#  $U_{q}(sl(2))$-quantum invariants from an intersection of two Lagrangians
in a symmetric power of a surface

Cristina Anghel $/\\!\\!/$ 13th May 2022

###### Abstract

In this paper we show that coloured Jones and coloured Alexander polynomials
can both be read off from the same picture provided by two Lagrangians in a
symmetric power of a surface. More specifically, the $N^{th}$ coloured Jones
and $N^{th}$ coloured Alexander polynomials are specialisations of a graded
intersection between two explicit Lagrangian submanifolds in a symmetric power
of the punctured disc. The graded intersection is parametrised by the
intersection points between these Lagrangians, graded in a specific manner
using the diagonals of the symmetric power. As a particular case, we see the
original Jones and Alexander polynomials as two specialisations of a graded
intersection between two Lagrangians in a configuration space, whose geometric
supports are Heegaard diagrams.

00footnotetext: Key words and phrases: Quantum invariants, Topological models,
Graded intersections, Symmetric powers

###### Contents

  1. 1 Introduction
  2. 2 Construction of the two submanifolds and the definition of the grading
  3. 3 State sum model
  4. 4 Computation of the intersection pairing from the state sum formula in the symmetric power setting
  5. 5 Intersection model given by two Lagrangians
  6. 6 Unified intersection model for Jones and Alexander polynomials
  7. 7 Examples: Trefoil knot and $8_{19}$ knot

##  1 Introduction

Coloured Jones and coloured Alexander polynomials are two sequences of quantum
invariants coming from the representation theory of the quantum group
$U_{q}(sl(2))$. More specifically, the quantum group with generic $q$ gives
the family of coloured Jones polynomials $\\{J_{N}(L,q)\in\mathbb{Z}[q^{\pm
1}]\\}_{N\in\mathbb{N}}$ associated to a link $L$, recovering the original
Jones polynomial at the first term (corresponding to $N=2$). The same quantum
group at roots of unity leads to the sequence of coloured Alexander
polynomials $\\{\Phi_{N}(L,\lambda)\\}$ (or ADO invariants [1]), which are
non-semisimple invariants. This family recovers the Alexander polynomial at
its first term. These invariants came originally from representation theory
([9],[11]) but there are important open questions predicting that their
asymptotics with respect to the colour contain geometric information. At the
same time, there is an active research area concerning relations between the
categorifications of the Alexander polynomial and Jones polynomial provided by
Khovanov homology and knot Floer homology ([16], [10],[15],[5],[6]). The first
topological model was constructed by Bigelow [7] for the Jones polynomial, as
a graded intersection of submanifolds in configuration spaces, using Lawrence
representations [13].

In recent years, we studied topological models for the sequences of
$U_{q}(sl(2))$-quantum invariants, and we showed in [2] that $J_{N}(L,q)$ and
$\Phi_{N}(L,\lambda)$ are specialisations of a state sum of graded Lagrangian
intersections in a configuration space in the punctured disc. The main purpose
of this paper is to show a globalised result, which uses a single graded
intersection between two submanifolds and does not involve any state sum.

Our main result is that the $N^{th}$ coloured Jones and $N^{th}$ coloured
Alexander polynomials of the closure of a braid with $n$ strands can be read
off from a graded intersection between two Lagrangians in the
$(n-1)(N-1)^{th}$ symmetric power of a punctured disc. The feature of this
formula is that the Lagrangian submanifolds are explicit and their geometric
support in the surface is a multi-pointed Heegaard diagram for a knot. Also,
the colour $N$ of the invariants can be seen directly in this geometric
picture by the number of particles chosen on a fixed geometric support given
by the diagram. For the particular case when $N=2$, this gives a picture which
provides a framework for constructing geometric categorifications for Jones
and Alexander polynomials and a further perspective to relate two such
theories.

###  1.1 Main result

We look at links as being the closure of braids. Let us fix $n\in\mathbb{N}$
the number of the strands of such a braid and $N\in\mathbb{N}$ the colour of
our quantum invariants. We start with the $(3n-1)$-punctured disc and consider
$\Sigma^{n,N}$ to be its $(n-1)(N-1)$-symmetric power. The construction of the
Lagrangians will be done by drawing a collection of $(n-1)$ arcs/ circles in
the punctured disc, considering products of $(N-1)$-symmetric powers on each
of them and taking their quotient to the symmetric power. We call this set of
curves in the punctured disc the “geometric support” of the submanifold.
Moreover, we will use graded intersections, and the grading procedure will be
prescribed by certain curves in the punctured disc which connect the two
submanifolds.

Following this procedure, we define two Lagrangians $\mathscr{S}_{n}^{N}$ and
$\mathscr{T}_{n}^{N}$ in the symmetric power $\Sigma^{n,N}$, given by the
collections of red arcs and green circles respectively:

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{T}_{n}^{N}}$

Figure 1.1: Lagrangian submanifolds

Using that the braid group $B_{3n-1}$ is the mapping class group of the
$(3n-1)-$punctured disc, we construct a well-defined Lagrangian
$(\beta_{n}\cup\mathbb{I}_{2n-1})\ \mathscr{S}_{n}^{N}\subseteq\Sigma^{n,N}$
associated to a braid $\beta_{n}\in B_{n}$.

In definition 1.3 we define an intersection pairing $\langle,\rangle$ taking
values in $\mathbb{Z}[x^{\pm 1},y^{\pm 1},d^{\pm 1}]$, which will be computed
from the intersection points between such submanifolds in the symmetric power,
graded by Laurent polynomials using a collection of loops in $\Sigma^{n,N}$.

For $c\in\mathbb{Z}$, we use a specialisation
$\psi_{c,q,\lambda}:\mathbb{Z}[u^{\pm 1},x^{\pm 1},y^{\pm 1},d^{\pm
1}]\rightarrow\mathbb{Z}[q^{\pm 1},q^{\pm\lambda}]$ given by:

$\psi_{c,q,\lambda}(u)=q^{c\lambda};\ \ \psi_{c,q,\lambda}(x)=q^{2\lambda};\ \
\psi_{c,q,\lambda}(y)=-q^{-2};\ \ \psi_{c,q,\lambda}(d)=q^{-2}.$ (1.1)

Also, we denote $\xi_{N}=e^{\frac{2\pi i}{2N}}$, $w(\beta_{n})$ the writhe of
$\beta_{n}$ and $\mathbb{I}_{2n-1}$ the trivial braid with $2n-1$ strands.

######  Theorem 1.1 (Unified model as an intersection of two Lagrangians)

Let us fix $N\in\mathbb{N}$ the colour of our invariants. Let $L$ be an
oriented link and $\beta_{n}\in B_{n}$ such that $L=\hat{\beta}_{n}$. We
consider the Lagrangian submanifolds
$\mathscr{S}_{n}^{N},\mathscr{T}_{n}^{N}\subseteq\Sigma^{n,N}$ and define
$\Gamma_{N}(\beta_{n})(u,x,y,d)\in\mathbb{Z}[u^{\pm 1},x^{\pm 1},y^{\pm
1},d^{\pm 1}]$ to be the following graded intersection:

$\Gamma_{N}(\beta_{n})(u,x,y,d):=u^{-w(\beta_{n})}u^{-(n-1)}(-y)^{-(n-1)(N-1)}\langle(\beta_{n}\cup\mathbb{I}_{2n-1})\
\mathscr{S}_{n}^{N},\mathscr{T}_{n}^{N}\rangle.$ (1.2)

Then, $\Gamma_{N}$ recovers the $N^{th}$ coloured Jones and $N^{th}$ coloured
Alexander invariants through the following specialisations of coefficients:

$\displaystyle J_{N}(L,q)=\Gamma_{N}(\beta_{n})|_{\psi_{1,q,N-1}}$ (1.3)
$\displaystyle\Phi_{N}(L,\lambda)=\Gamma_{N}(\beta_{n})|_{\psi_{1-N,\xi_{N},\lambda}}.$

###  1.2 Construction of the graded intersection

Let ${\bf d}:=(d^{1}_{1},...,d^{1}_{N-1},...,d^{n-1}_{1},...,d^{n-1}_{N-1})$
be the base point from figure 1.1. We consider $\mathbf{L}$ to be a set of
loops in the symmetric power which are allowed to intersect the diagonal
transversely and they do intersect it at most once on the image of a
semicircle, as in definition 2.2. Then, in relation (2.1) we define a grading
on this set of loops

$\displaystyle\phi:\mathbf{L}\rightarrow\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$
(1.4) $\displaystyle\hskip 36.98857pt\langle x\rangle\ \langle y\rangle\
\langle d\rangle.$

The components of $\phi(l)$ for $l\in\mathbf{L}$ are defined as below:

  * •

$x$-coefficients count the winding of the loop $l$ around the $2n-1$ black
punctures

  * •

$y$-coefficients record the winding of $l$ around the $n$ blue punctures

  * •

$d$-coefficients count the relative winding of the loop $l$ in the symmetric
power, give by the winding around the diagonal.

The intersection pairing is indexed by the set of intersection points:
$I_{\Gamma_{N}(\beta)}=(\beta_{n}\cup\mathbb{I}_{2n-1})\
\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}.$ We define a colouring
$F=(f^{1},...,f^{n-1})$ of such an intersection point $\bar{x}$ as a
collection of functions $f^{k}:\\{1,...,N-1\\}\rightarrow\\{1,...,N-1\\}$ for
all $k\in\\{1,...,n-1\\}$ such that the partitions of the set
$\\{1,...,N-1\\}$ induced by the pre-images of points through each of these
functions have cardinalities prescribed by the multiplicities of the
components of $\bar{x}$ (definition 2.8). Let us denote by
$\mathrm{Col}(\bar{x})$ the set of such colourings. Then each colouring $F$
prescribes a loop $l_{\bar{x},F}$ in the symmetric power, based at $\bf d$, as
in definition 2.12. These loops are defined explicitly and use the dotted
paths which link the red geometric support with the green geometric support.

###### Definition 1.2

(Grading of an intersection point) Let $\bar{x}\in I_{\Gamma_{N}(\beta)}$. We
consider the polynomial associated to this intersection point given by the
gradings of the family of loops associated to $\bar{x}$ and all possible
colourings $F\in\mathrm{Col}(\bar{x})$:

$P(\bar{x}):=\sum_{F\in\mathrm{Col}(\bar{x})}\phi(l_{\bar{x},F})\in\mathbb{Z}[x^{\pm
1},y^{\pm 1},d^{\pm 1}].$ (1.5)

Also, we denote by $\epsilon(\bar{x})$ the product of the signs of the local
intersections between the supports of $(\beta_{n}\cup\mathbb{I}_{2n-1})\
\mathscr{S}_{n}^{N}$ and $\mathscr{T}_{n}^{N}$ at the $(n-1)(N-1)$ components
of $\bar{x}$.

###### Definition 1.3

(Intersection form) Then, the intersection pairing is given by the
intersection points graded by the above polynomials:

$\langle(\beta_{n}\cup\mathbb{I}_{2n-1})\
\mathscr{S}_{n}^{N},\mathscr{T}_{n}^{N}\rangle=\sum_{\begin{subarray}{c}{\bar{x}\in(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}}\end{subarray}}\epsilon(\bar{x})\cdot
P(\bar{x}).$ (1.6)

###### Definition 1.4

(Unified formula as a graded intersection) The graded intersection is given
by:

$\displaystyle\Gamma_{N}(\beta_{n})(u,x,y,d):=u^{-w(\beta_{n})}u^{-(n-1)}(-y)^{-(n-1)(N-1)}\sum_{\begin{subarray}{c}{\bar{x}\in(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}}\end{subarray}}\epsilon(\bar{x})\cdot
P(\bar{x}).$ (1.7)

###  1.3 State sum models and the globalised model

The advantage of this model is that it uses the intersection points between
two fixed Lagrangians, which, as we will see already for $N=2$, brings the
model close to Heegaard diagrams and tools used for categorifications like
Floer homologies. For higher colours, in the previous state sum model from [2]
we would get many terms, and for each of them one has to compute the graded
intersection of two Lagrangians. Here, when we increase the colour we just get
more possible multiplicities for the intersection points, which has the effect
of having more possible colourings. However, the difference between two
colourings of a fixed point is directly seen as a power of the variable $d$,
so it is easy to handle.

###  1.4 Role of the blue punctures

We remark that the grading in this formula counts the winding around the black
punctures, blue punctures and a relative winding. The monodromy around the
blue punctures encodes precisely the quantum trace from the definition of
coloured Jones and Alexander polynomials from the representation theory. Also,
as we will see below, the addition of blue punctures together with the
relative winding permits us to pass from the Alexander polynomial towards the
Jones polynomial.

###  1.5 Jones and Alexander polynomials

For the particular case when $N=2$, we have an intersection model for the
Jones and Alexander invariants, in Section 6. In this case, for each
intersection point we have a unique colouring and our intersection model is
obtained from the intersection points between two Lagrangians in the
configuration space, graded by monomials.

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{2}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{T}_{n}^{2}}$

$\Large{Conf_{n-1}(\mathscr{D}_{3n-1})}$

Figure 1.2: Jones and Alexander polynomials

###### Theorem 1.5

(Jones and Alexander polynomials from the same graded intersection) The Jones
and Alexander polynomials come from the intersection points between the two
Lagrangians in the configuration space of the punctured disc, graded using the
monodromy around punctures and the relative winding in the configuration
space:

$\displaystyle J(L,q)=\Gamma_{2}(\beta_{n})|_{u=q;x=q^{2},y=-q^{-2},d=q^{-2}}$
(1.8)
$\displaystyle\Delta(L,t)=\Gamma_{2}(\beta_{n})|_{u=t^{-\frac{1}{2}};x=t;y=1;d=-1}.$

###  1.6 Differences between the two specialisations for Jones and Alexander
cases

Further on, in Subsection 6.1 we study this model for the case of the
Alexander polynomial. We show that in this case the blue punctures play no
role, and the grading which encodes the relative winding, given by the
intersection with the diagonal of the symmetric power can be removed as well.
In Theorem 6.3 we show that the Alexander polynomial of the closure of
$\beta_{n}$ is an intersection between two Lagrangians in the configuration
space of $(n-1)$ points in the $(2n-1)$-punctured disc. The grading takes care
of just the winding around the punctures of this punctured disc.

Moreover, we remark that the picture that is used for this model, figure 6.2,
is actually a multi-pointed Heegaard diagram, if we consider the genus $(n-1)$
surface obtained by gluing $(n-1)$ handles at the ends of each red curve.
Also, the intersection model of Corollary 6.3 is very close to the one
encountered in knot Floer homology. The advantage of the unified model of
Theorem 1.5 is that it gives the Jones polynomial through the same tools, if
one takes care of an extra grading coming from the intersections with the
diagonal of the symmetric power.

###  1.7 Further directions- Categorifications

For the case $N=2$ we expect that there exists a definition of a Floer type
construction starting from the model
$\Gamma_{2}(\beta_{n})|_{\psi_{-1,i,\lambda}}$ which gives knot Floer
homology. Our Lagrangians are very similar to the pictures encountered in this
theory and the gradings are very closely related. Our next part of the program
is to construct a categorification for the Jones polynomial starting from
$\Gamma_{2}(\beta_{n})|_{\psi_{1,q,-1}}$. There are examples suggesting that
this is related to Khovanov homology and the fact that we have both theories
from the same picture would create a proper framework to define a spectral
sequence from Khovanov homology towards knot Floer homology.

Further on, the form of the Lagrangians gives a natural framework to study
categorifications also for higher colours, for the families of coloured Jones
and coloured Alexander polynomials. This improves on and is different from the
previous results from [2], where we had sums of Lagrangian intersections,
which were not yet globalised in a geometric manner.

Asymptotics This result brings together the $U_{q}(sl(2))$ invariants of
colour $N$ as an intersection between two Lagrangians in a symmetric power of
order $(n-1)(N-1)$. The colour can be seen in the number of particles of this
symmetric power and formula 1.2 gives the possibility to investigate
geometrically the behaviour of this form when we increase the colour. This
provides a starting point for investigating geometric models for asymptotics
given by Melvin-Morton-Rozansky’s loop expansions ([14]) and Gukov-Manolescu
([8]).

### Structure of the paper

In Section 2 we define the Lagrangian submanifolds, and we construct the
intersection pairing in the symmetric power. In the following part, Section 3,
we recall the state sum model from [2]. Then, in Section 4, we show that we
can read the state sum model from our grading in the symmetric power. In
Section 5 we prove the intersection model. Section 6 concerns the intersection
model for the case given by Jones and Alexander polynomials. The last part,
Section 7, shows two concrete examples of computations for the Alexander and
Jones polynomials of the trefoil knot and the knot $8_{19}$.

###  1.8 Acknowledgments

I would like to thank András Juhász for discussions concerning the case $N=2$
and Jacob Rasmussen for asking me this main question of globalising the
classes that appeared in the previous state sum. I acknowledge the support of
SwissMAP, a National Centre of Competence in Research funded by the Swiss
National Science Foundation.

##  2 Construction of the two submanifolds and the definition of the grading

In this part, for $n,N\in\mathbb{N}$ we present the homological set-up and the
construction of the two Lagrangians which will lead to the colour $N$ quantum
invariants.

We start with the two dimensional disc with boundary, with $n_{1}+n_{2}$
punctures:

$\mathscr{D}_{n_{1},n_{2}}=\mathbb{D}\setminus\\{p_{1},...,p_{n_{1}},q_{1},...,q_{n_{2}}\\}.$

Further on, we consider the $m^{\text{th}}$ symmetric power of this punctured
disc:

$\displaystyle\Sigma_{n_{1},n_{2}}^{m}:=\mathscr{D}_{n_{1},n_{2}}^{\times
m}/Sym_{m}$ $\displaystyle\bar{\Sigma}^{m}:=\mathbb{D}^{\times m}/Sym_{m}$

(here $Sym_{m}$ is the $m^{\text{th}}$ symmetric group).

Now, let us fix $d_{1},..d_{m}\in\mathscr{D}_{n_{1},n_{2}}$ and ${\bf
d}=(d_{1},...,d_{m})\in\Sigma_{n_{1},n_{2}}^{m}$ the associated base point in
the symmetric power.

$\hskip 34.1433pt\Delta^{k}_{P}\ \ \ \ \ \ \ \ \ \ \ \ \Delta^{-k}_{P}$

$\hskip 39.83385pt\Delta\hskip 113.81102pt\Delta_{Q}$

Figure 2.1: Grading on the symmetric power of the punctured disc

###### Notation 2.1

(Diagonals) Let us denote the following submanifolds in the symmetric power:

$\displaystyle\Delta:=\\{(x_{1},...,x_{m})\in\Sigma_{n_{1},n_{2}}^{m}\mid\exists\
1\leqslant i,j\leqslant m,x_{i}=x_{j}\\}$
$\displaystyle\Delta^{k}_{P}:=\\{(x_{1},...,x_{m})\in\
\bar{\Sigma}^{m}\mid\exists\ 1\leqslant i\leqslant m,1\leqslant j\leqslant
k,x_{i}=p_{j}\\}$
$\displaystyle\Delta^{-k}_{P}:=\\{(x_{1},...,x_{m})\in\bar{\Sigma}^{m}\mid\exists\
1\leqslant i\leqslant m,n_{1}-k+1\leqslant j\leqslant n_{1},x_{i}=p_{j}\\}$
$\displaystyle\Delta_{Q}:=\\{(x_{1},...,x_{m})\in\bar{\Sigma}^{m}\mid\exists\
1\leqslant i\leqslant m,1\leqslant j\leqslant n_{2},x_{i}=q_{j}\\}$

for $k\in\\{1,...,n_{1}\\}$.

In the following part, we consider a particular set of loops in the symmetric
power, which will be used for the grading of our intersection pairing. For
this, we denote by $S^{1}$ the radius $1$ circle in $\mathbb{R}^{2}$ and by
$S^{1}_{-}$, $S^{1}_{+}$ the semicircles given by points from $S^{1}$ with a
negative or non-negative $x-$coordinate respectively.

######  Definition 2.2 (Isotopy classes of loops)

We will work with the following classes of loops:

$\displaystyle\mathbf{L}:=\\{l:S^{1}\rightarrow\Sigma_{n_{1},n_{2}}^{m}\text{
smooth }\mid l^{-1}(\Delta)\cap S^{1}_{+}\subseteq\\{1\\},\ l(S^{1}_{-})\text{
is transverse to }\Delta\\}$
$\displaystyle\Omega:=\pi_{0}\left(\mathbf{L}\right).$

###  2.1 Local system

######  Definition 2.3 (Grading on the space $L$)

Let $l\in\mathbf{L}$ be a loop and consider
$\bar{l}:S^{1}\rightarrow\bar{\Sigma}^{m}$ the associated loop in the
symmetric power of the disc. Since $\pi_{1}(\bar{\Sigma}^{m})\simeq 1$ there
exists

$\sigma_{l}:(\mathbb{D}^{2},\partial\mathbb{D}^{2})\rightarrow(\bar{\Sigma}^{m},\bar{l})$

a disc which has as boundary the loop $\bar{l}$. Moreover, by a version of the
Whitney Approximation Theorem, one can choose a smooth representative of such
a disc. Furthermore, by the Thom Transversality Theorem, we choose a smooth
representative of this disc which is transverse to the diagonals
$\Delta$,$\Delta^{k}_{P}$,$\Delta^{-k}_{P}$ and $\Delta_{Q}$. We remark that
$\Delta=\cup_{i,j=1}^{m}\Delta_{i,j}$ where

$\Delta_{i,j}:=\\{(x_{1},...,x_{m})\in\Sigma_{n_{1},n_{2}}^{m}\mid
x_{i}=x_{j}\\}.$

Let us fix an orientation for all components $\Delta_{i,j}$, and also for the
components of $\Delta^{k}_{P}$, $\Delta^{-k}_{P}$ and $\Delta_{Q}$. We define
the following grading on the space $\mathbf{L}$:

$\displaystyle\phi:\mathbf{L}\rightarrow\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$
(2.1) $\displaystyle\hskip 36.98857pt\langle x\rangle\ \langle y\rangle\
\langle d\rangle$
$\displaystyle\phi(l):=x^{(\sigma_{l},\Delta_{P}^{k})-\left(\sigma_{l},\Delta_{P}^{-(n_{1}-k)}\right)}\cdot
y^{(\sigma_{l},\Delta_{Q})}\cdot d^{(\sigma_{l}\cup~{}l(S^{1}_{-}),\Delta)}.$

The group $\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$ should be thought of as
the multiplicative group of monomials. Picture 2.1 shows a pictorial
representation of this grading.

Here $(\cdot,\cdot)$ is the geometric intersection number in
$\bar{\Sigma}^{m}$, which is defined as the sum of the intersection numbers
with each of the components. The grading $\phi$ is well defined on the space
$\mathbf{L}$ and it does not depend on the choice of a representative of the
disc. So we have the following induced grading.

######  Definition 2.4 (Grading on the set $\Omega$)

The function $\phi$ on $\mathbf{L}$ induces a well defined grading on its set
of connected components, which we denote by:

$\displaystyle\varphi:\Omega\rightarrow\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$
$\displaystyle\hskip 36.98857pt\langle x\rangle\ \langle y\rangle\ \langle
d\rangle$ $\displaystyle\varphi([l]):=\phi(l).$

###  2.2 Braid group action

In the following part we restrict to the context that we need and define a
graded intersection between submanifolds of the symmetric power, which have
the form that occurs in our situation. More precisely, we fix
$n,N\in\mathbb{N}$ and we use the following parameters:

$\begin{cases}&n_{1}=2n-1\ \ \ n_{2}=n\\\ &k=n\\\ &m=(n-1)(N-1).\end{cases}$
(2.2)

We denote the symmetric power of the (punctured) disc as follows:

$\begin{cases}&\Sigma^{n,N}:=\Sigma_{2n-1,n-1}^{(n-1)(N-1)}\\\
&\bar{\Sigma}^{n,N}:=\bar{\Sigma}^{(n-1)(N-1)}.\end{cases}$ (2.3)

The intersection pairing will be a twisted version of the geometric
intersection in $\Sigma^{n,N}$, graded by the function $\varphi$. In order to
prescribe the grading, we need a base point and we choose it on the
submanifold $\mathscr{S}_{n}^{N}$ and denote it by ${\bf
d}:=(d^{1}_{1},...,d^{1}_{N-1},...,d^{n-1}_{1},...,d^{n-1}_{N-1})$, as in
picture 2.2.

###### Definition 2.5

We denote by
$s\mathscr{S}_{n}^{N},s\mathscr{T}_{n}^{N}\subseteq\mathscr{D}_{2n-1,n}$ the
collections of red curves and green circles that are the geometric supports of
$\mathscr{S}_{n}^{N}$ and $\mathscr{T}_{n}^{N}$ respectively.

###### Definition 2.6

(Paths to the base points) Let us choose a collection of $(n-1)$ paths in the
punctured disc, which we denote by $\eta_{1},...,\eta_{(n-1)}$ which start
from the red curves and end on the left hand side of the circles, as in figure
2.2. Similarly, let $\eta^{\prime}_{1},...,\eta^{\prime}_{(n-1)}$ be the paths
which start from the red curves and end on the right hand side of the circles.

We use that the braid group $B_{n}$ is the mapping class group of the
punctured disc and so for $\beta_{n}\in B_{n}$ we get a diffeomorphism
$h_{\beta_{n}\cup\mathbb{I}_{2n-2}}\in\mathrm{Diff}(\mathscr{D}_{2n-1,n})$.

###### Remark 2.7

Up to isotopy we can choose this diffeomorphism such that it is the identity
outside the dark grey disc around the punctures labeled by $0,...,n-1$ from
picture 2.2. Also, we choose a representative
$h_{\beta_{n}\cup\mathbb{I}_{2n-2}}$ such that there are no trivial discs in

$\mathscr{D}_{2n-1,n}\setminus\left(h_{\beta_{n}\cup\mathbb{I}_{2n-2}}(s\mathscr{S}_{n}^{N})\cup
s\mathscr{T}_{n}^{N}\right)$

in the interior of this dark grey disc, with half of their boundary on
$(h_{\beta_{n}\cup\mathbb{I}_{2n-2}})(s\mathscr{S}_{n}^{N})$ and the other
half on $s\mathscr{T}_{n}^{N}$.

This map will induce a diffeomorphism on the symmetric power of the punctured
disc and we act by this on the submanifold $\mathscr{S}_{n}^{N}$.

###  2.3 Loops associated to intersection points

We will define a graded intersection between submanifolds of the form

$h_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N})\text{ and
}\mathscr{T}_{n}^{N},\text{ for }\beta_{n}\in B_{n}$

for $h_{\beta_{n}\cup\mathbb{I}_{2n-2}}$ that satisfies the above mentioned
conditions. This pairing will be parametrised by the intersection points
between $h_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N})$ and
$\mathscr{T}_{n}^{N}$ and let us denote this set by $I_{h_{\beta_{n}}}$. Since
the diffeomorphism $h_{\beta_{n}\cup\mathbb{I}_{2n-2}}$ was trivial outside
the dark grey disc from the picture we have that:

${\bf d}\in h_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N}).$

Figure 2.2: Braid action

###### Definition 2.8

(Colourings associated to an intersection point) Let
$\bar{x}=(x_{1},...,x_{(n-1)(N-1)})\in I_{h_{\beta_{n}}}$. Let
$k\in\\{1,...,n-1\\}$ and now we look at the $k^{th}$ red curve which ends in
the puncture labeled by $2n-1-k$. There are precisely $N-1$ components of
$\bar{x}$ which belong to this curve (counted with multiplicities) and denote
them and their multiplicities by:

$\displaystyle(x^{k}_{1},...,x^{k}_{l_{k}})$
$\displaystyle(m^{k}_{1},...,m^{k}_{l_{k}}),\text{ where
}m^{k}_{1}+...+m^{k}_{l_{k}}=N-1.$

(we denote them using the opposite orientation of the red segment: we start
from the left hand side of the disc and go towards the right hand side)

In order to define a colouring of the point $\bar{x}$, we need a collection of
functions $f^{k}:\\{1,...,N-1\\}\rightarrow\\{1,...,l_{k}\\}$ for all
$k\in\\{1,...,n-1\\}$ with the property that:

  * •

$\mathrm{card}\left((f^{k})^{-1}\\{i\\}\right)=m^{k}_{i}$.

  * •

$(f^{k})^{-1}\\{l_{k}\\}=\Bigl{\\{}N-m^{k}_{l_{k}},N-(m^{k}_{l_{k}}-1),...,N-2,N-1\Bigr{\\}}$
if $x^{k}_{l_{k}}$ belongs to the right hand side of the disc.

Let us denote:

$\left(f^{k}\right)^{-1}\\{i\\}=\Bigl{\\{}f^{k}[i,1],...,f^{k}[i,m^{k}_{i}]\Bigr{\\}}.$
(2.4)

We consider the family of these functions $F:=(f^{1},...,f^{n-1})$ and call
this a colouring of the multipoint $\bar{x}$.

###### Definition 2.9

For a given intersection point $\bar{x}$, we denote the set of associated
colourings by:

$\mathrm{Col}(\bar{x}):=\\{F=(f^{1},...,f^{n-1})\mid F\text{ colouring with
respect to the multiplicities of the multipoint }\bar{x}\\}.$

In the following part, for a given intersection point $\bar{x}$ and a
colouring $F$ we will construct a loop $l_{\bar{x},F}$ in the symmetric power
of the disc, based in $\bf{d}$. This will be done in two steps:

  1. •

First, we construct a path from $\bf d$ towards $\bar{x}$, which does not
depend on the colouring and uses the green submanifold $\mathscr{T}_{n}^{N}$.

  2. •

Secondly, we continue with a path from $\bar{x}$ back to the basepoint $\bf
d$, using the colouring $F$ and the red submanifold $\mathscr{S}_{n}^{N}$.

Then, we will concatenate these paths and obtain a loop in the symmetric
power. Let us make this precise, as follows.

###### Definition 2.10

(Path from the base point towards an intersection point)
Let $\bar{x}=(x_{1},...,x_{(n-1)(N-1)})\in I_{h_{\beta_{n}}}$. Let
$k\in\\{1,...,n-1\\}$ and now we look at the $k^{th}$ green circle which goes
around the punctures labeled by $(k,2n-1-k)$. There are precisely $N-1$
components of $\bar{x}$ which belong to this circle (counted with
multiplicities) and denote them and their multiplicities by:

$\displaystyle(x^{k}_{1},...,x^{k}_{l_{k}})$
$\displaystyle(\bar{m}^{k}_{1},...,\bar{m}^{k}_{l_{k}}),\text{ where
}\bar{m}^{k}_{1}+...+\bar{m}^{k}_{l_{k}}=N-1.$

by ordering them as they occur on the circle minus the maximal point using the
opposite of its orientation. If $x^{k}_{1}$ is in the left half of the disc,
let $\bar{\nu}^{k}_{1},...,\bar{\nu}^{k}_{\bar{m}_{1}}$ be $\bar{m}^{k}_{1}$
paths in the punctured disc which start in
$d^{k}_{1},...,d^{k}_{\bar{m}^{k}_{1}}$, follow the path $\eta_{k}$ and then
they follow the green curve until they reach the point $x^{k}_{1}$.

Figure 2.3: Paths from the base points

Now we look at the point $x^{k}_{2}$. If this is in the left hand side of the
disc as well, then we construct in a similar manner $\bar{m}^{k}_{2}$ paths
starting from the base points
$d^{k}_{\bar{m}^{k}_{1}+1},...,d^{k}_{\bar{m}^{k}_{1}+\bar{m}^{k}_{2}}$ which
all pass through $x^{k}_{2}$. We continue this procedure up to the point where
we have $\bar{m}^{k}_{l-1}$ paths passing through $x^{k}_{l-1}$.

Looking at the picture we notice that we have maximum one intersection point
in the right hand side of the disc, namely $x^{k}_{l_{k}}$. If so, then the
last set of paths will be constructed in a slightly different manner. Let
$\bar{\nu}^{k}_{N-\bar{m}^{k}_{l_{k}}},...,\bar{\nu}^{k}_{N-1}$ be a family of
paths starting from the last base points from the $k^{th}$ red curve:
$d^{k}_{N-\bar{m}^{k}_{l_{k}}},...,d^{k}_{N-1}$, follow the path
$\eta^{\prime}_{k}$ towards the green circle and then continue on the green
support up to the point $x^{k}_{l_{k}}$.

We do this procedure for all $k\in\\{1,...,n-1\\}$ and we obtain a collection
of $(n-1)(N-1)$ paths in the punctured disc. Let us consider

$\bar{\gamma}_{\bar{x}}:=\\{\bar{\nu}^{1}_{1},...,\bar{\nu}^{1}_{N-1},...,\bar{\nu}^{n-1}_{1},...,\bar{\nu}^{n-1}_{N-1}\\}$
(2.5)

the path in $\Sigma^{n,N}$ given by the collection of these paths (in the
disc), which will start from $\bf d$ and end in $\bar{x}$.

###### Definition 2.11

(Path associated to a coloured intersection point, ending in the base point)
Let $\bar{x}=(x_{1},...,x_{(n-1)(N-1)})\in I_{h_{\beta_{n}}}$ and suppose that
$F$ is a colouring of $\bar{x}$. Let $k\in\\{1,...,n-1\\}$ and this time we
use the $k^{th}$ red curve which ends in the puncture labeled by $2n-1-k$. On
this curve, there are exactly $N-1$ components of $\bar{x}$ (counted with
multiplicities) and we denote them as below:

$\displaystyle(x^{k}_{1},...,x^{k}_{l_{k}})$
$\displaystyle(m^{k}_{1},...,m^{k}_{l_{k}}),\text{ where
}m^{k}_{1}+...+m^{k}_{l_{k}}=N-1.$

The colouring will induce a partition of the base points
$d^{k}_{1},...,d^{k}_{N-1}$ into $l_{k}$ sets, which we denote by:

$\begin{cases}\\{d^{k}_{f^{k}[1,1]},...,d^{k}_{f^{k}[1,m^{k}_{1}]}\\}\\\
\\{d^{k}_{f^{k}[l_{k},1]},...,d^{k}_{f^{k}[l_{k},m^{k}_{l_{k}}]}\\}.\end{cases}$
(2.6)

To simplify the notation, we denote this partition of $N-1$ base points into
$l_{k}$ sets of cardinals $m^{k}_{1},...,m^{k}_{l_{k}}$ as follows:

$\begin{cases}\\{d^{k}_{[1,1]},...,d^{k}_{[1,m^{k}_{1}]}\\}\\\
\\{d^{k}_{[l^{k},1]},...,d^{k}_{[l^{k},m^{k}_{l_{k}}]}\\}.\end{cases}$ (2.7)

Let $\nu^{k}_{1},...,\nu^{k}_{m^{k}_{1}}$ be $m^{k}_{1}$ paths in the
punctured disc which start in $x^{k}_{1}$ and follow the red curve back to the
base points, arriving in $d^{k}_{[1,1]},...,d^{k}_{[1,m^{k}_{1}]}$
respectively.

Figure 2.4: Paths to the base points

Now we look at the point $x^{k}_{2}$. We continue with an analogous
construction, by defining $m^{k}_{2}$ paths starting from $x^{k}_{2}$ and
going to the base points $d^{k}_{[2,1]},...,d^{k}_{[2,m^{k}_{2}]}$. We
continue this procedure up to the point where we have $m^{k}_{l_{k}}$ paths
passing through $x^{k}_{l^{k}}$. We do this procedure for all
$k\in\\{1,...,n-1\\}$ and we obtain a collection of $(n-1)(N-1)$ paths in the
punctured disc. Let us consider

$\gamma_{\bar{x},F}:=\\{\eta^{1}_{1},...,\eta^{1}_{N-1},...,\eta^{n-1}_{1},...,\eta^{n-1}_{N-1}\\}$
(2.8)

the path in $\Sigma^{n,N}$ which is given by the above set of paths, starting
in $\bar{x}$ and ending in $\bf d$.

Now, we put together the previous definitions and we define a loop in the
symmetric power.

###### Definition 2.12

(Loop associated to a coloured intersection point)
Let $\bar{x}=(x_{1},...,x_{(n-1)(N-1)})\in I_{h_{\beta_{n}}}$ be an
intersection point and consider $F$ to be a colouring of $\bar{x}$. We define
the following loop based in $\bf d$, which passes through $\bar{x}$ and is
determined by the colouring $F$:

$l_{\bar{x},F}:=\gamma_{\bar{x},F}\circ\bar{\gamma}_{\bar{x}}\in\mathbf{L}.$
(2.9)

###  2.4 Graded intersection

###### Definition 2.13

We use the set of loops which pass through our intersection points in order to
define a graded intersection $\langle(\beta_{n}\cup\mathbb{I}_{2n-1})\
\mathscr{S}_{n}^{N},\mathscr{T}_{n}^{N}\rangle\in\mathbb{Z}[x^{\pm 1},y^{\pm
1},d^{\pm 1}]$ as follows:

$\langle
h_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N}),\mathscr{T}_{n}^{N}\rangle:=\sum_{\bar{x}\in
I_{h_{\beta_{n}}}}\epsilon_{x_{1}}\cdot...\cdot\epsilon_{x_{(n-1)(N-1)}}\cdot\left(\sum_{F\in\mathrm{Col}(\bar{x})}\varphi(l_{\bar{x},F})\right).$
(2.10)

Here, we denote by $\epsilon_{x_{i}}$ the sign of the local intersection in
the punctured disc at the point $x_{i}$ between the circle and the red curve
that $x_{i}$ belong to.

In the following part we show that the intersection form does not depend on
the choice of representative $h_{\beta_{n}\cup\mathbb{I}_{2n-2}}$.

###### Proposition 2.14

(Intersections given by different representatives)
Let $\beta_{n}\in B_{n}$ a braid and consider two associated diffeomorphisms
$h_{\beta_{n}\cup\mathbb{I}_{2n-2}}$ and
$h^{\prime}_{\beta_{n}\cup\mathbb{I}_{2n-2}}$ which satisfy the assertions
from the previous subsection. Then

$\langle
h_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N}),\mathscr{T}_{n}^{N}\rangle=\langle
h^{\prime}_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N}),\mathscr{T}_{n}^{N}\rangle.$
(2.11)

###### Proof.

The two intersection pairings are parametrised by the set of intersection
points:

$\begin{cases}I_{h_{\beta_{n}}}=h_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N})\cap\mathscr{T}_{n}^{N}\\\
I_{h^{\prime}_{\beta_{n}}}=h^{\prime}_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N})\cap\mathscr{T}_{n}^{N}.\end{cases}$
(2.12)

Now, following the condition that we do not have trivial discs (bigons)
between the red curves and the green ones inside the grey disc neither in the
picture corresponding to $h_{\beta_{n}}$ nor in the one for
$h^{\prime}_{\beta_{n}}$ we can find an isotopy between these two
diffeomorphisms such that at each level there are no bigons in the disc.

It means that the isotopy between these two diffeomorphisms gives a one to one
correspondence between the sets of intersection points:

$f:I_{h_{\beta_{n}}}\rightarrow I_{h^{\prime}_{\beta_{n}}}.$

Now, let us look at an intersection point $\bar{x}\in I_{h_{\beta_{n}}}$ and
its correspondent $f(\bar{x})\in I_{h^{\prime}_{\beta_{n}}}$. We notice that
the multiplicities of the point $\bar{x}$ are the same as the multiplicities
of the point $f(\bar{x})$. We are interested in the gradings which are
associated to these two intersection points.

Let us fix a colouring $F\in\mathrm{Col}(\bar{x})=\mathrm{Col}(f(\bar{x}))$.
We consider the loops associated to the points $\bar{x}$ and $f(\bar{x})$,
coloured with $F$:

$l_{\bar{x},F},l_{f(\bar{x}),F}\in\mathbf{L}.$

Using that $h_{\beta_{n}}$ and $h^{\prime}_{\beta_{n}}$ are isotopic and the
construction of $l_{\bar{x},F}$ and $l_{f(\bar{x}),F}$ from definition 2.8, we
conclude that these loops are also isotopic in $\mathbf{L}$ and so they are in
the same connected component of $L$:

$[l_{\bar{x},F}]=[l_{f(\bar{x},F})]\in\Omega.$

Following the definition of the grading we have that

$\varphi(l_{\bar{x},F})=\varphi\left(l_{f(\bar{x},F)}\right),\forall x\in
I_{h_{\beta_{n}}}.$ (2.13)

Using this relation and the definition of the graded intersection from
equation (2.10) we conclude the equality (2.12) . ∎

The last discussion shows that our intersection form is well defined and it
does not depend on the diffeomorphism that is chosen to represent the braid.
For the next part we want to use the symplectic structure. We take the
symplectic structure on the symmetric power $\Sigma^{N,n}$ which comes from
the restriction of the symplectic structure of the symmetric power of the
complex plane. Then, we represent $s\mathscr{S}_{n}^{N}$ and
$s\mathscr{T}_{n}^{N}$ by collections of smooth curves, which are Lagrangians
in the punctured disc. Thus $\mathscr{S}_{n}^{N}$ and $\mathscr{T}_{n}^{N}$,
which are given by the symmetric powers of these Lagrangians in the punctured
disc, will be Lagrangians in $\Sigma^{N,n}$.

###### Proposition 2.15

For $\beta_{n}\in B_{n}$ there is a representative
$h_{\beta_{n}\cup\mathbb{I}_{2n-2}}$ which is a symplectomorphism of
$\Sigma^{n,N}$ such that the two conditions from remark 2.7 are satisfied.

###### Definition 2.16

(Graded intersection) For $\beta_{n}\in B_{n}$, let
$h_{\beta_{n}\cup\mathbb{I}_{2n-2}}$ be a symplectomorphism of $\Sigma^{n,N}$
as above. Following the above discussion, we have a well defined pairing as
below:

$\displaystyle\langle{(\beta_{n}\cup\mathbb{I}_{2n-2}})\mathscr{S}_{n}^{N},\mathscr{T}_{n}^{N}\rangle:=$
$\displaystyle\langle(h_{\beta_{n}\cup\mathbb{I}_{2n-2}}(\mathscr{S}_{n}^{N}),\mathscr{T}_{n}^{N}\rangle=$
(2.14) $\displaystyle=$ $\displaystyle\sum_{\bar{x}\in
I_{h_{\beta_{n}}}}\epsilon_{x_{1}}\cdot...\cdot\epsilon_{x_{(n-1)(N-1)}}\cdot\left(\sum_{F\in\mathrm{Col}(\bar{x})}\varphi(l_{\bar{x},F})\right).$
(2.15)

##  3 State sum model

In this part, we present the setting from [2] where we constructed a state sum
model recovering the coloured Jones and coloured Alexander polynomials. There
we used graded intersections in configuration spaces rather than symmetric
powers.

We consider the unordered configuration space of $(n-1)(N-1)$ particles in the
punctured disc $\mathcal{D}_{2n-1}$.

###  3.1 Homology classes

For this model we have used the family of Lawrence representations which are
homological representations of the braid group $B_{n}$ on the homology of a
$\mathbb{Z}\oplus\mathbb{Z}$-covering of the configuration space of $m$
particles in the $n$-punctured disc $\mathscr{D}_{n}$ (which we denote by
$C_{n,m}$). More precisely, we have:

  * 1)

Lawrence representations $H_{n,m},H^{\partial}_{n,m}$ which are
$\mathbb{Z}[x^{\pm 1},d^{\pm 1}]$-modules that carry a $B_{n}$-action.

  * 2)

Intersection pairing $\ll,\gg:H_{n,m}\otimes
H^{\partial}_{n,m}\rightarrow\mathbb{Z}[x^{\pm 1},d^{\pm 1}]$ (Poincaré-
Lefschetz type duality).

The precise definition of these homology groups is presented in [2] Section 3.

Let us fix $n,N\in\mathbb{N}$. In the following we will use the space
$C_{2n-1,(n-1)(N-1)}$ and let ${\bf d}=\\{d_{1},...,d_{(n-1)(N-1)}\\}$ a base
point as in picture 3.1. The construction of the homology classes in the
covering of the configuration space will be done by drawing a set of disjoint
curves in the punctured disc, considering products of ordered configuration
spaces on those and then taking their quotient to the unordered configuration
space. Using this, for any multi-index
$\bar{i}=(i_{1},...,i_{n-1}),i_{k}\in\\{0,...,N-1\\},k\in\\{1,...,n-1\\}$ we
define two Lagrangians denoted by:

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F^{\prime}_{\bar{i}}\subseteq
C_{2n-1,(n-1)(N-1)}}\ \ \text{ and }\ \
{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}L^{\prime}_{\bar{i}}\subseteq
C_{2n-1,(n-1)(N-1)}}.$

Figure 3.1: State sum model

$\hskip
42.67912pt{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta^{F_{\bar{i}}}}\hskip
150.79959pt{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\eta^{L_{\bar{i}}}}\hskip
39.83385pt$

In order to lift these submanifolds to the covering of this configuration
space, we use two paths to the base point, which we denote by
$\eta^{F_{\bar{i}}}$ and $\eta^{L_{\bar{i}}}$. Then, we consider

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime}_{\bar{i}}\in
H_{2n-1,(n-1)(N-1)}}\ \ \text{ and }\ \
{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime}_{\bar{i}}\in
H^{\partial}_{2n-1,(n-1)(N-1)}}$

to be the homology classes given by the lifts of these submanifolds, using the
lifts of the paths to the base points. All this construction is presented in
Definition 5.0.2, Definition 5.0.3 and Definition 6.2.1 from [2].

######  Definition 3.1 (Specialisations)

Let $c\in\mathbb{Z}$ and consider the morphism:

$\displaystyle\gamma_{c,q,\lambda}:\mathbb{Z}[u^{\pm 1},x^{\pm 1},d^{\pm
1}]\rightarrow\mathbb{Z}[q^{\pm 1},q^{\pm\lambda}]$ (3.1)
$\displaystyle\gamma_{c,q,\lambda}(u)=q^{c\lambda};\ \
\gamma_{c,q,\lambda}(x)=q^{2\lambda};\ \ \gamma_{c,q,\lambda}(d)=q^{-2}.$

Then, we have the following model from [2], Corollary 8.1.1.

######  Theorem 3.2 (Unified embedded state sum model [2])

Let $L$ be an oriented link and $\beta_{n}\in B_{n}$ such that
$L=\hat{\beta}_{n}$. Let us consider the polynomial in $3$ variables given by
the following state sum:

$\displaystyle\Lambda^{\prime}_{N}(\beta_{n})(u,x,d)$
$\displaystyle:=u^{-w(\beta_{n})}u^{-(n-1)}\sum_{i_{1},...,i_{n-1}=0}^{N-1}d^{-\sum_{k=1}^{n-1}i_{k}}$
(3.2)
$\displaystyle\ll(\beta_{n}\cup{\mathbb{I}}_{n-1}){\mathscr{F}^{\prime}_{\bar{i}}},{\mathscr{L}^{\prime}_{\bar{i}}}\gg\
\in\mathbb{Z}[u^{\pm 1},x^{\pm 1},d^{\pm 1}].$

Then $\Lambda^{\prime}_{N}$ gives the $N^{th}$ coloured Jones and $N^{th}$
coloured Alexander polynomials:

$\displaystyle J_{N}(L,q)=\Lambda_{N}^{\prime}(\beta_{n})|_{\gamma_{1,q,N-1}}$
(3.3)
$\displaystyle\Phi_{N}(L,\lambda)=\Lambda^{\prime}_{N}(\beta_{n})|_{\gamma_{1-N,\xi_{N},\lambda}}.$

##  4 Computation of the intersection pairing from the state sum formula in
the symmetric power setting

The intersection pairing

$\ll,\gg:H_{2n-1,(n-1)(N-1)}\otimes
H^{\partial}_{2n-1,(n-1)(N-1)}\rightarrow\mathbb{Z}[x^{\pm 1},d^{\pm 1}]$
(4.1)

is defined on the homology of the covering, but it can actually be computed
from the geometric intersection in the base configuration space, graded by a
certain local system, as in Proposition 3.3.2 from [2]. In the following part,
we show that we can use the grading of loops from section 2.1 in order to
compute this pairing.

We start by presenting the computation from [2]. We fix the parameters from
section 2.1 as below:

$n_{1}\rightarrow 2n-1,\ \ n_{2}\rightarrow 0,\ \ m\rightarrow(n-1)(N-1).$

We recall that the base configuration space that we work with is
$C_{2n-1,(n-1)(N-1)}$. Now, for each intersection point

$\bar{w}=(w_{1},...,w_{(n-1)(N-1)})\in(\beta_{n}\cup{\mathbb{I}}_{n-1}){F^{\prime}_{\bar{i}}}\cap
L^{\prime}_{\bar{i}}$

we construct a loop, denoted by $l_{\bar{w}}\subseteq C_{n,m}$. We start with
the path $\eta^{L_{\bar{i}}}$ which goes from $\bf d$ and it ends on
$L^{\prime}_{\bar{i}}$. We continue this path, following the geometric support
given by the green circles (excluding the maximal point on each of them) in
the punctured disc and construct a path in the configuration space from
$\eta^{L_{\bar{i}}}(1)$ towards the intersection point $\bar{w}$, such that
the image of this path is included in $L^{\prime}_{\bar{i}}$. Basically, on
each circle we choose a collection of $N-1$ paths in the punctured disc which
go towards the $N-1$ components of $\bar{w}$ that lie on that circle, without
passing through the maximal point of that circle. We denote the new path
obtained by following the two previous paths by
$\eta^{L_{\bar{i}}}_{\bar{w}}$, which goes from $\bf d$ to $\bar{w}$.

In the next step we aim to go back to the base point. We start from $\bar{w}$
and go towards the point $\eta^{F_{\bar{i}}}(1)$ by a path in the
configuration space included in
$(\beta_{n}\cup{\mathbb{I}}_{n-1}){F^{\prime}_{\bar{i}}}$. In order to do
this, we look at each red curve from the support of
$(\beta_{n}\cup{\mathbb{I}}_{n-1}){F_{\bar{i}}}$ in the punctured disc, and
consider a family of $N-1$ paths which start from the $N-1$ components of
$\bar{w}$ which lie on this red curve and go towards $\eta^{F_{\bar{i}}}(1)$.
We continue this path in the configuration space with
$(\eta^{F_{\bar{i}}})^{-1}$ and go back to $\bf d$. We denote the path
obtained by gluing these two paths by $\eta^{F_{\bar{i}}}_{\bar{w}}$, which
goes from $\bar{w}$ back to $\bf d$.

Then, we define the loop which based in $\bf d$ and it is associated to the
intersection point $\bar{w}$:

$l_{\bar{w}}=\left(\eta^{F_{\bar{i}}}_{\bar{w}}\right)^{-1}\circ\eta^{L_{\bar{i}}}_{\bar{w}}.$

This is a loop in the configuration space $C_{2n-1,(n-1)(N-1)}$. Further on,
we use the canonical inclusion and see this loop in the symmetric power
$\Sigma_{2n-1,0}^{(n-1)(N-1)}$. This means that we can use the grading defined
in proposition 2.4 in order to evaluate the loop using the function $\varphi$.
This leads to the intersection form as follows.

######  Proposition 4.1 (Pairing between homology classes computed via the
symmetric power)

The intersection pairing from (4.1) has the following formula (which uses the
loops $l_{\bar{w}}$ and the grading):

$\ll(\beta_{n}\cup{\mathbb{I}}_{n-1}){\mathscr{F}^{\prime}_{\bar{i}}},{\mathscr{L}^{\prime}_{\bar{i}}}\gg~{}=\sum_{\bar{w}\in(\beta_{n}\cup{\mathbb{I}}_{n-1}){F^{\prime}_{\bar{i}}}\cap
L^{\prime}_{\bar{i}}}\epsilon_{y_{1}}\cdot...\cdot\epsilon_{y_{(n-1)(N-1)}}\cdot\varphi(l_{\bar{w}})\in\mathbb{Z}[x^{\pm
1},d^{\pm 1}].$ (4.2)

Here, for the sum from the right hand side of the equation we look at
$F^{\prime}_{\bar{i}}$ and $L^{\prime}_{\bar{i}}$ in the symmetric power
$\Sigma_{2n-1,0}^{(n-1)(N-1)}$ rather than in the configuration space. Also,
$\epsilon_{w_{1}},...,\epsilon_{w_{n}}$ are signs of local intersections in
the punctured disc, which are defined as the ones from definition 2.13.

###### Proof.

Since we are working in the symmetric power of the punctured disc in the
situation where we have no blue punctures ($n_{2}=0$), it means that we have
no $y$-variables in the grading $\varphi$. Moreover we have that

$\varphi(l_{\bar{w}})=\phi^{-(n-1)}(l_{\bar{w}})$ (4.3)

where $\phi^{-(n-1)}$ is the local system from [2] presented in Definition
3.1.2. Then, using the formula from the pairing $\ll,\gg$ from Remark 3.3.3
([2]) we have:

$\ll(\beta_{n}\cup{\mathbb{I}}_{n-1}){\mathscr{F}^{\prime}_{\bar{i}}},{\mathscr{L}^{\prime}_{\bar{i}}}\gg~{}=\sum_{\bar{w}\in(\beta_{n}\cup{\mathbb{I}}_{n-1}){F^{\prime}_{\bar{i}}}\cap
L^{\prime}_{\bar{i}}}\epsilon_{w_{1}}\cdot...\cdot\epsilon_{w_{(n-1)(N-1)}}\cdot\phi^{-(n-1)}(l_{\bar{w}})\in\mathbb{Z}[x^{\pm
1},d^{\pm 1}].$

This definition combined with the property from equation (4.3) concludes the
formula from the statement. ∎

##  5 Intersection model given by two Lagrangians

In this section we prove the intersection model from Theorem 5.1. We remind
the statement below.

######  Theorem 5.1 (Unified model as an intersection of two Lagrangians)

Let $L$ be an oriented link and $\beta_{n}\in B_{n}$ such that
$L=\hat{\beta}_{n}$. We consider the following graded intersection:

$\Gamma_{N}(\beta_{n})(u,x,y,d)=u^{-w(\beta_{n})}u^{-(n-1)}(-y)^{-(n-1)(N-1)}\langle(\beta_{n}\cup\mathbb{I}_{2n-1})\
\mathscr{S}_{n}^{N},\mathscr{T}_{n}^{N}\rangle\in\mathbb{Z}[u^{\pm 1},x^{\pm
1},y^{\pm 1},d^{\pm 1}].$ (5.1)

Then, $\Gamma_{N}$ recovers the $N^{th}$ coloured Jones and $N^{th}$ coloured
Alexander invariants:

$\displaystyle J_{N}(L,q)=\Gamma_{N}(\beta_{n})|_{\psi_{1,q,N-1}}$ (5.2)
$\displaystyle\Phi_{N}(L,\lambda)=\Gamma_{N}(\beta_{n})|_{\psi_{1-N,\xi_{N},\lambda}}.$

We split the proof in four main steps. First, looking at the two types of
Lagrangians that we introduced so far, in figure 1.1 and figure 3.1, we notice
that the support of $\mathscr{F}^{\prime}_{\bar{i}}$ and the one of
$\mathscr{S}_{n}^{N}$ are very different. In the first step, we will consider
a Lagrangian which is given by the geometric support of
$\mathscr{F}^{\prime}_{\bar{i}}$ by pushing the point from the boundary
towards the middle of the disc, giving a homology class which we denote by
$\mathscr{F}_{\bar{i}}$. We show that we have a state sum model using this
family of homology classes. This model is useful on its own for pursuing
computations.

Then, we will investigate the new state sum model and the graded intersection
model that we are interested in and write their formulas, which will be
parametrised by certain intersection points in the symmetric power of the
disc, graded in two different ways. In the last two steps, we show that these
models lead to the same polynomial.

### Step 1-New state sum model

Let us consider the Lagrangian submanifolds given by the geometric supports
from the picture below:

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i}}\subseteq
C_{2n,(n-1)(N-1)}}\ \ \ \ \text{ and }\ \ \ \
{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}L_{\bar{i}}\subseteq
C_{2n,(n-1)(N-1)}}.$

Figure 5.1: State sum model

$\hskip
56.9055pt{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\eta^{F_{\bar{i}}}}\hskip
150.79959pt{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\eta^{L_{\bar{i}}}}\hskip
39.83385pt$

Then, let us consider the homology classes given by the lifts of these
submanifolds in the associated covering space (constructed by the same
procedure as the one from section 3.1) and denote them by:

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i}}\in
H_{2n,(n-1)(N-1)}}\ \ \text{ and }\ \
{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i}}\in
H^{\partial}_{2n,(n-1)(N-1)}}.$

######  Theorem 5.2 (Unified embedded state sum model with new classes)

Let us consider the following state sum for $\beta_{n}\in B_{n}$:

$\displaystyle\Lambda_{N}(\beta_{n})(u,x,d):=u^{-w(\beta_{n})}u^{-(n-1)}\sum_{i_{1},...,i_{n-1}=0}^{N-1}$
$\displaystyle d^{-\sum_{k=1}^{n-1}i_{k}}\cdot$ (5.3)
$\displaystyle\cdot\ll(\beta_{n}\cup{\mathbb{I}}_{n}){\mathscr{F}_{\bar{i}}},{\mathscr{L}_{\bar{i}}}\gg\
\in\mathbb{Z}[u^{\pm 1},x^{\pm 1},d^{\pm 1}].$

Then $\Lambda_{N}$ gives the $N^{th}$ coloured Jones and $N^{th}$ coloured
Alexander polynomials of an oriented link $L$ such that $L=\hat{\beta}_{n}$:

$\displaystyle J_{N}(L,q)=\Lambda_{N}(\beta_{n})|_{\gamma_{1,q,N-1}}$ (5.4)
$\displaystyle\Phi_{N}(L,\lambda)=\Lambda_{N}(\beta_{n})|_{\gamma_{1-N,\xi_{N},\lambda}}.$

###### Proof.

We will show that
$\Lambda_{N}(\beta_{n})(u,x,d)=\Lambda^{\prime}_{N}(\beta_{n})(u,x,d)$. This
will be done in two parts.

  * •

First, we study the action of the braid group on the first family of classes
which are used in the definition of $\Lambda_{N}^{\prime}(\beta_{n})(u,x,d)$
and the ones which appear in the formula for $\Lambda_{N}(\beta_{n})(u,x,d)$.
We will see that these actions are very similar.

  * •

Secondly, we present a criterion stating the requirements for which family of
dual classes one has to use in order to obtain the state sum
$\Lambda_{N}^{\prime}(\beta_{n})(u,x,d)$.

  * •

Then, we prove that the second family of dual homology classes
$\mathscr{L}_{\bar{i}}$ satisfies this property.

###### Definition 5.3

For a set of natural numbers $j_{0},...,j_{2n-2}\in\mathbb{N}$ such that

$j_{0}+...+j_{2n-2}=(n-1)(N-1)$

we denote by $\mathscr{U}_{j_{0},...,j_{2n-2}}$,
$\mathscr{U}^{\prime}_{j_{0},...,j_{2n-2}}$ the homology classs given by the
lifts of the geometric supports from the picture below:

$\mathscr{U}^{\prime}_{j_{0},...,j_{2n-2}}\in H_{2n-1,(n-1)(N-1)}\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathscr{U}_{j_{0},...,j_{2n-2}}\in
H_{2n,(n-1)(N-1)}$

Figure 5.2: Change of the first family of clases

Let us fix $\bar{i}=(i_{1},...,i_{n-1})$. Using the above notation we have:

$\displaystyle\mathscr{F}^{\prime}_{\bar{i}}=\mathscr{U}^{\prime}_{0,i_{1},...,i_{n-1},N-1-i_{n-1},...,N-1-i_{1}}$
$\displaystyle\mathscr{F}_{\bar{i}}=\mathscr{U}_{0,i_{1},...,i_{n-1},N-1-i_{n-1},...,N-1-i_{1}}.$

Looking at the action of braids on these classes and using the structure of
these homology groups discussed in [3] (Subsection 7.5), we conclude that
there are the following decompositions:

$\displaystyle(\beta_{n}\cup{\mathbb{I}}_{n-1})\mathscr{F}^{\prime}_{\bar{i}}=\sum_{\begin{subarray}{c}j_{0},...,j_{n-1}=0\\\
j_{0}+...+j_{n-1}=(n-1)(N-1)\end{subarray}}^{N-1}\alpha^{\prime}(j_{0},...,j_{n-1})\
\mathscr{U}^{\prime}_{j_{0},j_{1},...,j_{n-1},N-1-i_{n-1},...,N-1-i_{1}}$
(5.5)
$\displaystyle(\beta_{n}\cup{\mathbb{I}}_{n})\mathscr{F}_{\bar{i}}=\sum_{\begin{subarray}{c}j_{0},...,j_{n-1}=0\\\
j_{0}+...+j_{n-1}=(n-1)(N-1)\end{subarray}}^{N-1}\alpha(j_{0},...,j_{n-1})\
\mathscr{U}_{j_{0},j_{1},...,j_{n-1},N-1-i_{n-1},...,N-1-i_{1}}$ (5.6)

where
$\alpha^{\prime}(j_{0},...,j_{n-1}),\alpha^{\prime}(j_{0},...,j_{n-1})\in\mathbb{Z}[x^{\pm
1},d^{\pm 1}]$. Now we want to understand what is the relation between these
coefficients and we will actually prove that:

$\alpha^{\prime}(j_{0},...,j_{n-1})=\alpha(j_{0},...,j_{n-1}),\forall\
j_{0},...,j_{n-1}.$ (5.7)

This relation can be seen by looking at the formula for these coefficients,
which can be computed by pairing with a dual basis (as in [4], Section 3 and
Proposition 7.6). This formula uses the red geometric supports, the paths to
the boundary and the local system. Up to isotopy we can assume that the braid
action is non-trivial just on a grey disc which goes around the first
$n$-punctures. Now, the red geometric support inside the grey disc is the the
same in both pictures, the only change is part of the geometric support
outside that disc. However, any path from the first picture corresponds to a
path in the second picture which has the same winding around punctures and the
same relative winding of the particles, and so any such paths (which compute
the coefficients) are evaluated in the same way by the local system.

We pass to the second part and investigate the intersections with the dual
classes. In [2] (Section 6) we computed the following intersection:

$\ll\mathscr{U}_{j_{0},...,j_{2n-2}},{\mathscr{L}^{\prime}_{\bar{i}}}\gg=\begin{cases}1,\text{
if }(j_{0},...,j_{2n-2})=(0,i_{1},...,i_{n-1},N-1-i_{n-1},...,N-1-i_{1})\\\
0,\text{ if }j_{0}\neq 0\text{ or }\exists\ k\in\\{1,...,n-1\\}:j_{k}\neq
N-1-j_{2n-1-k}.\end{cases}$ (5.8)

Moreover, following the argument from Step III and Step IV from Section 6
([2]), we conclude the following criterion.

###### Lemma 5.4

If we have a family of homology classes $\mathscr{W}^{\prime}_{\bar{i}}\in
H^{\partial}_{2n-1,(n-1)(N-1)}$ such that:

$\ll\mathscr{U}^{\prime}_{j_{0},...,j_{2n-2}},{\mathscr{W}^{\prime}_{\bar{i}}}\gg=\begin{cases}1,\text{
if }(j_{0},...,j_{2n-2})=(0,i_{1},...,i_{n-1},N-1-i_{n-1},...,N-1-i_{1})\\\
0,\text{ if }j_{0}\neq 0\text{ or }\exists\ k\in\\{1,...,n-1\\}:j_{k}\neq
N-1-j_{2n-1-k}\end{cases}$ (5.9)

then they we can replace the classes $\mathscr{L}^{\prime}_{\bar{i}}$ by
$\mathscr{W}_{\bar{i}}$ in equation (3.2) without changing the value of the
state sum.

On the other hand, we know that after we act with the braids and get the
classes $(\beta_{n}\cup{\mathbb{I}}_{n-1})\mathscr{F}^{\prime}_{\bar{i}}$ and
$(\beta_{n}\cup{\mathbb{I}}_{n})\mathscr{F}_{\bar{i}}$, they have the same
coefficients in the decomposition with respect to the classes of the form
$\mathscr{U}^{\prime}_{j_{0},...,j_{2n-2}}$ and
$\mathscr{U}_{j_{0},...,j_{2n-2}}$ respectively (as presented in the formulas
(5.5) and (5.7)). This shows the following property.

###### Lemma 5.5

(Family of classes which lead to the same state sum)
Suppose that we have a family of homology classes
$\mathscr{L}^{W}_{\bar{i}}\in H^{\partial}_{2n,(n-1)(N-1)}$ such that:

$\ll\mathscr{U}_{j_{0},...,j_{2n-2}},{\mathscr{L}^{W}_{\bar{i}}}\gg=\begin{cases}1,\text{
if }(j_{0},...,j_{2n-2})=(0,i_{1},...,i_{n-1},N-1-i_{n-1},...,N-1-i_{1})\\\
0,\text{ if }j_{0}\neq 0\text{ or }\exists k\in\\{1,...,n-1\\}:j_{k}\neq
N-1-j_{2n-1-k}.\end{cases}$ (5.10)

We consider the state sum using these dual classes:

$\Lambda^{W}_{N}(\beta_{n})(u,x,d):=u^{-w(\beta_{n})}u^{-(n-1)}\sum_{i_{1},...,i_{n-1}=0}^{N-1}d^{-\sum_{k=1}^{n-1}i_{k}}\ll(\beta_{n}\cup{\mathbb{I}}_{n}){\mathscr{F}_{\bar{i}}},{\mathscr{W}_{\bar{i}}}\gg.$
(5.11)

Then by the procedure of replacing the $\mathscr{F}^{\prime}$-classes by
$\mathscr{F}$-classes and the duals $\mathscr{L}^{\prime}$ by
$\mathscr{L}^{W}$ in $\Lambda^{\prime}_{N}(\beta_{n})$ we keep the same value
of the state sum:

$\Lambda^{\prime}_{N}(\beta_{n})=\Lambda^{W}_{N}(\beta_{n}).$

In the next part we show that the family $\mathscr{L}_{\bar{i}}$ has the above
property. Let us fix $j_{0},...,j_{2n-2}\in\mathbb{N}$ such that
$j_{0}+...+j_{2n-2}=(n-1)(N-1)$. The pairing
$\ll\mathscr{U}_{j_{0},...,j_{2n-2}},{\mathscr{L}_{\bar{i}}}\gg$ is encoded by
the geometric intersection in the configuration space between the
submanifolds: $U_{j_{0},...,j_{2n-2}}$ and $L_{\bar{i}}$, as in the picture
below.

Figure 5.3: Intersection between the geometric supports

We will show that if

$U_{j_{0},...,j_{2n-2}}\cap L_{\bar{i}}\neq\emptyset$

then $j_{0}=0$ and $j_{k}=N-1-j_{2n-1-k},\forall k\in\\{1,...,n-1\\}.$ Let
$\bar{w}=(w_{0},...,w_{(n-1)(N-1)})\in U_{j_{0},...,j_{2n-2}}\cap
L_{\bar{i}}$. There are $N-1$ components of $w$ lying on the green circle
around the punctures labeled by $(n-1,n)$ and also on the red support from the
above picture. This means that: $j_{n-1}+j_{n}\geqslant N-1$. Similarly, the
$k^{th}$ circle intersects just the red curves which end in the punctures
labeled by $(k,2n-1-k)$ and so we have that:

$\begin{cases}j_{1}+j_{2n-2}\geqslant N-1\\\ ...\\\ j_{n-1}+j_{n}\geqslant
N-1.\\\ \end{cases}$ (5.12)

On the other hand, we know that $j_{0}+...+j_{n-1}=(n-1)(N-1)$. Putting these
relations together we conclude that: $j_{0}=0\text{ and
}j_{k}=N-1-j_{2n-1-k},\forall k\in\\{1,...,n-1\\}.$ For the last part of the
argument, it is a similar computation as the one from [2] (Proposition 6.4.3)
to see that

$\ll\mathscr{U}^{\prime}_{j_{0},...,j_{2n-2}},{\mathscr{L}_{\bar{i}}}\gg=1,\text{
if }(j_{0},...,j_{2n-2})=(0,i_{1},...,i_{n-1},N-1-i_{n-1},...,N-1-i_{1}).$

This shows that the family of classes $\\{\mathscr{L}_{\bar{i}}\\}$ satisfies
the requirements (5.11) from Lemma 5.5 and we conclude the equality of the two
state sums:

$\Lambda^{\prime}_{N}(\beta_{n})=\Lambda_{N}(\beta_{n}).$ (5.13)

∎

### Step 2- Formulas for the two intersection pairings

We will prove that the graded intersection $\Gamma_{N}$ leads to the same
polynomial as the state sum $\Lambda_{N}(\beta_{n})(u,x,d)$, namely:

$\Gamma_{N}(\beta_{n})|_{y=-d}=\Lambda_{N}(\beta_{n})(u,x,d),\forall\beta_{n}\in
B_{n}.$ (5.14)

First of all, we choose the action of the braid group to be trivial outside a
grey disc around the first punctures and also we choose two representatives of
the classes $\mathscr{F}_{\bar{i}}$ and $\mathscr{L}_{\bar{i}}$ which have the
geometric supports that intersect inside the grey disc, in the left hand side
of the punctured disc.

To begin with we remind the formulas that give the above intersection
pairings, following relations (2.14), (4.2) and (5.13):

$\displaystyle\langle{(\beta_{n}\cup\mathbb{I}_{2n-1}})\mathscr{S}_{n}^{N},\mathscr{T}_{n}^{N}\rangle=\sum_{\bar{x}\in(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}}\epsilon_{x_{1}}\cdot...\cdot\epsilon_{x_{(n-1)(N-1)}}\cdot\left(\sum_{F\in\mathrm{Col}(\bar{x})}\varphi(l_{\bar{x},F})\right)$
(5.15)
$\displaystyle\ll(\beta_{n}\cup{\mathbb{I}}_{n}){\mathscr{F}_{\bar{i}}},{\mathscr{L}_{\bar{i}}}\gg=\sum_{\bar{w}\in(\beta_{n}\cup{\mathbb{I}}_{n}){F_{\bar{i}}}\cap
L_{\bar{i}}}\epsilon_{w_{1}}\cdot...\cdot\epsilon_{w_{(n-1)(N-1)}}\cdot\varphi(l_{\bar{w}})\in\mathbb{Z}[x^{\pm
1},d^{\pm 1}].$ (5.16)

This means that the graded intersections have the following form:

$\displaystyle\Gamma_{N}(\beta_{n})(u,x,y,d)$ $\displaystyle=\
u^{-w(\beta_{n})}u^{-(n-1)}(-y)^{-(n-1)(N-1)}\cdot$
$\displaystyle\sum_{\bar{x}\in(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}}\epsilon_{x_{1}}\cdot...\cdot\epsilon_{x_{(n-1)(N-1)}}\cdot\left(\sum_{F\in\mathrm{Col}(\bar{x})}\varphi(l_{\bar{x},F})\right)$
$\displaystyle\Lambda_{N}(\beta_{n})(u,x,d)$
$\displaystyle~{}=u^{-w(\beta_{n})}u^{-(n-1)}\cdot$
$\displaystyle\sum_{i_{1},...,i_{n-1}=0}^{N-1}d^{-\sum_{k=1}^{n-1}i_{k}}\cdot\sum_{\bar{w}\in(\beta_{n}\cup{\mathbb{I}}_{n}){F_{\bar{i}}}\cap
L_{\bar{i}}}\epsilon_{w_{1}}\cdot...\cdot\epsilon_{w_{(n-1)(N-1)}}\cdot\varphi(l_{\bar{w}})\in\mathbb{Z}[x^{\pm
1},d^{\pm 1}].$

We will prove that these two expressions become equal after we replace the
$d$-coefficients from the second sum by the variable $-y$ as below:

$\displaystyle(-y)^{-(n-1)(N-1)}\sum_{\bar{x}\in(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}}\epsilon_{x_{1}}\cdot...\cdot\epsilon_{x_{(n-1)(N-1)}}\cdot\left(\sum_{F\in\mathrm{Col}(\bar{x})}\varphi(l_{\bar{x},F})\right)=$
(5.17)
$\displaystyle=\sum_{i_{1},...,i_{n-1}=0}^{N-1}(-y)^{-\sum_{k=1}^{n-1}i_{k}}\cdot\sum_{\bar{w}\in(\beta_{n}\cup{\mathbb{I}}_{n}){F_{\bar{i}}}\cap
L_{\bar{i}}}\epsilon_{w_{1}}\cdot...\cdot\epsilon_{w_{(n-1)(N-1)}}\cdot\varphi(l_{\bar{w}})\in\mathbb{Z}[x^{\pm
1},y^{\pm 1},d^{\pm 1}].$

In the next part we investigate the terms that appear in the two previous
sums. One first remark is that these pairings are given by intersections in
different symmetric powers and also that the indexing sets provided by the
intersection points between Lagrangian submanifolds are different. We
introduce the following notations:

$\mathscr{I}_{\beta_{n}}:=(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}$
$\mathscr{K}_{\beta_{n}}(\bar{i}):=(\beta_{n}\cup{\mathbb{I}}_{n}){F_{\bar{i}}}\cap
L_{\bar{i}}.$

Then, the indexing sets for the two state sums are the following:

$\displaystyle
I_{\Gamma_{N}(\beta_{n})}:=\\{(\bar{x},F)\mid\bar{x}\in\mathscr{I}_{\beta_{n}},F\text{
colouring associated to the multiplicities of }\bar{x}\\}$ $\displaystyle
I_{\Lambda_{N}(\beta_{n})}:=\\{(\bar{i},\bar{w})\mid\bar{i}=(i_{1},...,i_{n-1}),0\leqslant
i_{1},...,i_{n-1}\leqslant N-1,\bar{w}\in\mathscr{K}_{\beta_{n}}(\bar{i})\\}.$

We will split the next part of the proof into two steps, as follows:

  * •

First, we want to create a bijective correspondence between these indexing
sets:

$\chi:I_{\Lambda_{N}(\beta_{n})}\rightarrow I_{\Gamma_{N}(\beta_{n})}.$

  * •

Secondly, once this is constructed, we will investigate the gradings of the
loops associated to two points which correspond one another through the
function $\chi$.

### Step 3- Correspondence between the indexing sets using the intersection
points

Let $(\bar{i},\bar{w})\in I_{\Lambda_{N}(\beta_{n})}$. In particular, we have
that

$\bar{w}=(w_{1},...,w_{(n-1)(N-1)})\in(\beta_{n}\cup{\mathbb{I}}_{n}){F_{\bar{i}}}\cap
L_{\bar{i}}.$

This means that for any $k\in\\{1,...,n-1\\}$, there are $N-1$ components the
point $\bar{w}$ which belong to the $k^{th}$ green circle from the support of
$L_{\bar{i}}$. Now, if we compress the red curves from the support of
${F_{\bar{i}}}$ which end in the same puncture we obtain the picture in the
punctured disc which is on the left side of figure 6.1. On the right hand side
we have the geometric support of the submanifolds
$(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}$ and
$\mathscr{T}_{n}^{N}$.

We remark that those supports are exactly the same except the interior of the
cream vertical band from the picture. The $N-1$ components of $w$ belonging to
the $k^{th}$ circle will transform through the compression process into a
number of points, which we denote by

$y^{k}_{1},...,y^{k}_{l(k)}.$

Looking at the two pictures, we see that the intersection points between the
red and the green supports are in one to one correspondence, since they are
all in the exterior of the horizontal cream band.

Figure 5.4: Intersections between geometric supports

Then, we denote by

$x^{k}_{1},...,x^{k}_{l(k)}$

the points from the right picture associated to $y^{k}_{1},...,y^{k}_{l(k)}$,
which belong to the intersection
$(\beta_{n}\cup{\mathbb{I}}_{2n-1}){\mathscr{S}_{n}^{N}}\cap\mathscr{T}_{n}^{N}$.
In order to define a point in the symmetric power of the disc using these
points in the punctured disc, we have to prescribe their multiplicities. In
order to do this, we go back and remember that the points
$y^{k}_{1},...,y^{k}_{l(k)}$ came from red curves with multiplicities, which
we compressed during the above procedure. This means that for a fixed point
$y^{k}_{i}$ there will be many points from the support of
$(\beta_{n}\cup{\mathbb{I}}_{n}){F_{\bar{i}}}\cap L_{\bar{i}}$ which get
compressed to that point, and we denote this number by $m^{k}_{i}$.

We consider the point in the symmetric power given by the following elements
in the punctured disc (with the associated multiplicities)

$x^{(\bar{i},\bar{w})}:=\left((x^{1}_{1},m^{1}_{1}),...,\left(x^{n-1}_{l(n-1)},m^{n-1}_{l(n-1)}\right)\right).$
(5.18)

In the next part we will define the colouring associated to this multipoint,
in order to complete the definition of the function $\chi$. This time, we will
use the multiplicities of the red curves which give
$(\beta_{n}\cup{\mathbb{I}}_{n})F_{\bar{i}}$. We want to define a colouring as
in definition 2.8: $F:=(f^{1},...,f^{n-1})$ where
$f^{k}:\\{1,...,N-1\\}\rightarrow\\{1,...,l(k)\\}$ for $k\in\\{1,...,n-1\\}$
such that:

$\mathrm{card}\left((f^{k})^{-1}\\{i\\}\right)=m^{k}_{i}.$ (5.19)

Let us fix $k\in\\{1,...,N-1\\}$. We look at the geometric support of the
$k^{th}$ pair of red multicurves from the support of
$(\beta_{n}\cup{\mathbb{I}}_{n})F_{\bar{i}}$ (this is the image through
($\beta_{n}\cup{\mathbb{I}}_{n}$) of the set of red curves which end in the
punctures $k$ and $2n-1-k$ ). We begin from the puncture from left hand side
of the disc where this pair ends. We start to follow this red geometric
support and we encounter elements of $\bar{w}$ precisely in the places where
the smoothing has elements of $\bar{y}$.

This means that for such a point $y^{i}_{j}$ which belongs to the $k^{th}$
smoothened support, correspondingly we have $m_{j}^{i}$ points at the
intersection between the supports of
$(\beta_{n}\cup{\mathbb{I}}_{n})F_{\bar{i}}$ and $L_{\bar{i}}$. If $y^{i}_{j}$
is in the left half of the disc, those $m_{j}^{i}$ points will lie on the
$i_{k}$ red arcs (which all start and end in the same punctures) that give
$y^{i}_{j}$ after the compression procedure. We remark that we have maximum
one point $y^{i}_{j}$ in the right hand side of the disc, and in this
situation the $m_{j}^{i}=N-1-i_{k}$ components will be precisely the points of
intersection between the $N-1-i_{k}$ red arcs which end in the puncture
$2n-1-k$ and the green circle, as in the picture from the top ($1)$ and $2)$)
of figure 6.2.

We denote the set of such components of $\bar{w}$ which correspond to the
points $y^{i}_{j}$ from the $k^{th}$ red support by:

$(c_{1},...c_{N-1})$

(counted by using the ordering the $N-1$ curves from the $k^{th}$ red support
of $F_{\bar{i}}$ from left to right around their common intersection point).

These points can be connected to $N-1$ points from the boundary of the disc
following the paths that give $\eta^{F_{\bar{i}}}$. We think of these boundary
points as being labeled by $\\{1...,N-1\\}$ from left to right.

Now, we remember that we can partition the set of points $(c_{1},...c_{N-1})$
into $l(k)$ subsets, where each subset gets compressed the same point through
the compression procedure. This means that we can colour the points
$(c_{1},...c_{N-1})$ by $l(k)$ colours, and looking at the correspondence to
the points on the boundary, we have an induced colouring of the set
$\\{1,...,N-1\\}$ into $l(k)$ colours. We define
$f^{k}:\\{1,...,N-1\\}\rightarrow\\{1,...,l(k)\\}$ to be this function and
consider the associated colouring:

$F^{(\bar{i},\bar{w})}:=\left(f^{1},...,f^{n-1}\right).$ (5.20)

###### Definition 5.6

(Correspondence between the indexing sets)

We consider the function $\chi:I_{\Lambda_{N}(\beta_{n})}\rightarrow
I_{\Gamma_{N}(\beta_{n})}$ given by:

$\chi\left((\bar{i},\bar{w})\right):=\left(x^{(\bar{i},\bar{w})},F^{(\bar{i},\bar{w})}\right).$
(5.21)

Further on, it is possible to reverse this procedure and starting from a point
$(x,F)\in I_{\Gamma_{N}(\beta_{n})}$ to construct an associated element in
$I_{\Lambda_{N}(\beta_{n})}$ and so we conclude that the function $\chi$ is a
well defined bijection.

### Step 4- Gradings in the two models

In this last step we want to show that the gradings behave well with respect
to the correspondence $\chi$. Following the notations for the indexing sets,
we express relation (5.17) as below:

$\displaystyle\sum_{(\bar{x},f)\in
I_{\Gamma_{N}(\beta_{n})}}\epsilon_{x_{1}}\cdot...\cdot\epsilon_{x_{(n-1)(N-1)}}\cdot\varphi(l_{\bar{x},F})=$
(5.22) $\displaystyle=\sum_{(\bar{i},\bar{w})\in
I_{\Lambda_{N}(\beta_{n})}}\epsilon_{w_{1}}\cdot...\cdot\epsilon_{w_{(n-1)(N-1)}}\cdot(-y)^{(n-1)(N-1)-\sum_{k=1}^{n-1}i_{k}}\cdot\varphi(l_{\bar{w}}).$

We will prove this relation. More specifically we will show that for
$(\bar{i},\bar{w})\in I_{\Lambda_{N}(\beta_{n})}$ we have:

$\epsilon_{w_{1}}\cdot...\cdot\epsilon_{w_{(n-1)(N-1)}}\cdot(-y)^{(n-1)(N-1)-\sum_{k=1}^{n-1}i_{k}}\cdot\varphi(l_{\bar{w}})=\epsilon_{x_{1}}\cdot...\cdot\epsilon_{x_{(n-1)(N-1)}}\cdot\varphi(l_{\bar{x},F})$
(5.23)

where $(\bar{x},F)=\chi\left((\bar{i},\bar{w})\right)$. Now we look at the
associated loops in the symmetric power: $l_{\bar{w}}$ and $l_{\bar{x},F}$. We
remark that each of these has in total

$(N-1-i_{1})+...+(N-1-i_{n-1})=(n-1)(N-1)-(i_{1}+...i_{n-1})$

components in the punctured disc which are on the right hand side of it.
However, $l_{\bar{w}}$ does not have any winding, but the components of
$l_{\bar{x},F}$ wind around the blue punctures. This means that we get a
factor of

$y^{(n-1)(N-1)-(i_{1}+...i_{n-1})}$

when we evaluate the grading $\varphi(l_{\bar{x},F})$. On the other hand,
looking at the signs of the local orientations, we notice that the
$(n-1)(N-1)-(i_{1}+...i_{n-1})$ components from

$(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}$

which belong to the right hand side of the disc have negative orientations,
and so we get an extra $(-1)^{(n-1)(N-1)}$ factor this way.

It remains to show that the contributions from the left hand side of the disc
to the gradings of the two loops are equal. We start with the loop
$l_{\bar{w}}$ and we look at its components in the punctured disc which start
from the base points that are connected to the $k^{th}$ circle. The recipe for
constructing this loop says that we have to start from these base points, go
to the green circle and then continue on this green support towards the
components of the intersection point $\bar{w}$ which belong to this circle,
then go back following the red support, as in the first picture $1)$ from
figure 6.2. We remark that we can slide these base points vertically,
preserving their ordering, and consider them fixed on the set of red curves as
in the picture $1^{\prime})$. This change will preserve the evaluation of the
grading on this loop. From now on we use these base points, from picture
$1^{\prime})$ for the grading of the loops associated to the state sum
coefficients.

Further on, we want to understand the effect of the compression procedure with
respect to the grading of the loop. This procedure does not affect the first
part of the construction where we start from the base points from picture 6.2
($1^{\prime})$ and $2^{\prime})$) and then we follow the dotted curves to get
to the green support. What changes is the construction of the second part of
the loop, regarding the path that comes back from the intersection points
towards the base points, following the red curves. Now, for
$t\in\\{1,...,n-1\\}$ we imagine that we compress the set of $i_{t}$ and
$N-1-i_{t}$ red curves from the support of
$(\beta_{n}\cup{\mathbb{I}}_{n-1}){F_{\bar{i}}}\cap L_{\bar{i}}$ towards the
associated red curve from figure 6.2 $2^{\prime})$. The base points will be
sent to $d^{t}_{1},...,d^{t}_{N-1}$ and the order will be preserved through
this procedure.

This shows that for any $k,l\in\\{1,..,n-1\\}$ with $k\neq l$, the
contribution of relative grading (given by the intersection with the diagonal
$\Delta$) coming from parts of the loops $l_{\bar{w}}$ and $l_{\bar{x},F}$
which go back towards base points belonging to the $k^{th}$ and $l^{th}$ red
curves is the same in the two pictures.

$\Lambda_{N}(\beta_{n})\hskip 170.71652pt\Gamma_{N}(\beta_{n})$

$(\beta_{n}\cup{\mathbb{I}}_{n}){F_{\bar{i}}}\cap L_{\bar{i}}\hskip
113.81102pt(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}$

Figure 5.5: Gradings in the two models

Now we fix any $t\in\\{1,...,n-1\\}$. The last subtlety to be discussed
concerns the contribution to the relative grading coming from points which are
chosen on the set of $i_{t}$ curves from figure $1^{\prime})$ and the
associated points with multiplicities from the compressed picture
$2^{\prime})$. More precisely, let us consider two different points
$x^{t}_{p}$ and $x^{t}_{q}$ which are on the same red curve from the support
$(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}$ from figure
$2^{\prime})$. Then they come from the compression of $m^{t}_{p}$ and
$m^{t}_{q}$ points respectively, which belong to the set of red curves with
multiplicities $i_{t}$ from the support
$(\beta_{n}\cup{\mathbb{I}}_{n}){F_{\bar{i}}}\cap L_{\bar{i}}$. We notice that
all the $m^{t}_{p}$ points belong to the same circle and also the $m^{t}_{q}$
points are all on a different circle. Then following the construction of the
loop $l_{\bar{w}}$, we have to go from these points to the base points from
figure $1^{\prime})$, which now lie on the red curves. The $m^{t}_{p}$ points
are sent to a subset $S_{p}$ of cardinality $m^{t}_{p}$ of the set of $i_{t}$
base points which are on these red curves. Using the constructions of the
colouring $F$ and the loop $l_{(\bar{x},F)}$ we see that in the compressed
picture the multipoint $(x^{t}_{p},m^{t}_{p})$ is sent to the $m^{t}_{p}$
components of $\\{d^{t}_{1},...,d^{t}_{N-1}\\}$ obtained from $S_{p}$ by
compression. Similarly, the point $x^{t}_{q}$ with multiplicity $m^{t}_{q}$
will be connected through $l_{(\bar{x},F)}$ to a set of base points from
$\\{d^{t}_{1},...,d^{t}_{N-1}\\}$ obtained from $S_{q}$ by compression. We
remark that different choices of fixing $m^{t}_{p}$ points from the $i_{t}$
points which get compressed to $x^{t}_{p}$ and $m^{t}_{q}$ points from the
$i_{q}$ points which get compressed to $x^{t}_{q}$ induce different colourings
on the right picture.

With this we see that the loops $l_{(\bar{x},F)}$ and $l_{\bar{w}}$ induce the
same permutation of the set of ordered base points. The difference between
these loops comes from the fact that $l_{\bar{w}}$ does not intersect the
diagonal, but it winds around it and it has a contribution to the relative
winding coming from the choice of the above $m^{t}_{p}$ and $m^{t}_{q}$ points
on the $i_{t}$ red curves. On the other hand the loop $l_{(\bar{x},F)}$
encodes directly this relative winding induced by the multiplicity $m^{t}_{p}$
and $m^{t}_{q}$ points by the fact that it intersects the diagonal thanks to
the colouring. We obtain that overall the two loops have equal contributions
to the relative winding.

In other words, the grading on the left hand side changes according to the
relative winding number induced by such choices of $m^{t}_{p}$ and $m^{t}_{q}$
points, which on the right hand side will be encoded precisely by the
intersection of the loop $l_{(\bar{x},F)}$ with the diagonal, which will be
created by the colouring.

We conclude that the contribution of the gradings coming from the parts of the
loops $l_{(\bar{x},F)}$ and $l_{(\bar{w})}$ from the left hand side of the
disc is the same, and so relation (5.23) holds.

This concludes the proof of the intersection model in the symmetric power.

##  6 Unified intersection model for Jones and Alexander polynomials

This part concerns the particular case given by $N=2$, which shows a unified
model for the original Jones and Alexande polynomials. In this situation, the
ambient manifold is the $(n-1)$-symmetric power of the $(3n-1)$-punctured
disc. On the other hand, we remark that the submanifolds $\mathscr{S}_{n}^{2}$
and $\mathscr{T}_{n}^{2}$ are given directly by the products of the red curves
and green circles respectively, as in the picture below. This means that these
submanifolds do not intersect the diagonal of the symmetric power and so they
belong to the configuration space $Conf_{n-1}(\mathscr{D}_{3n-1})$.

$\ \ \ \ \ \mathscr{S}_{n}^{2}\ \ \ \text{ and }\ \ \ \mathscr{T}_{n}^{2}$

$\Large{Conf_{n-1}(\mathscr{D}_{3n-1})}$

Figure 6.1: Jones and Alexander polynomials

Also, on each red curve there is just one base point, so for each intersection
point $\bar{x}$ there is a unique associated colouring $F_{\bar{x}}$. We
conclude that there are no choices for colourings in this case. This means
that for each intersection point
$\bar{x}\in(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{N}\cap\mathscr{T}_{n}^{N}$
we have a unique loop in the symmetric power $l_{\bar{x},F_{\bar{x}}}$, which
is constructed following the recipe from definition 2.12, and we denote it by
$l_{\bar{x}}$.

We notice that these loops $l_{\bar{x}}$ are included in the configuration
space, since they do not intersect the diagonal $\Delta$, and so we can
compute the intersection pairing directly in the configuration space
$Conf_{n-1}(\mathscr{D}_{3n-1})$.

###### Theorem 6.1

(Jones and Alexander polynomials from two Lagrangians in a configuration
space)
We have the intersection pairing, given by the graded intersection points
between the two Lagrangians in the configuration space of $(n-1)-$particles in
the punctured disc:

$\Gamma_{2}(\beta_{n})(u,x,y,d):=u^{-w(\beta_{n})}u^{-(n-1)}(-y)^{-(n-1)(N-1)}\sum_{\bar{x}\in(\beta_{n}\cup\mathbb{I}_{2n-1})\mathscr{S}_{n}^{2}\cap\mathscr{T}_{n}^{2}}\epsilon(\bar{x})\cdot\phi(l_{\bar{x}}).$
(6.1)

This specialises to the Jones and Alexander polynomials of the closure of the
braid, as below:

$\displaystyle J(L,q)=\Gamma_{2}(\beta_{n})|_{u=q;x=q^{2},d=q^{-2}}$ (6.2)
$\displaystyle\Delta(L,x)=\Gamma_{2}(\beta_{n})|_{u=x^{-\frac{1}{2}};y=1;d=-1}.$

###  6.1 Alexander polynomial from the configuration space

We end this section with some remarks concerning the difference between the
Jones polynomial and Alexander polynomials seen through this model. Following
the above relation, we see that the variable $y$ gets specialised to $1$ for
the Alexander polynomial. This means that in this case the blue punctures play
no role, so we can remove them and then we have a pairing in the punctured
disc $\mathscr{D}_{2n-1}$. Also, the relative grading counted by $d$ gets
evaluated to $-1$. Following [2], we can count the relative grading by $-d$,
if we use the intersection of the submanifolds in the configuration space
rather than the product of local intersections in the punctured disc, using
the discussion from Remark 3.3.3 and the going back to the model from
Proposition 3.3.2 from [2].

###### Definition 6.2

For $n\in\mathbb{N}$ we consider the morphism given by:

$\displaystyle\varphi:\pi_{1}(Conf_{n-1}(\mathscr{D}_{2n-1}))\rightarrow\mathbb{Z}$
(6.3) $\displaystyle\hskip 120.92421pt\langle x\rangle$
$\displaystyle\varphi(\sigma)=x^{W(\sigma,\\{0,...,n-1\\})-W(\sigma,\\{n,...,2n-2\\})}$

where $W(\sigma,\\{p_{1},...,p_{i}\\})$ is the winding number of the loop
$\sigma$ around the punctures $p_{1},...,p_{i}$.

This comes from the local system $\phi$ defined in subsection 2.1, but here
$\varphi$ counts just the intersection with the diagonals $\Delta^{n}_{P}$ and
$\Delta^{-(n-1)}_{P}$ of the symmetric power (which are related to the
punctures of the puntured disc), and does not record the relative winding
given by the intersections with the diagonal $\Delta$, as in picture 6.2.

$(\beta_{n}\cup\mathbb{I}_{n-1})\mathscr{S}_{n}^{2}\cap\mathscr{T}_{n}^{2}$

$\Large{Conf_{n-1}(\mathscr{D}_{2n-1})}$ $\varphi$

Figure 6.2: Alexander polynomial from the intersection pairing Local system

In the next formula, for
$\bar{z}\in(\beta_{n}\cup\mathbb{I}_{n-1})\mathscr{S}_{n}^{2}\cap\mathscr{T}_{n}^{2}$
we denote by $\alpha_{\bar{z}}$ the sign of the local intersection between the
submanifolds $(\beta_{n}\cup\mathbb{I}_{n-1})\mathscr{S}_{n}^{2}$ and
$\mathscr{T}_{n}^{2}$ of the configuration space at the point $\bar{z}$.
Putting all these remarks together, we conclude following model for the
Alexander polynomial.

###### Corollary 6.3

(Alexander polynomial from a Lagrangian intersection)
The Alexander polynomial of a closure of a braid $\beta_{n}\in B_{n}$ is given
by the intersection between the following two Lagrangians in the configuration
space of $(n-1)-$particles in the punctured disc:

$\Delta(L,x)=x^{-\frac{w(\beta_{n})+(n-1)}{2}}\sum_{\bar{z}\in(\beta_{n}\cup\mathbb{I}_{n-1})\mathscr{S}_{n}^{2}\cap\mathscr{T}_{n}^{2}}\alpha_{\bar{z}}\cdot\varphi(l_{\bar{z}}).$
(6.4)

##  7 Examples: Trefoil knot and $8_{19}$ knot

###  7.1 Trefoil knot

Let $T$ be the trefoil knot, which is the closure of $\sigma^{3}$ for
$\sigma\in B_{2}$. We fix $N=2$, so we will work directly in the punctured
disc $\mathscr{D}_{5}$. Then, we look at the two submanifolds and compute the
gradings as in the picture below:

$(\sigma^{3}\cup\mathbb{I}_{3})\mathscr{S}_{2}^{2}\cap\mathscr{T}_{2}^{2}$

Figure 7.1: Trefoil knot

Then, the intersection pairing has the following expression:

$\displaystyle\Gamma_{2}(\sigma^{3})(u,x,y,d):=u^{-4}(-y)^{-1}\left(-x^{-3}+x^{-2}-x^{-1}+1-y\right).$
(7.1)

Following relation (6.2), for the Jones polynomial we have to compute the
specialisation given by $u=q,\ x=q^{2},\ y=-q^{-2}\ d=q^{-2}$ and we get the
normalised version of this invariant:

$\displaystyle
J(T,q)=\Gamma_{2}(\sigma^{3})(u,x,y,d)|_{\psi_{1,q,1}}=-q^{-8}+q^{-2}+q^{-6}.$
(7.2)

For the Alexander polynomial, we have the specialisation
$u={\xi_{2}}^{-\lambda},\ x={\xi_{2}}^{2\lambda},y=-{\xi_{2}}^{-2}=1\
d={\xi_{2}}^{-2}=-1$ and we obtain:

$\displaystyle\Phi_{2}(T,\lambda)=\Gamma_{2}(\sigma^{3})(u,x,d)|_{\psi_{-1,\xi_{2}=i,\lambda}}={\xi_{2}}^{2\lambda}-1+{\xi_{2}}^{-2\lambda}.$
(7.3)

Or, if we write it in terms of $x=\xi^{2\lambda}$ we get the usual form of the
Alexander polynomial:

$\displaystyle\Delta(T,x)=x-1+x^{-1}.$ (7.4)

###  7.2 Knot $8_{19}$

In this part we will compute the intersection pairing for the knot $8_{19}$,
seen as the closure of the braid

$\beta:=\sigma^{3}_{1}\sigma_{2}\sigma^{3}_{1}\sigma_{2}\in B_{3}.$

In this case, we have to work in the configuration space of $(2-1)(3-1)=2$
points in the punctured disc with $8$ punctures. We have drawn below the
geometric supports of

$(\beta\cup\mathbb{I}_{5})\mathscr{S}_{3}^{2}\ \ \ \ \ \ \ \ \text{ and }\ \ \
\ \ \ \ \ \mathscr{T}_{3}^{2}.$

$\Large{Conf_{2}(\mathscr{D}_{8})}$

Figure 7.2: Knot $8_{19}$

We have the following set of generators which belong to the intersection
between the submanifolds $(\beta\cup\mathbb{I}_{5})\mathscr{S}_{3}^{2}$ and
$\mathscr{T}_{3}^{2}$ in the configuration space of two points in the
punctured disc, which carry the associated gradings from the tabel below.

$(x_{1},z_{1})$ | $(x_{1},z^{{}^{\prime}}_{1})$ | $(x_{1},z^{"}_{1})$ | $(x_{1},z_{4})$ | $(x_{1},z_{5})$
---|---|---|---|---
$y^{2}$ | $-y$ | $yx^{-1}$ | $-yx^{-3}$ | $yx^{-4}$
$(x_{4},z_{1})$ | $(x_{4},z^{{}^{\prime}}_{1})$ | $(x_{4},z^{"}_{1})$ | $(x_{4},z_{4})$ | $(x_{4},z_{5})$
$-yx^{-1}$ | $x^{-1}$ | $-x^{-2}$ | $d^{-2}x^{-4}$ | $-d^{-2}x^{-5}$
$(x_{5},z_{1})$ | $(x_{5},z^{{}^{\prime}}_{1})$ | $(x_{5},z^{"}_{1})$ | $(x_{5},z_{4})$ | $(x_{5},z_{5})$
$yx^{-2}$ | $-x^{-2}$ | $x^{-3}$ | $-d^{-2}x^{-5}$ | $d^{-2}x^{-6}$
$(x_{6},z_{1})$ | $(x_{6},z^{{}^{\prime}}_{1})$ | $(x_{6},z^{"}_{1})$ | $(x_{6},z_{4})$ | $(x_{6},z_{5})$
$-yx^{-3}$ | $x^{-3}$ | $-x^{-4}$ | $d^{-2}x^{-6}$ | $-d^{-2}x^{-7}$
$(x_{7},z_{1})$ | $(x_{7},z^{{}^{\prime}}_{1})$ | $(x_{7},z^{"}_{1})$ | $(x_{7},z_{4})$ | $(x_{7},z_{5})$
$yx^{-4}$ | $-x^{-4}$ | $x^{-5}$ | $-d^{-2}x^{-7}$ | $d^{-2}x^{-8}$
$(x_{2},z_{2})$ | $(x_{2},z_{3})$ | $(x_{2},z_{6})$ | $(x_{2},z_{7})$ | $(x_{2},z_{8})$
$d^{-1}x^{-1}$ | $-d^{-1}x^{-2}$ | $d^{-1}x^{-4}$ | $-d^{-1}x^{-5}$ | $d^{-1}x^{-6}$
$(x_{3},z_{2})$ | $(x_{3},z_{3})$ | $(x_{3},z_{6})$ | $(x_{3},z_{7})$ | $(x_{3},z_{8})$
-$d^{-1}x^{-2}$ | $d^{-1}x^{-3}$ | -$d^{-1}x^{-5}$ | $d^{-1}x^{-6}$ | -$d^{-1}x^{-7}$

Then the intersection pairing has the following form:

$\displaystyle\Gamma_{2}(\beta)(u,x,y,d)=u^{-10}y^{-2}\cdot($ $\displaystyle
y^{2}-y+yx^{-1}-yx^{-3}+yx^{-4}$ (7.5) $\displaystyle-
yx^{-1}+x^{-1}-x^{-2}+d^{-2}x^{-4}-d^{-2}x^{-5}$
$\displaystyle+yx^{-2}-x^{-2}+x^{-3}-d^{-2}x^{-5}+d^{-2}x^{-6}$
$\displaystyle-yx^{-3}+x^{-3}-x^{-4}+d^{-2}x^{-6}-d^{-2}x^{-7}$ $\displaystyle
yx^{-4}-x^{-4}+x^{-5}-d^{-2}x^{-7}+d^{-2}x^{-8}$ $\displaystyle
d^{-1}x^{-1}-d^{-1}x^{-2}+d^{-1}x^{-4}-d^{-1}x^{-5}+d^{-1}x^{-6}$
$\displaystyle-d^{-1}x^{-2}+d^{-1}x^{-3}-d^{-1}x^{-5}+d^{-1}x^{-6}-d^{-1}x^{-7}).$

Replacing $y=-d$, we have the form in $3$-variables as below:

$\displaystyle\Gamma_{2}(\beta)(u,x,-d,d)=u^{-10}\Bigl{(}$ $\displaystyle
1+d^{-1}+(d^{-2}+d^{-3})x^{-1}+$ (7.6)
$\displaystyle+(-d^{-1}-2d^{-2}-2d^{-3})x^{-2}+$
$\displaystyle+(2d^{-1}+2d^{-2}+d^{-3})x^{-3}$
$\displaystyle+(-2d^{-1}-2d^{-2}+d^{-3}+d^{-4})x^{-4}$
$\displaystyle+(d^{-2}-2d^{-3}-2d^{-4})x^{-5}$
$\displaystyle+(-2d^{-3}+2d^{-4})x^{-6}-$
$\displaystyle-(d^{-3}+2d^{-4})x^{-7}+d^{-4}x^{-8}\Bigr{)}.$

Specialising $u=x^{\frac{1}{2}},d=x^{-1},x=q^{2}$, we get the Jones polynomial
of the knot $8_{19}$:

$\displaystyle\Gamma_{2}(\beta)(u,x,y,d)|_{\psi_{1,q,1}}$
$\displaystyle=q^{-10}(1-q^{-6}+q^{4})=$ (7.7)
$\displaystyle=q^{-10}-q^{-16}+q^{-6}.$

Also, if we specialise $u=x^{-\frac{1}{2}},d=-1$, we get the Alexander
polynomial of the knot $8_{19}$:

$\displaystyle\Gamma_{2}(\beta)(u,x,y,d)|_{u=x^{-\frac{1}{2}},y=-d=1}$
$\displaystyle=x^{5}(x^{-2}-x^{-3}+x^{-5}-x^{-7}+x^{-8})=$ (7.8)
$\displaystyle=x^{3}-x^{2}+1-x^{-2}+x^{-3}.$

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