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## The coloured Jones polynomial JN(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. Definition 1.2.4.1. (The coloured Jones polynomial-V. Jones) Let N be a natural number and L a link. Then L ∈ HomTRepf.dim(U) (∅, ∅).

The N′th coloured Jones polynomial is defined from the Reshetikhin-Turaev functor, using the representation VN ∈ Rep(U ) as colour, in the following way: JN(L, q) := FVN (L) ∈ Z[q±] (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 JN(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 ∈ B2n. Then, we will have to study what is happening with F at three levels:

1) the evaluation ∩ ∩ ∩ ∩.

2) braid level β.

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([60],[57]). This relation is established using the notion of highest weight spaces.

### 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 table 1.2.7). These subspaces are rich objects and carry very interesting braid group representations, as we will see. Definition 1.2.5.1. Consider the set of indexes: En,m := {e = (e1, …, en−1) ∈ Nn−1|e1 + … + en−1 = m}.

#### Basis in heighest weight spaces

In [44], there were studied certain bases in the highest weight spaces from the Verma module, as well as connections between highest weight spaces and weight spaces corresponding to different parameters n and m. Jackson and Kerler proved that for the parameter m = 2, the braid group action onto the highest weight space ˆWn,m corresponds to the homological Lawrence-Bigelow- Krammer ([59], [15],[55],[56])representation and conjectured that this identification is true for any natural number m. Later on, Kohno ([41],[57]) proved this conjecture. We will discuss in details this identification in section 1.5. Now, we will present from [41] 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.

**Identifications with q and λ complex numbers**

We are interested in understanding the quantum representations with natural parameter λ = N − 1 ∈ N. 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.

Remark 1.5.2.1. Let R be a ring and M an R-module with a fixed basis B of cardinal d. Consider a group action G y M and a representation of G using the basis B: 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.

**Homological model for the Coloured Jones Polynomial**

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.

**Table of contents :**

**Introduction **

0.1 Quantum invariants – Historical context

0.2 Main Results

0.3 Summary of the PhD contents

0.4 Invariants quantiques – Contexte historique

0.5 R´esultats principaux

0.6 R´esum´e du contenu de la these

**1 A Homological Model for the coloured Jones polynomials **

1.1 Introduction

1.2 Representation theory of Uq(sl(2))

1.2.1 Uq(sl(2)) and its representations

1.2.2 Specialisations

1.2.3 The Reshetikhin-Turaev functor

1.2.4 The coloured Jones polynomial JN(L, q)

1.2.5 Highest weight spaces

1.2.6 Basis in heighest weight spaces

1.2.7 Quantum representations of the braid groups

1.3 Lawrence representation

1.3.1 Local system

1.3.2 Basis of multiforks

1.3.3 Braid group action

1.4 Blanchfield pairing

1.4.1 Dual space

1.4.2 Graded Intersection Pairing

1.4.3 Pairing between Hn,m and H@n ,m

1.4.4 Specialisations

1.4.5 Dualizing the algebraic evaluation

1.5 Identifications between quantum representations and homological representations

1.5.1 KZ-Monodromy representation

1.5.2 Identifications with q and λ complex numbers

1.5.3 Identifications with q indeterminate

1.6 Homological model for the Coloured Jones Polynomial

1.7 Topological model with non-specialised Homology classes

1.7.1 Identifications with q, s indeterminates

1.7.2 Lift of the homology classes FN n and G N n

**2 Modified Turaev-Viro Invariants from quantum sl(2|1) **

2.1 Context

2.2 Categorical Preliminaries

2.2.1 k-categories

2.2.2 Colored ribbon graph invariants

2.2.3 G-graded and generically G-semi-simple categories

2.2.4 Traces on ideals in pivotal categories

2.3 Quantum sl(2|1) at roots of unity

2.3.1 Notation

2.3.2 Superspaces

2.3.3 The superalgebra Uq(sl(2|1))

2.3.4 Representations of Uq(sl(2|1))

2.3.5 The subcategory C of D

2.4 The right trace and its modified dimension

2.4.1 The existence of the right trace

2.4.2 The modified trace

2.4.3 Computations of modified dimensions

2.5 The relative C/Z-spherical category

2.5.1 Purification of C

2.5.2 Generically finitely semi-simple

2.5.3 Trace

2.5.4 T-ambi pair

2.5.5 The b map

2.5.6 Main theorem

**3 A combinatorial description of the centralizer algebras connected to the Links-Gould Invariant **

3.1 The quantum group Uq(sl(2|1))

3.2 The Links-Gould invariant

3.3 The centralizer algebra LGn

3.4 Proof of the Conjecture

3.4.1 Combinatorial description for the intertwiners of V (0, α)⊗n

3.4.2 Computation for the dimension of LGn+1(α)

**4 Further directions **

**Bibliography**