e f ++ D e f − D e f then again we have to deal with several cases: If e is a chord, then β m ( u ) is a sum of O4T relations.Similarly, if e is a labeled edge, and its label is not “touched” by β m in either of the summandsof u , then again β m ( u ) is a sum of O4T relations. And if e is labeled and its label is touched by β m in some (or all) of the summands of u , then β m ( u ) is an O4T relation + some terms which areequivalent to 0 according to remark 5.3.It is easy to see that β m is the inverse of o m − , which completes the proof.The map ˜ u s : RU ˆ t , → SR A
The obvious map f : F OSR A
There is no obvious multiplication in F OSR A
Let D ∈ OSR D
OSR A D' If only B. applies to e , define: D' v e f α m (cid:55)−→ ' v f + * (cid:13) D' v f − * (cid:13) D' v f In the last 2 summands (marked by * (cid:13) ) we should specify the location of the label v in the linearorder of the labels in D . This is determined as follows: If there is a 2 labeled edge above the vertexof f , locate v as the label immediately below it. Otherwise, locate v as the highest vertex. Thischoice guaranties that α m ( D ) is indeed in OSR A D' We have n f ( u ) > n f ( u ) , n f ( u ). Assume n f ( u ) = m . If v is the highest label with propertiesA. or B. then by definition α m ( u ) = 0. Otherwise α m ( u ) is equivalent to a sum of STU-likerelations.Assume now that u is an O4T relation: u = u + u + u + u == D' v e f − D' v f ++ D' v f − D' v f Clearly we have n f ( u ) > n f ( u ) and n f ( u ) > n f ( u ). Therefore, potentially we might have45 f ( u i ) = m only for i = 1 and i = 3. Assume first that only n f ( u ) = m . If v is not the highestlabel in u with properties A. or B., then α m ( u ) is equivalent to an O4T relation. If v is the highestsuch label and only property B. applies to it, then by definition α m ( u ) = 0. And if property A.also applies to it, then we have: u = D' vw − D' vw + D' vw −− D' vw α m (cid:55)−→ (cid:13) D' vw + < v, w > (cid:13) D' −− (cid:13) D' vw − < v, w > (cid:13) D' + 5 (cid:13) D' vw −− (cid:13) D' vw ≈ (cid:13) , 3 (cid:13) , 5 (cid:13) and 6 (cid:13) are an O4T relation, and 2 (cid:13) and 4 (cid:13) areequivalent to 0 by remark 5.3. Note that in some of the diagrams involved in this calculation wemight actually have a different order of the labels or a different order along strand 2, but this doesnot affect the actual calculation. 46f only n f ( u ) = m the argument is similar. If n f ( u ) = n f ( u ) = m then the computationis more complicated, since we need to consider several possibilities, but the principles of thecalculation are the same, and we leave it to the reader.It is easy to see that α m is the inverse of f m − , thus we have completed the proof.The map ˜ u o : RU ˆ t ,n → OSR A
2) and˜ u f ( t ). Therefore, there is an obvious map (cid:94) F OSR A t Those relations indeed hold in U ˆ t , (see [16] Definition 2.1.1 and Lemma 2.1.2). This completesthe proof of theorem 5.4. 47 Elliptic Associators and the LMO Functor In this section we introduce the concept of elliptic associators and the specific associator defined in[8] and [13]. We then study the relation between this elliptic associator and the elliptic structurerelative to A ∂ → A
2) (6.2) Y ( x + x , y + y ) = φ ( t , t ) − Y ( x , y ) φ ( t , t ) exp( − t / ·· φ ( t , t ) − Y ( x , y ) φ ( t , t ) exp( − t / 2) (6.3) φ ( t , t ) − Y ( x , y ) φ ( t , t ) exp( t / φ ( t , t ) − ·· X ( x , y ) φ ( t , t ) exp( t / 2) = exp( t / φ ( t , t ) − ·· X ( x , y ) φ ( t , t ) exp( − t / φ ( t , t ) − Y ( x , y ) φ ( t , t ) (6.4)If X ( A, B ) , Y ( A, B ) is an elliptic associator, then it is easy to see that ∆ ++ ω ,ω ( u ( X ( x , y )))and ∆ ++ ω ,ω ( u ( Y ( x , y ))) define an elliptic structure relative to A ∂ → A
In the completed Lie algebra ˆ f ( A, B ), denote: T := [ B, A ]˜ A := ad Be ad B − A ) = A − 12 [ B, A ] + 112 [ B, [ B, A ]] + · · · Note: The coefficients which appear in this expansion are the Bernoulli numbers, which aredenoted by B i . 48 efinition 6.2. Given a Drinfel’d associator φ , let e ( φ ) = ( X φ , Y φ ) be defined by: X φ ( A, B ) = φ ( ˜ A, T ) exp( ˜ A ) φ ( ˜ A, T ) − Y φ ( A, B ) = exp( T / φ ( − ˜ A − T, T ) exp( B ) φ ( ˜ A, T ) − Theorem 6.1. e ( φ ) is an elliptic associator relative to φ . A proof of this theorem is given in [13] (Proposition 3.8, see also [8] Proposition 5.3). Our goalin this section is to give a different proof of this theorem, based on the following theorem, whichrelates e ( φ ) to the elliptic structure relative to A ∂ → A
Using the same techniques one might be able to define associators for higher genus,by pulling back the value of the LMO functor on the right tangles.The rest of this section is dedicated to proving theorems 6.1 and 6.2. We begin with several lemmas. Here and in the following proofs we denote by t ij ∈ A ( ↑ n , S ) thediagram with a single edge connecting the i strand and the j strand. Lemma 6.3. Given a word ω of length and words ω , ω and ω in { + , −} , ∆ ωω ,ω ,ω φ ( t , t ) =1 (i.e. the empty diagram) in A ( ↑ | ω | + | ω | + | ω | ) (as defined in section 3.2).Proof. It is enough to show that φ ( t , t ) = 1 in A ( ↑ ). Indeed, using an I relation we get: φ ( t , t ) = φ (cid:18) , (cid:19) = φ (cid:18) − − , (cid:19) = 1because both and commute with (using the STU relation). Lemma 6.4. Assume that in the pattern ↑ n +1 the left strand is labeled x , and let a ∈ A ( ↑ n +1 , { y } ) .Recall the map j : A y → A defined in section 3.4, and the map k : A y −→ A < definedin section 3.5. Then we have χ − x (cid:18) exp (cid:18) y (cid:19) · a (cid:19) ∈ Im ( j ) ⊂ ts A ( ↑ n ) , and k ◦ j − ◦ − x (cid:18) exp (cid:18) y (cid:19) · a (cid:19) is the element obtained from a by replacing each vertex on x by the fol-lowing sum: D x (cid:55)−→ ∞ (cid:88) i =0 B i D xyy { itimes where B i are the Bernoulli numbers.Proof. Recall the element λ ( a, b ; r ) ∈ A ( ∅ , { a, b, r } ) defined in [10] by: λ ( a, b ; r ) = χ − r exp(a )exp(b )r == exp (cid:96) br + (cid:88) n ≥ u , ..., u n ∈ { a, b } r ( u , ..., u n ) auu r where r ( u , ..., u n ) are some coefficients determined by the Baker-Campbell-Hausdorff formula. Inparticular we have r ( b, ..., b (cid:124) (cid:123)(cid:122) (cid:125) n times ) = B n (follows from [12], formula (12)). λ ( a, b ; r ) has the following property: If D ∈ D ( P ∪ {↑ r } , S ) is of the following form: D = D'r χ − r ( D ) = (cid:42) χ − r ,r D'rr , λ ( r , r ; r ) (cid:43) r , r where the form < D , D > r ,r is defined on diagrams D , D to be the sum of all ways to glueall vertices labeled by r in D to all vertices labeled r in D and all vertices labeled by r in D to all vertices labeled r in D .Now, it is enough to prove the theorem for a which is a diagram. Suppose we have m ≥ x (the case m = 0 is trivial), and denote x ( m ) = x . χ − x ( m ) D'x (m) exp(y ) == (cid:42) χ − x m ,x ( m − D'xx m(m-1) exp(y ) , λ ( x m , x ( m − ; x ( m ) ) (cid:43) x m , x ( m − == ψ x ( m − , ˜ x ( m − (cid:88) k m ≥ B k m · χ − x ( m − exp ˜ x ( m − x ( m ) · D'x (m-1) exp(y )xxx (m)(m-1)(m-1) ~~ { k m times where ψ z, ˜ z ( D ) for a diagram D which contains the labels z and ˜ z is defined to be the sum of allways to glue all vertices labeled z to all vertices labeled ˜ z .51epeating this process recursively we get: χ − x ( m ) D'x (m) exp(y ) == ψ x ( m − , ˜ x ( m − ◦ · · · ◦ ψ x (0) , ˜ x (0) ( (cid:88) k ,...,k m ≥ B k · · · B k m exp ˜ x ( m − x ( m ) · · · exp ˜ x (0) x (1) · exp yx (0) · * (cid:13) D'xxx (m)(m-1)(m-1) ~~ { k m times xxx (1)(0)(0) ~~ { k times )For a specific choice of k , ..., k m , we wish to describe the element we get by applying ψ x ( m − , ˜ x ( m − ◦ · · · ◦ ψ x (0) , ˜ x (0) to the corresponding summand. A careful examination shows thatthe element we get is a product of exp yx ( m ) with the sum of all the diagrams which can beproduced from * (cid:13) by the following process:1. Change all the ˜ x (0) labels to y .2. For i = 1 , ..., m − • Attach some of the x ( i ) labels to some of the ˜ x ( i ) labels. • Change all the remaining x ( i ) labels to x ( i +1) . • Change all the remaining ˜ x ( i ) labels to y This sum can be described more shortly as the sum of all ways to glue some of the x ( i ) labelsto ˜ x ( j ) labels with j ≥ i , and then change all the remaining x ( i ) labels to x ( m ) = x and all theremaining ˜ x ( i ) labels to y .The result of the above calculation clearly belongs to the image of j . It is not difficult to see52hat j − ◦ χ − x (cid:18) exp (cid:18) y (cid:19) · D (cid:19) has a simpler presentation when mapped by k to A < ( ↑ n ), as: (cid:88) k ,...,k m ≥ B k · · · B k m D'xyy { k m times xyy { k times Thus we have completed the proof of the lemma. Lemma 6.5. Recall the notation ˜ A := ad Be ad B − ( A ) in ˆ f ( A, B ) . Similarly we have in U ˆ t , : ˜ x = ad y e ad y − x ) . Then we have: ˜ u (˜ x ) ≈ ∞ (cid:88) i =0 B i xyy { itimes mod H Proof. ˜ x = (cid:80) ∞ i =0 B i [ y [ · · · [ y (cid:124) (cid:123)(cid:122) (cid:125) i times , x ] · · · ]] by definition. We will prove by induction on i ≥ u ([ y [ · · · [ y (cid:124) (cid:123)(cid:122) (cid:125) i times , x ] · · · ]]) = xyy { itimes + yy { i-1times For i = 1 we have: u ([ y , x ]) = xy − yx = xy − xy + xy − yx == xy +Assume we proved the lemma for i . Then for i + 1: u ([ y [ · · · [ y (cid:124) (cid:123)(cid:122) (cid:125) i + 1 times , x ] · · · ]]) = xyy { itimes y − xyy { itimes y + =0 (cid:122) (cid:125)(cid:124) (cid:123) yy { times yi -1 − yy { i -1times y = xyy { itimes y −− xyy { itimes y + xyy { itimes y − xyy { itimes y == xyy { timesi +1 + yy { timesi We are now ready to prove theorem 6.2. 54 roof of theorem 6.2. We represent the tangle X + , + as follows: ( ) ( ) ( ( ))( ) ( ) (( ) ) (( ) ) (( ) ) ( ( )) ( ) ( ) ( ) ( )xy We will calculate χ x ( LM O ( X + , + )) ∈ A ( ↑ x ↑ , { y } ). Inside A ( ↑ x ↑ , { y } ) we have the subspace H which is spanned by all diagrams with a component which has more than one vertex on thesecond strand from the right, or has a loop. For an element a ∈ H ⊂ A ( ↑ x ↑ , { y } ) which ismapped by χ − x to the image of j : A yp → A p , we have k ◦ j − ◦ χ − x ( a ) ∈ H ⊂ A
2) exp( t / Z (cid:32) ( ) (( ) ( ))(( ) ( )) ( ) (cid:33) = 1 by lemma 6.3 Z (cid:32) ( ) ((( ) ) ) ( ) (( ) ( )) (cid:33) = φ ( t − t , t ) ≈≈ φ ( t , t ) mod H (after composition) Z (cid:32) ( ) (( ( )) ) ( ) ((( ) ) ) (cid:33) = φ ( − t , − t ) − ≈ H (after composition) Z (cid:32) ( ) ( ) ( ) (( ( )) ) (cid:33) ≈ H Z (cid:32) ( ( )) ( ) ( ) (cid:33) = 1 by lemma 6.3 Z (cid:32) ( ( )) ( ( )) (cid:33) = exp (cid:32) (cid:33) = (using an I relation)= exp (cid:32) − (cid:33) ≈ H Z (cid:32) (( ) ) ( ( )) (cid:33) = 1 by lemma 6.3 Z (cid:32) ( ( )) (( ) ) (cid:33) = φ ( t , t ) − = φ ( − t − t , t ) − t + t + t commutes with t and t ) Z (cid:32) ( ) ( ) ( ( )) (cid:33) = 1 by lemma 6.3 Z (cid:32) ( ) ( ) ( ) ( ) (cid:33) = exp( − t / A ( ↑ x ↑ , { y } ): χ x ( LM O ( Y − , + )) ≈ exp (cid:32) y x (cid:33) ·· φ ( t , t ) exp (cid:32) − y x (cid:33) φ ( − t − t , t ) − exp( − t / 2) mod H Note that when we travel along strand x we encounter exp (cid:32) y x (cid:33) after the asso-ciator φ ( t , t ), whereas when we travel along the second strand we encounter exp (cid:32) − y x (cid:33) before this associator.By Lemmas 6.4 and 6.5 we get: LM O < ( Y − , + ) = ˜ u ( φ (˜ x , t ) exp( − y ) φ ( − ˜ x − t, t ) − exp( − t/ LM O < ( Y + , + ) = ˜ u (exp( t/ φ ( − ˜ x − t, t ) exp( y ) φ (˜ x , t ) − )which is the expression we wanted to get.In order to complete the proof we only need to show that C − + ≈ H . C − + is thecomposition of with φ ( − t , − t ) φ ( − t , − t + t ) ∈ A ( ↓↑↓↑ ). t is in H (after the composition), therefore we are left with φ ( − t , t ). This can be written as the exponentof a sum of iterated commutators, where the innermost commutator is: − Using several STU and IHX relations we can “transfer” the nodes on the rightmost strand to nodeson the leftmost strand, in the price of adding many more diagrams in which this node is transferredto the lower capped strand. But all those extra diagrams are in H . Therefore we are left withthe commutator: − H . Thus we have completed theproof. Remark 6.2. Looking at the above calculation we see why the H relations were needed. For exam-ple, in LM O < ( X + , + ) ∈ A
Let ∆ : RU ˆ t , → RU ˆ t , be the map defined by : v (cid:55)→ v + v and v (cid:55)→ v ( v = x or v = y ). As explained above, we need to calculate ˜ u − r, ◦ LM O < ( Z ++ , + ) for Z = X, Y ,and show that they are equal to ∆ ◦ ˜ u − r, LM O < ( Z + , + ). We will use the same decomposition of X + , + and Y + , + to simple tangles as we used above. For most of those simple tangles T we canshow that: ˜ u − r, ◦ LM O < (∆ ++++ , + ( T )) = ∆ ◦ ˜ u − r, ( LM O < ( T )) (6.5)In the few cases where this identity does not hold, the extra components we get will eventuallycancel each other.For a tangle T with no cups or caps, we can use the identity LM O < (∆ ++++ , + ( T )) = ∆ ++++ , + ◦ LM O < ( T ). Decompose LM O < ( T ) as LM O < ( T ) u + LM O < ( T ) H , where LM O < ( T ) u is in theimage of u , and LM O < ( T ) H is in H . Clearly, ˜ u − r, ◦ ∆ ++++ , + ( LM O < ( T ) u ) = ∆ ◦ ˜ u − r, ( LM O < ( T ) u ).Therefore, in order to prove identity (6.5) for such T , it is enough to show that ∆ ++++ , + ( LM O < ( T ) H )is in H .In the calculation of the LM O < functor of X + , + and Y + , + we encountered several contributionsto the H part. First, according to lemma 6.4, each edge with a vertex on the x strand, after59pplying k ◦ j − ◦ χ − x , became: ∞ (cid:88) i =0 B i D xyy { itimes (6.6)The H part of this sum is, according to lemma 6.5: (cid:80) ∞ i =1 B i yy { i-1times . For i even wehave: ∆ ++++ , + yy { i-1times = yy { i-1times + yy { i-1times ++ yy { i-1times + yy { i-1times == yy { i-1times + yy { i-1times ∈ H For i odd, the only non-zero coefficient in the sum (6.6) is B = − . When this sum appearsin an associator, the − summand cancels, because it commutes with everything else.In LM O < ( X + , + ) the sum (6.6) appears twice inside an associator, so the H part of thosetangle-parts indeed maps to H . There is also one occurance of this sum which appears insidean exponent. Therefore we are left with exp( − ) which is in H , but is not mapped by∆ ++++ , + to H . However, we will immediately see that this part eventually cancels.In LM O < ( X + , + ) we also have Z (cid:32) (cid:33) which has a cup and a cap. We need tocalculate Z (cid:32) ∆ ++++ , + (cid:32) (cid:33)(cid:33) = Z . This can be written as exp ( t + u ),where u is in H . So we are left with exp ( t ). But this part cancels with exp ( − t ) coming from60pplying ∆ ++++ , + to the exponent of the sum (6.6) (because they commute with everything inbetween). This concludes the proof for LM O < ( X + , + ).In LM O < ( Y + , + ), all the occurences of the sum (6.6) are in associators, so their H part ismapped to H . But there are several more contributions to the H part. First, we had: Z (cid:32) ( ( )) ( ( )) (cid:33) = exp (cid:32) − (cid:33) Applying ∆ ++++ , + to exp (cid:32) − (cid:33) we get exp( − t ) which is not in H . However,we will soon see that this exponent cancels with another exponent.Another contribution to the H part comes from Z (cid:32) ( ) ( ) ( ) (( ( )) ) (cid:33) . So we need tocalculate Z (∆ ++++ , + (cid:32) ( ) ( ) ( ) (( ( )) ) (cid:33) = Z . This is equalto exp( t + u ) with u ∈ H . So we are left with exp( t ) which is not in H . But this cancels outwith the above exp( − t 56) (because they commute with everything in between).We will now deal with the H parts which come from: Z (cid:32) ( ) ((( ) ) ) ( ) (( ) ( )) (cid:33) and Z (cid:32) ( ) (( ( )) ) ( ) ((( ) ) ) (cid:33) , which are: φ (cid:32) − , (cid:33) · φ − (cid:32) , (cid:33) · (cid:32) (cid:33) (For convenience we ignored the 2 left strands.) φ − (cid:32) , (cid:33) is an exponent of a sum of iterated commutatorsin and . The innermost commutator in each of those iteratedcommutators is (cid:34) , (cid:35) , so it is enough to show that applying ∆ ++++ , + to this commutator multiplied by (cid:32) (cid:33) is in H . (Note that we apply ∆ ++++ , + tothe right strands. The left strand will disappear when we apply χ − x .) And indeed:∆ ++++ , + − =61 + (cid:124) (cid:123)(cid:122) (cid:125) ∈ H −− − (cid:124) (cid:123)(cid:122) (cid:125) ∈ H + − ++ − (cid:124) (cid:123)(cid:122) (cid:125) canceling + − (cid:124) (cid:123)(cid:122) (cid:125) canceling ++ − ≈ + = 0We are left with summands in which the empty diagram of: φ − (cid:32) , (cid:33) · (cid:32) (cid:33) is multiplied by the H part of: φ (cid:32) − , (cid:33) This associator is an exponent of a sum of iterated commutators of ,and . If does not appear in this commutator, it does not con-tribute to the H part. So we may consider only commutators in which appears.We may assume the commutator is one sided. It is also enough to consider commutators of the type (cid:34) , u (cid:35) where u is a commutator in and (becauseall the commutators we consider here have this type of commutator as their inner part). So we62eed to consider elements of the following form: u − u where u has no vertices on the down-going strand ↓ . It is easy to prove (by induction) that byapplying ∆ to the strands 2 and 3 of u we get u + u , where u is obtained by putting all thevertices of strand 3 (from the left) in ↑↓↑↑ on strand 4 of ↑↓↓↑↑↑ , and u is obtained by puttingall the vertices of strand 3 in ↑↓↑↑ on strand 5 of ↑↓↓↑↑↑ . So we have:∆ ++++ , + u − u == u +u − u +u + u +u − u +u (cid:124) (cid:123)(cid:122) (cid:125) ∈ H ++ u − u (cid:124) (cid:123)(cid:122) (cid:125) canceling + u − u (cid:124) (cid:123)(cid:122) (cid:125) canceling ++ u − u + u − u u and u can be written as a sum of connected diagrams. The diagrams in this sum which havemore than 1 vertex on strand 4 (for u ) or on strand 5 (for u ) are already in H . For diagramswith only one vertex on those strands, the above sum equals 0 via the STU relation.At last we have to deal with the H part coming from C − + . Most of the summands in C − + are mapped to H by the exact same argument we have just seen. We only need to show that∆ ++++ , + (cid:32) ◦ φ ( − t , t ) (cid:33) is in H . φ ( − t , t ) is an exponent of a linear combination of commutators in t and t . In each sum-mand of the exponent, one of the commutators is closest to the caps at the top. This commutatorcan be written as a one-component diagram with only one vertex on strand 2. So after applying∆ ++++ , + we get two copies of this diagram, each with a vertex on each copy of strand 2. The sameargument from the end of the proof of theorem 6.2, which showed that C − + is in H , shows now63hat each of these copies is in H . This completes the proof for LM O < ( Y + , + ), and hence the proofof theorem 6.1. References [1] D. Bar-Natan, On the Vassiliev knot invariants, Topology (1995), 423-472.[2] D. Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications (1995), 13-32.[3] D. Bar-Natan, Non-associative tangles, Geometric topology (Athens, GA, 1993), 139-183.[4] D. 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