A Homological model for the coloured Jones polynomials

In this paper we will present a homological model for Coloured Jones Polynomials. For each color $N \in \N$, we will describe the invariant $J_N(L,q)$ as a graded intersection pairing of certain homological classes in a covering of the configuration space on the punctured disk. This construction is based on the Lawrence representation and a result due to Kohno that relates quantum representations and homological representations of the braid groups.


Introduction
The theory of quantum invariants of knots started in 1984 with the discovery of the Jones polynomial. Later on, in 1989, Reshetikhin and Turaev constructed a general method that having as input any ribbon category leads to coloured link invariants. This method uses techniques which are purely algebraic and combinatorial. The coloured Jones polynomials J N (L, q) are a family of quantum link invariants, constructed from the representation theory {V N |N ∈ N} of the quantum group U q (sl (2)). The N th coloured Jones polynomial J N (L, q) is defined using a Reshetikhin-Turaev type construction: In this paper, we give a topological interpretation for the coloured Jones polynomials.
The first invariant from this sequence, corresponding to N = 2, is the original Jones polynomial J(q). This invariant has many different flavours, it is a quantum invariant, but it has also a different characterisation, namely it can be described by skein relations. In 1993, R. Lawrence constructed a sequence of representations of the braid group {H n,m } n,m∈N using the homology of a certain covering of the configuration space on the punctured disc. Later on, Bigelow [1] and Lawrence [10] constructed a homological model for the original Jones polynomial J(L, q), describing it as a graded intersection pairing between homology classes in a covering of a configuration space on the punctured disk. They used the Lawrence representation and the skein nature of the invariant for the proof.
We give a homological model for all coloured Jones polynomials. Unlike the original case, they can not be described directly by skein relations. Our strategy is to use their definition as quantum invariants, to study more deeply the Reshetikhin-Turaev functor and to construct step by step a homological counterpart. The main tools in our construction are the Lawrence representation H n,m , a dual Lawrence representation H ∂ n,m constructed using the homology relative to the boundary of the same covering space and a geometric graded intersection form <, > between the Lawrence representation and its dual. The main result is the following: (Homological model for the coloured Jones polynomial J N (L, q)) Let n ∈ N. Then, for any colour N ∈ N there exist two homology classes F N n ∈ H 2n,n(N −1) | α N −1 and G N n ∈ H ∂ 2n,n(N −1) | α N −1 such that for any link L for which there exists β 2n ∈ B 2n with L =β 2n (oriented plat closure), the N th coloured Jones polynomial is given by:

is a certain specialisation of the coefficients.)
Our strategy is to start with a link L and consider a braid β 2n ∈ B 2n such that L =β or 2n ( oriented plat closure). We study the Reshetikhin-Turaev functor at three main levels: the union of cups, the braid level β 2n and the union of caps. The first remark is the fact that even if a priori, the Reshetikhin-Turaev construction applied on the previous diagram passes through the tensor power of the finite dimensional representation V ⊗2n N , actually the whole functor passes through a specific highest weight space W 2n,n(N −1) of V ⊗n N . Secondly, we use the property that that highest weight spaces of tensor power of quantum representations carry homlogical information. More specifically, in 2012, Kohno ( [8] [4]) proved that the highest weight spaces of the Verma module for U q (sl (2)) are isomorphic as braid group representations with certain specialisations of the homological Lawrence representations.
Let N ∈ N be the colour of the coloured Jones polynomial that we want to study. We use the Lawrence representation corresponding to the configuration space of n(N − 1) points in the 2n-punctured disc. We consider an element F N n ∈ H 2n,n(N −1) | α N −1 that corresponds to the coevaluation on the algebraic side (the cups from the diagram). In a dual way, we describe an element in the dual Lawrence representation G N n ∈ H ∂ 2n,n(N −1) | α N −1 that corresponds to the evaluation (union of caps). The braid part from the picture corresponds to the action on the Lawrence representation. Finally the construction of the invariant J N (L, q) through the Reshetikhin-Turaev functor on the quantum side will correspond to the geometric intersection form between the Lawrence representation and its dual, with specialised coefficients. We would like to mention that the homology classes F N n and G N n are intrinsic in the sense that they do not depend on the link. The only part where the link plays a role in this model is in the action of the corresponding braid onto the homological representation. We would like to stress the fact that the specialisation of the coefficients depends on the choice of the colour N .
A further continuation that we are interested in is to study the Floer type homology coming from this model and wether this theory is invariant with respect to the choice of the braid. If it is, this will lead to a geometrical categorification for the coloured Jones polynomials. After that, the aim would be to understand the relation between this categorificaton and the symplectic Khovanov homology studied by , [11]) for the case of the Jones polynomial.
The Lawrence representation H n,m and its dual H ∂ n,m are generated by homology classes of m-dimensional Lagrangian submanifolds in a certain covering a configuration spaceC n,m , called "multiforks" and "barcodes". Both are lifts of Lagrangian submanifolds from C n,m . This means that β 2n F N n and G N n are given by linear combinations of homology classes of Lagrangian submanifolds inC n,m . The further project is to apply graded Floer homology to each graded intersection from before, and to study the invariance of the corresponding Floer homology groups HF N m (β) with respect to the choice of the braid.
Structure of the paper: The paper has five main sections. In Section 2, we present the quantum group U q (sl (2)) that we work with, properties about its representation theory and the definition of the coloured Jones polynomials. Then, Sec-tion3 contains the details about the homological Lawrence representation. Further on, in Section4 we define the dual Lawrence representation and we present a graded geometric intersection form that relates the two representations, with emphasise on the way of computing this form and on the non-degeneracy of this pairing. After that, in Section5, we present the identifications between quantum and homological representations of the braid group and discuss in detail the specialisation at natural parameters. The last part, Section6, is devoted to the construction and the proof of the homological model for the coloured Jones polynomials.
Acknowledgements: I would like to thank very much my advisor, Professor Christian Blanchet for asking this beautiful problem of finding homological interpretations for quantum invariants, at the beginning of my PhD. I am very grateful for many useful and nice discussions and for his continuous support. Also, I am thankful to Dr Martin Palmer for discussions about homology with twisted coefficients and Lawrence representations. This research was supported by grants from Région Ile-de-France. I would also like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme "Homology theories in low dimensional topology" where work on this paper was undertaken between 12 February-28 March 2017. During this time, I was also supported by EPSRC grant no EP/K032208/1.

2.
Representation theory of U q (sl(2)) 2.1. U q (sl(2)) and its representations. Definition 2.1.1. Let q, s parameters and consider the ring Consider the quantum enveloping algebra U q (sl(2)), the algebra over L s generated by the elements E, F (n) , K ±1 with the folowing relations: This is a Hopf algebra with the following comultiplication, counit and antipode: We will use the following notations: Now we will describe the representation theory of U q (sl(2)).
Definition 2.1.2. (The Verma module) In the sequel the abstract variable s will be thought as being the weight of the Verma module. ConsiderV be the L s -module generated by an infinite family of vectors {v 0 ,v 1 , ...}. The following relations define an U q (sl(2)) action onV :

2.2.
Specialisations. For our purpose, to arrive at the definition of the coloured Jones polynomial, it is needed to consider some specialisations of the previous quantum groups and its Verma representations.
Definition 2.2.1. Consider the following spacialisations of the coefficients: 2) Let h, λ ∈ C and q = e h . In this case, we spacialise both variable q and the highest weight s to concrete complex numbers: 3) This is the case where the coloured Jones polynomial will be defined. Consider q still as a parameter (it will be the parameter from the coloured Jones polynomial), and specialise the highest weight using λ = N − 1 ∈ N a natural parameter: Using these specialisations, we will consider the corresponding specialised quantum groups and their representation theory. We obtain the following:

Ring
Quantum Group Representations Specialisation If we specialise as above, U λ and U q,λ become Hopf algebras and V q,λ a U q,λ -representation andV λ a U λ -representation.
Proof. We can see that K acts by scalars and E decreases the indexes on the basis from before. We only have to see what the generators F (n) do on this space.
If n < N − i, from the definition, the action of F (n) will remain inside the module: This shows us that in the previous formula with the action of F (n) , there is a term corresponding to k = N − 1 − i and its coefficient vanishes.
We obtain that F (n)v i = 0, for any n ≥ N − i. This concludes the existence of the N -dimensional submodule V N .
2.3. The Reshetikhin-Turaev functor. In this section, we will present the general construction due to Reshetikhin and Turaev, that having as input any ribbon category C gives a functor from the category of tangles to C . In particular, this machinery leads to link invariants. We will present this, using the category of representations of U . Proposition 2.3.1. There exist an element R ∈ U q (sl(2))⊗U q (sl(2)) called Rmatrix which leads to a braid group representations. For any representation V of U q (sl(2)), we have that the morphism: (2)), the category of U q (sl(2)) representations Rep(U q (sl(2))) becomes a braided category. More precisely, for any V, W ∈ Rep Uq(sl(2)) , the braidingR V,W : V ⊗ W → W ⊗ V is defined using the R-matrix in the following way: 2) The subcategory of finite dimensional U -representations Rep f. dim (U ) becomes a ribbon category. In this case, the braiding comes from the action of the specialisation of the R-matrix R| η λ ⊗η λ ∈ U⊗U . For any representations V, W ∈ Rep U , the braiding R V,W : V ⊗ W → W ⊗ V is defined as: The dualities of this category have the following form: Remark 2.3.3. The action ofR on the standard basis of the Verma moduleV ⊗V is given in [4](Section 4.1): In the previous formula F i,j,n ∈ Z[q ± ] has the expression: For the Verma moduleV N −1 , since it is infinite dimensional, we do not have a well defined coevaluation. However, there is the finite dimensional submodule inside it V N ⊆V N −1 , which has a coevaluation defined on it. In the sequel, we will define a kind of evaluation onV N −1 , which will be supported on V N . Let us make this precise. From the classification of finite dimensional representations of U , it is known that the finite dimensional representations are self-dual.
given by the expression and extended it by linearity. In other words, this is an extension of the previous coevalation: In the following part we will present the Reshetikhin-Turaev method of obtaining link invariants starting from any ribbon category. Firstly, we will see the definition for the category of coloured tangles. Definition 2.3.7. Let C be a category. The category of C -colored framed tangles T C is defined as follows: δ 1 ), ..., (W n , δ n )/isotopy Remark: The tangles T have to respect the colors V i which are at their boundaries. Once we have such a tangle, it has an induced orientation, coming from the signs i , using the following conventions: which contains two objects V and V * and all tangles are coloured just with these two colours. Denote by the restriction of the Reshetikhin-Turaev functor F onto this subcategory.
2.4. The coloured Jones polynomial J N (L, q). So far we have seen the algebraic structure coming from U and the Reshetikhin-Turaev construction. Now, we will present how this machinery is actually a tool that leads to quantum invariants for links. The N th coloured Jones polynomial is defined from the Reshetikhin-Turaev functor, using the representation V N ∈ Rep(U ) as colour, in the following way: (here by applying the functor we get a morphism from Z[q ± ] to Z[q ± ], which is identified with a scalar ) As we have seen so far, the construction that leads to the definition of J N (L, q) is purely algebraic and combinatorial. We are interested in a geometrical interpretation for this invariant. The method that we are thinking of is to study what is happening with the Reshetikhin-Turaev functor at the intermediary levels of the link diagram. More precisely, we will start with L as a plat closure of a braid β ∈ B 2n . Then, we will have to study what is happening with F at three levels: 1) the evaluation ∩ ∩ ∩ ∩ 2) braid level β ⊗ I n 3)the coevaluation ∪ ∪ ∪ ∪ The interesting part and the starting point in our description is the fact that at the level of braid group representation, there is a homological counterpart for the quantum representation, called Lawrence representation( [10], [8]). This relation is established using the notion of highest weight spaces.
2.5. Highest weight spaces. In this part, we will introduce and discuss the properties of some certain vector subspaces which live in the tensor power of a certain representation (we will refer to the ones defined in the tabel 2.7). These subspaces are rich objects and carry very interesting braid group representations, as we will see. The highest weight space of the generic Verma moduleV ⊗n corresponding to the weight m:Ŵ n,m :=V n,m ∩ KerE 2) Specialisation with two complex numbers Let h, λ ∈ C and q = e h . The weight space ofV ⊗n q,λ corresponding to the weight m: V q,λ n,m := {v ∈V ⊗n q,λ |Kv = q nλ−2m v} The highest weight space of the Verma moduleV ⊗n q,λ corresponding to the weight m: W q,λ n,m :=V q,λ n,m ∩ KerE 3)The case with q parameter and λ natural number The highest weight space of the finite dimensional representation V ⊗n N corresponding to the weight m: if and only if e ∈ E N n,m . This will happen also at the level of (highest) weight spaces: and W N n,m ⊆Ŵ N −1 n,m . Remark 2.5.4. 1) Basis for the weight spaces from Verma module One can see easily that: 2) Basis for the weight space of the finite dimensional module V N : From the previous remark and 1), we conclude that: n,m ∩ KerE then we have the following splitting as vector spaces: 2.6. Basis in heighest weight spaces. Now, we will present from [4] some "good" bases for the highest weight spaces, that will have a role in the identification between quantum and homological representations of the braid groups.
In the sequel, the highest weight spacesŴ n,m will be identified with a certain subspace of the weight spacesV n,m .
Let ι : E n,m → E n+1,m the inclusion: Then φ is an isomorphism of L s -modules. The set BŴ n,m = {φ(v s ι(e) )|e ∈ E n,m } will describe a basis for the generic highest weight spaceŴ n,m .

2.7.
Quantum representations of the braid groups. In the following part, we will see that the braid group action on (generic) Verma module and on the finite dimensional module V ⊗n N , passes at the level of highest weight spaces.
Proposition 2.7.1. Since ϕV n commutes with the actions of K, E, it will induce a well defined action on the generic highest weight spaces ϕŴ n,m : B n → Aut(Ŵ n,m ). This action in the basis BŴ n,m will lead to a representation: ϕŴ n,m : B n → GL(d n,m , L s ) This is called the generic quantum representation on highest weight spaces of the Verma module. Proposition 2.7.2. Similarly, using the previous specialisations we have induced braid group actions : well defined action induced by ϕV N −1 n called the quantum representation on highest weight spaces of the Verma module.
called the quantum representation on highest weight spaces of the finite dimensional module.
As a summary we have the following highest weights spaces, which carry braid group actions and live inside the n th tensor power of different spacialisations of the Verma moduleV :

Braid group action
Highest weight space Representation Specialisation ϕŴ n,mŴn,mV

Lawrence representation
3.1. Local system. In this section we will present certain braid group representations introduced by Lawrence ( [10]). These are defined on the middle homology of a certain covering of the configuration space in the punctured disk. They are called homological Lawrence representations and they have a topological description. Let n ∈ N. Consider D 2 ⊆ C the unit disk with its boundary and {p 1 , ..., p n }-n points in its interior, on the real axis.
Let D n := D 2 \ {p 1 , ..., p n } and fix m ∈ N a natural number. Let C n,m be the unordered configuration space of m points in the n-punctured disk: Then H 1 (C n,m ) Z n ⊕ Z, where each i-th element of Z n is generated by a loop around the puncture p i and the last component counts the total winding number of the loop with respect to itself, viewed in the configuration space C m (D 2 ). Consider the function aug : Z n ⊕ Z → Z ⊕ Z < x > < d > by taking the sum on the first components: aug((x 1 , ..., x n ), y) = (x 1 +...+x n , y). By composing the previous maps, define the local system : We denoteC n,m be the covering of C n,m corresponding to Ker(φ) and its associated projection map π :C n,m → C n,m .
Then each deck transformation will induce a cellular chain map forC n,m and moreover this map will pass at the level of homology. So we have an action . Moreover, this action will be defined at the level of the group ring: It follows that the homology groups H lf m (C n,m , Z) and H m (C n,m , Z; ∂) have a structure of Z[x ± , d ± ]-modules ( here H lf means the Borel-Moore homology/ the homology of locally finite chains).

3.2.
Basis of multiforks. So far, we have seen the covering of the configuration space whose homology will lead to the Lawrence representation. For that, we will define certain subspaces in the homology (Borel-Moore/relative to the boundary) ofC n,m .
1) Submanifolds Let e = (e 1 , ..., e n−1 ) ∈ E n,m as in the definition 2.5.1. To each e it will be associate an m-dimensional submanifold inC n,m which will give a class in the Borel Moore homology.
Fix d 1 , ..., d m ∈ ∂D n . For each i ∈ {1, ..., n − 1}, consider e i -disjoint horizontal segments in D n , between p i and p i+1 (which meet just at their boundary). Denote those segments by I e 1 , ..., I e e1 , ..., I e m . Also, for each k ∈ {1, ..., m}, choose a vertical path γ e k between the segment I e k and d k . Think each segment as a map I e i : (0, 1) → D n . Since these segments are disjoint, their product gives a map: Let the projection defined by the quotient of with respect to the Sym m -action: Composing the two previous maps we obtain: 2) Base Points: The paths to the base points d 1 , .., d m will help us to lift the submanifold F e inC n,m . The m-uple (d 1 , ..., d m ) defines a point d ∈ C n,m . Considerd ∈ π −1 (d). The product of the paths towards the boundary γ e k , will define a path in the configuration space. Let ThenF e will define a class in the Borel-Moore Homology is called the multifork corresponding to the element e ∈ E n,m The Lawrence representation will be a subspace of this Borel-Moore homology of the covering, spanned by these multiforks. More precisely we have the following definition: It is known the cardinal of this set: Then B n C n,m and it will induce an action B n π 1 (C n,m ).
Remark 3.3.1. Let π : E → B be a covering map corresponding to a local system φ : Suppose that G is a group that acts on B.
We will consider the fiber of E over a point x ∈ B in the following way: Then this action can be lifted to a G-action on E (constructed at the level of paths using the previous definition) in if and only if From the definition of the local system 3.1.1 it can be shown that ∀β ∈ B n : . Remark 3.3.2. 1) It follows that there is a well defined action of the braid group B n on the covering of the configuration spaceC n,m .
2)We are interested to study the homology of this covering(3.1.2). One can check that the action B n C n,m commutes with the action of the Deck transformations < x, d >.
Moreover, it can be shown that φ is the finest abelian local system such that the induced action of the braid group on the corresponding covering 3.3.1 commutes with the deck transformations given by H.
The subspace H n,m ⊆ H lf m (C n,m , Z) is invariant under the action of B n . Considering the braid group action on H n,m in the multifork basis B Hn,m , it is obtained a representation which is called the Lawrence representation:

Blanchfield pairing
In this section, we will present a non-degenerate duality between the Lawrence representation H n,m and a "dual" space, which we will denote by H ∂ n,m . This dual space lives in the homology of the covering relative to its boundary. Using this form, we will be able to express any element in the dual of H n,m , as certain geometric pairing, using elements from the dual space. This property will play an important role in the homological model from Section 6. 4.1. Dual space. Firstly we will define a subset in the homology of the covering of the configuration space relative to its boundary H m (C n,m , Z; ∂), by specifying a generating set, which we will think as a dual set to the multifork basis. 1) Submanifolds Let e = (e 1 , ..., e n−1 ) ∈ E n,m ( 2.5.1). For each such e, we will define an m-dimensional submanifold inC n,m , which will give a homology class in H m (C n,m , Z; ∂).
For each i ∈ {1, ..., n − 1}, consider e i -disjoint vertical segments in D n , between p i and p i+1 as in the picture above. Denote those segments by J e 1 , ..., J e e1 , ..., J e m . Also, for each k ∈ {1, ..., m}, choose a vertical path δ e k between the segment J e k and d k .
Each of these segments is a map J e i : [0, 1] → D n . Then the product of these segments leads to a map: Projecting onto the configuration space using π n we obtain a submanifold: [D e ] is called the barcode corresponding to the element e ∈ E n,m .
We will call this the "dual" representation of H n,m . Also, consider the set: We do not know yet that B H ∂ n,m is a basis for H ∂ n,m , but we will prove this in the next section, using a pairing between H n,m and H ∂ n,m . 4.2. Graded Intersection Pairing. In this part, we will describe how the Borel-Moore homology and the homology relative to the boundary ofC n,m are related by a pairing. More precisely we are interested to define a Blanchfield type pairing between H n,m and H ∂ n,m . We will follow [2], [3], especially the way of computing the pairing in the case when the homology classes are given by some manifolds. Let us take two homology classes [M ] ∈ H lf m (C n,m , Z) and [Ñ ] ∈ H m (C n,m , Z; ∂) which can be represented by the classes of lifts of two m-dimensional submanifolds M, N ⊆ C n,m . The idea is to fix the second submanifoldÑ in the covering and act with all deck transformations on the first submanifoldM . Each time, we will count the geometric intersection between the two submanifolds with the coefficient given by the element from the deck group. Recall that the local system is defined as φ : π 1 (C n,m ) → Z ⊕ Z and Deck(C n,m ) = Z ⊕ Z. Then the graded intersection between the submanifoldsM andÑ is defined by the formula: where (· ∩ ·) means the geometric intersection number between submanifolds.
Remark 4.2.2. For any ϕ ∈ Deck(C n,m ): This shows that the previous sum has a finite number of non-zero terms and the graded geometric intersection betweenM andÑ is well defined.
In the sequel, we will see that the graded intersection betweenM andÑ which is a priori defined in the coveringC n,m , can be computed in the base, using M and N and the local system for coefficients. More specifically, the pairing will be described as a sum parametrised by all intersection points of M and N in C n,m , where for each point it will be counted a coefficient which is prescribed by the local system.
Let us denoteỹÑ :=Ñ ∩ π −1 ({y}) andỹM :=M ∩ π −1 ({y}) (we use the same property for M as well). Then it follows that: From this, we conclude that if ϕ satisfies the required condition, then From the properties of the Deck transformation, this is a characterisation for an unique ϕ x .
The last two remarks show that the intersection points between all the translations ofM by the deck transformations andÑ are actually parametrised by the intersection points between M and N : We will fix a basepoint d ∈ C n,m andd ∈ π −1 (d). From the last part, we notice that in order to compute the pairing <<M ,Ñ >>, it is enough to consider a sum parametrised by the set M ∩ N and see which is the corresponding coefficient for each intersection point. Let x ∈ M ∩ N and ϕ x ∈ Deck(C n,m ) as in 4.2.3. Denote byx = (ϕM ∩Ñ ) ∩ π −1 (x). Now we will describe ϕ x using just the local system and the point x. We notice that we have the same sign of the intersection in the covering and in the base: Corollary 4.2.5. The pairing betweenM andÑ can be computed using just the submanifolds in the base space C n,m and the local system: This pairing <<, >> can be defined in a similar way for homology classes F ∈ H lf m (C n,m , Z) and G ∈ H m (C n,m , Z; ∂) that can be represented as linear combinations of homology classes of lifts of submanifolds of the type that we described above, with the condition of a finite set of intersection points.  For any e, f ∈ E n,m , the pairing has the following form: where p e ∈ N[d ± ] and p e = 0 with a non-zero free term.

Proof. We remark that working in the configuration space
Secondly, we notice that the previous intersection can be computed using the separate intersections between the submanifolds from F e and D e "supported" between punctures i and i + 1, in the following manner: where F ei := F (0,0,...,ei,...,0) and D ei := D (0,0,...,ei,...,0) . Now we will compute <<F ei ,D ei >>. We notice that each intersection point x ∈ F ei ∩ D ei is characterised by an e i -uple which pairs a horizontal line from the multifork with a vertical line from the barcode. In other words, x is determined by a permutation on the grid σ x ∈ S ei . It follows: The geometric intersection sign c σ counts whether F ei and D ei have a positive or a negative intersection in x = x (1,σ(1)) , ..., x (ei,σ(ei)) . The configuration space on the disc is orientable. Let us consider R = {v 1 , v 2 } the standard base for the tangent space of the disc. Let c = (c 1 , ..., c m ) ∈ C n,m and a tangent vector in this point w. We will define the orientation of w by writing it into the form (w 1 c1 , ..., w m cm , w 2 c1 , ..., w 2 cm ) and see if written in the canonical base R has the same sign or not as the vector (v 1 c1 , ...., v 1 cm , v 2 c1 , ..., v 2 cm ). This is well defined at the level of configuration space, because we are working on a manifold of even dimension, so if we change the order of points by a transposition, we will have to modify the matrix with an even number negative signs.
Following this recipe, we see that c σ is the sign of the tangent vector v σ obtained by taking the tangent vectors at the multiforks followed by the tangent vectors at the barcode: v σ = (v 1 x (1,σ(1)) , ...., v 1 x (e i ,σ(e i )) , v 2 x (1,σ(1)) , ..., v 2 x (e i ,σ(e i )) ) Here, we used that all segments of the multifork are oriented in the same way, and also, that all parts of the barcode have the same orientation. We conclude that c σ = 1. Now we will look at the polynomial part from the graded intersection. Following the previous description of computation, for any k ∈ 1, ..., m let: Then, the loop corresponding to σ has the following form: Firstly we see that l σ does not goes around any of the punctures, so the variable x from the local system will not appear. Secondly, for σ = Id the path l Id is the union of trivial loops and so φ(l Id ) = 1. From the formula 4.3 and the previous remarks, we conclude that: and has a nontrivial free term.
are all non-zero divisors).
2) The specialized Lawrence representation and its dual will have the following definition: and these multiforks will define a basis of H n,m | ψ λ over Z[q ± ], using 4.3.6 and these barcodes will define a basis of H n,m | ψ λ over Z[q ± ], using 4.3.4 Definition 4.4.2. Let us consider a specialised Blanchfield pairing, obtained from the generic pairing <, > by specialising its coefficients using ψ λ .
Remark 4.4.3. We notice that {p e |e ∈ E n,m } ∩ Ker(ψ λ ) = ∅. 1)Here we see that the choice of barcodes on the dual side of H n,m has an important role. They lead to the non-zero polynomials p ∈ N[d ± ] on the diagonal of the matrix M <,> of the geometric intersection pairing <, >. This fact, ensures that these polynomials become non-zero elements in Z[q ± ] through the specialization ψ λ .
2) It would be interesting to compare this situation with the case where we use dual-noodles (noodles with multiplicities) instead of barcodes. In that case, the generic pairing will have as coefficients on the diagonal, polynomials p ∈ Z[x ± , d ± ], which are really in 2 variables and moreover have Z coefficients not only N coefficients. In our geometric model for the coloured Jones polynomial J N (L, q), we will use the specialisation ψ N −1 with natural parameter λ = N − 1 ∈ N. In this case, some of these diagonal polynomials might become zero through the specialisation ψ N −1 because this change of coefficients essentially impose the relation x = −d −λ and λ = N − 1 ∈ N.
3)An interesting question is to understand the pairing in the noodle case and to compute its kernel.

4.5.
Dualizing the algebraic evaluation. This part is motivated by the fact that we are interested to describe the third level of a plat closure of a braid (the union of "caps") viewed through the Reshetikhin-Turaev functor, in a geometrical way using the geometric intersection pairing. We will see the details of this in the folowing section 6, but for this part the aim is to be able to understand an element of the dual of H n,m | ψ λ , as a geometric intersection < ·, G > for some G ∈ H n,m | ψ λ . Remark 4.5.1. The pairing <, > | ψ λ : H n,m | ψ λ ⊗ H ∂ n,m | ψ λ → Z[q ± ] is nondegenerate and has the matrix: ] are polynomials with non-zero free term).
In particular, this shows that the diagonal coefficients of the pairing are not necessary invertible elements in Z[q ± ]. Problem 4.5.2. From this, we see that a priori, not any element of F ∈ (H n,m | ψ λ ) * can be described as a geometric intersection pairing < ·, G F > for some G F ∈ H ∂ n,m | ψ λ . This issue comes from the fact that we are working over a ring and not over a field. In order to overcome this problem, in the sequel we will change the coefficients ring Z[q ± ] by passing to the field of fractions Q(q).
We remember the specialisation ψ λ : Let us consider the embedding i : Z[q ± ] → Q(q) and use Q(q) as field of coefficients.
2)Let the specialised Lawrence representations defined in a similar way as before: and the multiforks define a basis of H n,m | α λ over Q(q), from 4.3.6 |e ∈ E n,m ] > Q(q) and the barcodes define a basis of H n,m | α λ over Q(q), from 4.3.4 We notice that, in fact the specialisations are related in the following manner: Also, let us denote p λ : H n,m | ψ λ → ( · ⊗i1) H n,m | α λ the corresponding change of the coefficients.
Definition 4.5.4. Consider in a similar way as before a specialised Blanchfield pairing, by specialising the pairing <, > using α λ : Remark 4.5.5. For any e ∈ E n,m , α λ (p e ) ∈ Q(q) is a non-zero element, so it is invertible. This shows that <, > | α q,λ is a non-degenerate sesquilinear form.
Moreover, working on a field, we conclude that any element in the dual of the first space, will be described as a pairing with a fixed element from the second space.
Corollary 4.5.6. For any G ∈ (H n,m | α λ ) * , there exist a homology class G ∈ H ∂ n,m | α λ such that: If we consider the pairing with the dual element G of G given by 4.5.6, we obtain G 0 in a geometrical way:

Identifications between quantum representations and homological representations of the braid group
So far, we have presented two important constructions that lead to representations of the braid group: the quantum representation and the Lawrence representation. A priori, they are defined using totally different tools, the quantum representation comes from the algebraic world whereas the Lawrence representation has a homological description. In this section we will discuss about those, using a result due to Kohno that relates these two representations.
Let h, λ ∈ C and q = e h . Let us consider the folowing two specialisations of the coefficients using these complex numbers: 1) for the quantum representationŴ (defined over Z[q ± , s ± ]): Kohno relates these two representations, by connecting each of them with a monodromy representation of the braid group which arises using the theory of KZconnections. We will shortly describe these relations, following [4].
Notation 5.1.1. 1) For λ ∈ C * consider M λ to be the Verma module of sl(2), M λ =< v 0 , v 1 , ... > C with the following actions: 2)Denote by X n = C n \ 1≤i,j≤n Ker(z i = z j ) and Y n := X n /S n . 3) Let n ∈ N and consider the endomorphism Ω i,j ∈ End(M ⊗n λ ) to be the action of Ω onto the i th and j th components.
This describes a connection which is flat with values into the trivial bundle Y n ×M ⊗n λ over Y n .
After that, the monodromy of this connection will lead to a representation: Then, for any λ ∈ C * \ N, the following set describes a basis of N [nλ − 2m]: In this case, we are not anymore in the "generic parameters" case. For that, we will study the relation between the previous braid group representations specialised with any parameters.
We will start with some general remarks about the group actions on modules and how they behave with respect to specialisations. ρ : G → GL(d, R). Suppose that S is another ring and we have a specialisation of the coefficients, given by a ring morphism: ψ : R → S Denote: M ψ := M ⊗ R S and B M ψ := B ⊗ R 1 ∈ M ψ . From this, we will have an induced group action G M ψ .
Then, we have the following properties: 1) B M ψ is a basis for M ψ .
2)Let ρ ψ : G → GL(d, S) the representation of G on M ψ coming from the induced action, in the basis B M ψ . In this way, the two actions, before and after specialisation give the same action in the following sense: We are interested into the case of non-generic complex parameters (h, λ) ∈ C * × C. We would like to to emphasise that quantum representation and Lawrence representation on one side and the KZ-monodromy representation on the other have different natures with respect to the complex parameters (h, λ). Actually, both quantum representation ϕŴ q,λ n,m and Lawrence representation l n,m | ψ q,λ are coming from some generic braid group representations ϕŴ n,m and l n,m and then are specialised using the procedure from the previous remark for the fuctions η q,λ and ψ q,λ . On the contrary, in order to obtain the KZ-monodromy representation η h , one has to fix the complex numbers (λ, h) and do all the construction through this parameters. This is not globalised in a way that does not depend on the specific values, in the sense that we can't construct a representation over some abstract variables such that the KZ-representation at the complex parameters can be obtained from the abstract one by a specialisation, as in the previous remark.
Problem 5.2.3. Since the KZ representation does not comes from a specialisation procedure, we see that we do not have a well defined action in a well defined basis for any complex parameters. From the remark 5.1.6, for λ ∈ N a natural parameter, B N [nλ−2m] is not even a well defined set in N [nλ − 2m]. However, the isomorphism between the quantum and homological representations still works for any parameters, using a continuity argument.
Theorem 5.2.4. Let (h, λ) ∈ C * × C fixed parameters. Then the following braid group representations are isomorphic, using the following corresponding bases: Proof. 1) In the proof of 5.1.7, there are glued two identifications between representations of the braid group B n . Basically, the relation between the quantum representation and the Lawrence representation is established by passing from both of them to the monodromy of the KZ-connection. There are constructed two isomorphisms of braid group representations: More precisely, those isomorphisms are proved using correspondences between the following bases: ϕŴ 2)From this, Kohno proved that for any pair of parameters (h, λ) ∈ U : ϕŴ q,λ n,m (β) = l n,m | ψ q,λ (β), ∀β ∈ B n 3) Let us denote by Θ q,λ : H n,m | ψ q,λ →Ŵ q,λ n,m Θ q,λ ([F e ]) = φ(v s e )| η q,λ , ∀e ∈ E n,m This function is defined for all (h, λ) ∈ C × C. We notice that, having in mind that they are defined directly on the bases, the functions f W N q,λ , f N H q,λ are continuous with respect to the parameters (h, λ) ∈ U . This means that the function Θ q,λ is continuous with respect to the two complex parameters. Now, we will see what is happening with non-generic parameters. 4) We are interested to see what is happening to the specialisation of the quantum representation.
We know that BŴ n,m is a basis forŴ n,m . Making the specialisation η q,λ , means to take a tensor product, which will ensure that BŴ n,m | η q,λ will still describe a basis for the specialised module. We conclude that BŴ q,λ n,m := BŴ n,m | η q,λ is a well defined basis ofŴ q,λ n,m , for any (h, λ) ∈ C * × C. 5) Since the specialisation η q,λ is well defined for any complex parameters (h, λ) ∈ C * ×C, all the coefficients from ϕŴ n,m | η q,λ will become well defined complex numbers. In particular ϕŴ q,λ n,m in the basis BŴ n,m | η q,λ has all the coefficients well defined. 6) Using the previous steps 4) and 5), we conclude that for any braid β ∈ B n , the specialisation of the matrix obtained from the initial action ϕŴ n,m ontoŴ n,m in the basis BŴ n,m , is actually the matrix of the specialised action ϕŴ q,λ n,m in the specialised basis BŴ q,λ n,m : ϕŴ n,m (β)| η q,λ = ϕŴ q,λ n,m (β), The set B Hn,m| ψ q,λ is well defined and describes a basis for H n,m | ψ q,λ for any parameters (h, λ) ∈ C * × C (4.3.6).
8)This shows that for every β ∈ B n , the spacialisations of the matrices from the action on H n,m in the multifork basis, are actually the same as the matrices of the spacialised Lawrence action, in the specialised multifork basis B Hn,m | ψ q,λ : B Hn,m | ψ q,λ Combining the points 2), 3), 6), 8) we obtain that for any parameters (q, λ) ∈ C * × C, we have the identification: ϕŴ q,λ n,m (β) = l n,m | ψ q,λ (β), ∀β ∈ B n This concludes that the quantum representation and the Lawrence representation are isomorphic for any parameters.

5.3.
Identifications with q indeterminate. From the previous discussion, we know that the quantum representationŴ n,m and the Lawrence representation l n,m are isomorphic after appropriate identifications of the coefficients, as long as we fix (q, λ) complex numbers. In the sequel, we will state a similar result, but for the case where we keep q as an indeterminate.
Then we notice that the specialisations are related: Proof. We will basically use the Theorem 5.2.4, and just study a little more its proprieties. Let Θ λ : H n,m | ψ λ →Ŵ n,m | η λ Θ λ (F e ) = φ(v s e )|η λ , ∀e ∈ E n,m 1) We notice that BŴ n,m | η λ is well defined and, as in the proof of Theorem 5.2.4, it will define a basis inŴ λ n,m . 2)Similarly, B Hn,m | ψ λ is a basis of H n,m | ψ λ .
3) Actually we have the relations: If we take β ∈ B n , we notice that for any q ∈ C: ϕŴ q,λ n,m (β) = f q ϕŴ n,m (β)| η λ l n,m (β)| ψ q,λ = f q (l n,m (β)| ψ λ ) (here, the sense is that f q : M (d n,m , Z[q ± ]) → M (d n,m , C), by specialising every entry of the matrix using the function f q ). 5)This shows that ϕŴ n,m (β)| η λ = l n,m (β)| ψ λ , ∀β ∈ B n 6)Up to this point, this is just an equality of matrices. The question now, is wether these are matrices of the actions ϕŴ n,m | η λ and l n,m | ψ λ in some well defined basis. Puting everything together, we conclude that the vector spacesŴ λ n,m and H n,m | ψ λ have the spacialised subsets BŴ n,m | η λ , B Hn,m | ψ λ which are still well defined through the specialisation and moreover they describe two bases using the remarks (1) and 2)). From this, we get that: B Hn,m | ψ λ Combining the previous two conclusions, 5) and 6) we obtain that:  Proof. There is known that there exists an oriented braid β or n ∈ B or n such that L =β or . Then, we pick the first strand of L that continues as a trivial circle outside of the braid. We move the straight part such that it arrives between the first and second strand of β or n . We continue with the trivial part of the second strand, and pull it over the braid, until it arrives between the strands which were initially second and third. We continue the algorithm inductively. We obtain L as a plat closure of a braid, but the orientation of cups and caps can be in any way. If we see a cup oriented to the right, then we add a twist on top of it and transform it to be left-oriented. We do the same for caps. Finally we get the desired β 2n . 6. Homological model for the Coloured Jones Polynomial J N (q) In this section, we present a geometric model for the Coloured Jones polynomials. We will start with a link and consider a braid that leads to the link by plat closure. Firstly we will study the Reshetikhin-Turaev functor on a link diagram that leads to the invariant, by separating it on three main levels. Secondly, we will describe step by step for each of these levels a homological counterpart using the Lawrence representation and its dual. Finally, we will show that the evaluation of the Reshetikhin-Turaev functor on the whole link corresponds to the geometric intersection pairing between the homological counterparts.
Consider the colour N ∈ N. Let the parameter λ = N − 1 and the specialisations as in Section 5: Let n ∈ N. Then, for any colour N ∈ N there exist two homology classes such that for any link L for which there exists β 2n ∈ B 2n with L =β or 2n ( oriented plat closure 5.3.4), the N th coloured Jones polynomial has the formula: Proof. Let L be a link and β 2n ∈ B 2n such that L =β or 2n ( oriented plat closure). Consider the corresponding planar diagram for the link L, using the plat closure of the braid β 2n which has three main levels: 1) the upper closure with n caps ∩ ∩ ∩ ∩ 2) the braid β 2n 3) the lower closure with n cups ∪ ∪ ∪ ∪ Step I Following the definition of the Reshetikhin-Turaev construction, the Coloured Jones polynomial 2.4.1, can by obtained in the following way: As we have seen in section 5, quantum representations of the braid group have homological information. Therefore, for the braid part we are interested to have the action on V ⊗2n N but there is a difference with the orientation of the strands. In the previous picture, corresponding to the braid group action, we have the Reshetikhin-Turaev functor: We will compose at the first and the third level with isomorphisms that transform (V N ) * into V N and back, Then, at the middle level we will have the Reshetikhin-Turaev functor between V 2n N , which is exactly the quantum representation ϕ V N 2 n. Let us make this precise: Lemma 6.0.2. Let V, W finite dimensional representations of U and φ : V → V * isomorphism of representations. Then there is the following commutation relation: Proof.
Denote β or 2n the braid with the topological support β 2n and the orientation inherited from the oriented plat closure. Also letβ or 2n be the same braid where we put all the strands to be oriented upwards. The previous lemma tell us the following: We remark that for the diagram, we need the specific orientation for the braid, but through the Reshetikhin-Turaev functor it is enough to know the source and the target and that encodes the orientation: Using the previos two formulas, the first remark from Step I and 2, we conclude: Step II The important remark is the fact that, from algebraic properties, by following the first morphism ←− Coev ⊗n V N , we naturally arrive a particular highest weight space.
Proof. From the fact that N is an isomorphism U -modules, this will commute with the E and K-actions. Since Z[q ±1 ] is regarded as being the trivial representation, this shows that: N From this, one gets that for any vector v ∈ Im ←− Coev ⊗n V N : After writing q 0 = q 2n(N −1)−2(N−1)n , we conclude that: We know that the action B 2n V ⊗2n N preserves the highest weight spaces, in particular preserves W 2n,n(N −1) . Using this invariance, we notice that actually we can obtain J N (L, q) using the highest weight spaces, by composing the morphisms from the second column (2)).
Problem 6.0.6. The highest weight spaces W 2n,m of the finite dimensional module V ⊗2n N , do not have a geometric counterpart known yet. This is one of the reasons why there are not known geometric interpretations for these invariants. On the other hand, by Kohno's Theorem, we know a geometric flavor of the bigger highest weight spacesŴ 2n,n(N −1) , which live inside the power of the Verma moduleV N −1 .
Step III Having this in mind, we will look at the inclusion W 2n,n(N −1) → ιŴ2n,n(N −1) In this part, we will study the behaviour of this inclusion with respect to the braid group action. More precisely, we will see that if we start from the small highest weight spaces, and see them as subsets of the big ones, then these are left invariant by the braid group action. Firstly, we will prove that this action preserves V N ⊗ V N , insideV N −1 ⊗V N −1 (2.2.3). We will show this, by checking it on the basis {v i ⊗v j |0 ≤ i, j ≤ N −1}. Let 0 ≤ i, j ≤ N − 1. In the formula above, all the second componentsv i−n will remain in V N . For the first components, suppose that we pass over V N , and j + n ≥ N . The idea is that in this situation, the coefficient will vanish.
We notice that q (N −1)−k−j − q −((N −1)−k−j) = 0 if k = N − 1 − j. If j + n ≥ N , it follows that N − 1 − j ≤ n − 1, so the term corresponding to k = N − 1 − j will appear in the previous product, so the coefficient of v j+n ⊗ v i−n vanishes.
Secondly, we will look at the action: B n W n,m ⊆Ŵ n,m . We will show that each generator σ i preserves W n,m . Let w ∈ W n,m . Using that W n,m ⊆ V n,m , it is possible to write w = e∈E N n,m α eve1 ⊗v ei ⊗v ei+1 ⊗ ... ⊗v en We have that σ i w = (Id i−1 ⊗ R ⊗ Id n−i−1 )w. From the first part, σ i w will modify just the components i and i + 1 of w, and always the indexes of those vectors will remain smaller than N . This shows that: Since the action of B n is an action over U q -modules, this commutes with the action of E and K, so it preserves the weights and the kernel of E.
(b) As a conclusion, from (a) and (b), σ i w ∈Ŵ n,m ∩ V ⊗n N and so σ i w ∈ W n,m .
Step IV The strategy will be to use the highest weight spaces from the Verma module in the algebraic description of the coloured Jones polynomial.
We remember that the evaluation −→

EvV
N −1 on the Verma moduleV ⊗2 N −1 restricted to V ⊗2 N gives the evaluation −→ Ev V N (2). From this, we can obtain the invariant starting with the co-evaluation on the highest weight spaces W 2n,n(N −1) from V N , then following the inclusion intoŴ 2n,n(N −1) , the braid group action on those and finally close with the evaluation on the "big" highest weight spaces −→

EvV
N −1 (following the column 3) ). We conclude that the coloured Jones polynomial has the form: Step V Now, we will pas towards homological classes. The advantage of the "big" highest weight spacesŴ 2n,n(N −1) is the fact that they have an homological correspondent, given by the Lawrence representation H 2n,n(N −1) , due to Kohno's relation. We will consider the element corresponding to the image of 1 through the co-evaluation, followed by inclusion, which lives inŴ 2n,n(N −1) . After that, we will reverse it to the geometrical part given by the Lawrence construction, using Khono's function. In other words, we would like to stress the fact that the correspondence between v and F 0 one to the other through the fuction Θ N −1 , is preserved by the action of B 2n . Up to this point, we found the first homology class F 0 , such that encodes homologically the algebraic coevaluation ←− Coev ⊗n V N . Using this, the braid group action on the quantum side and on this class will correspond.
Step VI In the sequel, we are interested to find the second homology class G , which will be a geometric counterpart for the evaluation −→ Ev ⊗n V N defined on W n,m . Even if we are interested to do this, in practice we will find a description for the evaluation −→ Ev ⊗n V N −1 onŴ 2n,n(N −1) , but which encodes basically the evaluation on the highest spaces of the finite dimensional module. We will study −→ Ev ⊗n V N −1 as an element of the dual space (Ŵ 2n,n(N −1) ) * and make the correspondence with a geometric element in element in H ∂ 2n,n(N −1) | α N −1 . We will use the discussion from Section4.5 about different flovours of specialisations of the Blanchfield pairing.