q^{\ \sum_{k=1}^{n-1}2i_{k}}.$ (5.11)

Now we we would like to see this coefficient as coming from the specialisation
of the variables that we use for the local system that gives our homology
groups. For this, we remark the following:

$\displaystyle\psi_{q,\bar{N},\overline{[N]}}(x_{C(i)})=q^{N_{C(i)}-1}=q^{N^{C}_{i}-1},\forall
i\in\\{1,...,n\\}\ (\text{following }\eqref{eq:N})$ (5.12)
$\displaystyle\psi_{q,\bar{N},\overline{[N]}}(d^{-4})=q^{2}.$

This means that the coefficient from (5.11) is precisely the specialisation:

$\psi_{q,\bar{N},\overline{[N]}}\left(\left(\prod_{i=2}^{n}x^{-1}_{C(i)}\right)\cdot
d^{-4\sum_{k=1}^{n-1}i_{k}}\right).$ (5.13)

The last coefficient from the formula presented in (5.7) is given by the
framing contribution of the components of our link $L$. This completes the
intersection model for the coloured Jones polynomial
$J_{N_{1},...,N_{l}}(L,q)$ for the link coloured with colours
$N_{1},...,N_{l}$. ∎

##  6 Coloured Alexander polynomials for framed links

In this part we prove a topological intersection model for the coloured
Alexander polynomials for links.

This is the first topological model for these non-semisimple invariants for
links whose components are coloured with different colours. For the case of
links coloured with a single colour we showed a topological intersection model
in [3] and [5], where we used configuration spaces on figure eights and open
circles respectively as the geometric supports for the second homology class.

From the representation theory point of view, the change from the single
colour case to the multi-colour case is totally non-trivial and requires the
definition of so-called modified dimensions. In order to provide this model,
we will encode geometrically this extra data, in our configuration space in
the punctured disc.

Even for the case where we colour by a single colour we obtain a new
perspective, given by the fact that we have homology classes based on
configuration spaces on circles rather than an open support given by the
circles minus a puncture (as in [5]). This is achieved by our work on choosing
a more subtle local system than the one from [5].

As in the previous section, we fix a framed oriented link $L=K_{1}\cup...\cup
K_{l}$ with $l$ components, which has framings $f_{1},...,f_{l}\in\mathbb{Z}$.
Then we fix $\beta_{n}\in B_{n}$ to be a braid representative such that
$L=\widehat{\beta_{n}}$. Also, we fix $\mathcal{N}\in\mathbb{N}$,
$\mathcal{N}\geqslant 2$. We recall the notation for the set of generic
colours:
$\ddot{\mathbb{C}}:=(\mathbb{C}\setminus\mathbb{Z})\cup\mathcal{N}\mathbb{Z}$.

Let $\lambda_{1},...,\lambda_{l}\in\ddot{\mathbb{C}}$ a set of colours for the
strands of our link. The associated coloured multi-index is given by:

$\bar{\lambda}:=(\lambda_{1},...,\lambda_{l}).$

###### Notation 6.1

For $\lambda\in\ddot{\mathbb{C}}$ let us denote by $U_{\lambda}$ the
associated representation of the quantum group at roots of unity
$U_{\xi_{\mathcal{N}}}(sl(2))$ (here $\xi_{\mathcal{N}}$ is our fixed
primitive root of unity).

Then, we colour the components of $L$ with the representations
$U_{\lambda_{1}},...,U_{\lambda_{l}}$. Further, let us denote by
$\Phi_{\mathcal{N}}(L,\bar{\lambda})$ to be the coloured Alexander polynomial
of the framed link $L$ (which was defined in [1]).

Now we want to define the local system such that our geometric support given
by configuration spaces on circles leads to a well defined homology class in
the homology of the associated covering space.

###### Definition 6.2

(Induced colorings)
a) (Colourings associated to the braid) As in the case of the coloured Jones
polynomials, the colours of the link $\bar{\lambda}$ induce a colouring of the
strands of $\beta_{n}$. Let us denote the associated colours by:
$(C_{1},...,C_{n})$, as in Figure 6.1:

Figure 6.1: Colouring for the braid

Now, our the link $L$ is the closure of $\beta_{n}$ together with $n$ straight
strands. They induce an associated colouring of $2n$ points
$C:\\{1,...,2n\\}\rightarrow\\{1,...,l\\}$. Let us denote by

$\lambda^{C}_{i}:=\lambda_{C(i)}.$ (6.1)

We consider the colouring of the $2n$ points:

$\bar{C}^{\bar{\lambda}}:=(\lambda^{C}_{1},...,\lambda^{C}_{n},\lambda^{C}_{n},...,\lambda^{C}_{1}).$

###### Definition 6.3

(Set of states at level $\mathcal{N}$) In this case, our indexing set will be
given by the following set:

$C(\mathcal{N}):=\\{\bar{0},\dots,\overline{\mathcal{N}-1}\\}=\big{\\{}\bar{i}=(i_{1},...,i_{n-1})\in\mathbb{N}^{n-1}\mid
0\leqslant i_{k}\leqslant\mathcal{N}-1,\ \forall
k\in\\{1,...,n-1\\}\big{\\}}.$

We remark that in contrast to the case of coloured Jones polynomials, where
the set of states depends on the choice of colours (as in Definition 5.3),
here the set of states is intrinsec and it is determined just by the level and
not by the particular choice of colours.

######  Definition 6.4 ($\mathcal{N}$-coloured multi-level)

We consider the following indices:

$\begin{cases}&\mathcal{N}_{1}=1,\\\
&\mathcal{N}_{i}=\mathcal{N}-1,i\in\\{2,...,n\\}.\end{cases}$ (6.2)

Then, we define $\mathcal{N}$-coloured multi-level (which has $n$ components):

$\bar{\mathcal{N}}:=(2,\mathcal{N}-1,...,\mathcal{N}-1).$ (6.3)

###  6.1 Homology classes

In the next part we define the homology classes that we work with for this
second case, of the invariants at roots of unity. In contrast to the situation
for coloured Jones polynomials, where we used the induced colouring in order
to define the homology groups, here we need just the level $\mathcal{N}$ in
order to define these homologies.

More specifically, let us consider the configuration space of

$2+(n-1)(\mathcal{N}-1)$

points on the $(2n+2)$-punctured disc.

After this, we look at the local system $\Phi^{\bar{\mathcal{N}}}$ associated
to the following parameters:

$n\rightarrow n;\ \ \ m\rightarrow 2+(n-1)(\mathcal{N}-1);\ \ \
{\bar{l}}\rightarrow 1;\ \ \ \bar{\mathcal{N}}.$

We fix the homology groups associated to these parameters, as below:

$H^{\bar{\mathcal{N}}}_{n,2+(n-1)(\mathcal{N}-1)}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\text{ and }\ \ \ \ \ \ \ \ \ \ \ \ \ \
H^{\bar{\mathcal{N}},\partial}_{n,2+(n-1)(\mathcal{N}-1)}.$

As in the previous section, we erase the third component associated to
$\bar{l}$ from the indices of the homology groups.

Now we have all the tools needed in order to introduce the homology classes
that we use for the intersection model for coloured Alexander polynomials.

###### Definition 6.5

(Homology classes for ADO invariants)
Let us fix a multi-index $\bar{i}=(i_{1},...,i_{n})\in C(\mathcal{N})$. We
define the following homology classes associated to this multi-index, given by
the geometric supports from Figure 6.2:

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

$\hskip 14.22636pt\downarrow\text{ lifts }$

Figure 6.2: Lagrangians for Coloured Alexander link invariants coloured with
diferent colours

###### Proposition 6.6

The geometric support presented in Figure 6.2, induces a well-defined homology
class in the covering, which we denoted by
$\mathscr{L}_{\bar{i},\mathcal{N}}\in
H^{\bar{\mathcal{N}},\partial}_{n,2+(n-1)(\mathcal{N}-1)}$.

###### Proof.

We remark that the geometric support for the dual class from Figure 6.2 is the
the geometric support from Definition 4.2, associated to the multi-level
$\bar{\mathcal{N}}.$

This was denoted by $L(\bar{\mathcal{N}})$. Then, using Lemma 4.3 we know that
this submanifold gives well-defined lifts in the associated covering space.

As in the previous section, we use the procedure of lifting described in
Notation 3.30. This means that the collection of straight vertical paths which
connect the ovals to the boundary of the disc gives us a path in the
configuration space, from ${\bf d}$ to $L(\bar{\mathcal{N}})$, shown in the
right hand side of Figure 6.2 (see [4]-Section 5 for the procedure of
constructing such a path).

The base point of the covering, $\tilde{{\bf d}}$, gives us a base point for
lifting the submanifold, and we obtain a lift denoted by:

$\tilde{L}(\bar{\mathcal{N}})_{\bar{i}}.$

From Lemma 4.3, this submanifold gives a well-defined homology class in the
dual homology $H^{\bar{\mathcal{N}},\partial}_{n,2+(n-1)(\mathcal{N}-1)}$. We
denote this class by:

$\mathscr{L}_{\bar{i},\mathcal{N}}\in
H^{\bar{\mathcal{N}},\partial}_{n,2+(n-1)(\mathcal{N}-1)}.$

Figure 6.3: Monodromy of configurations on a circle for the coloured Alexander
classes

This concludes the proof of the existence of the well-defined homology class.
∎

Now, let us fix the indices:

$\begin{cases}\bar{\lambda}:=(\lambda_{1},...,\lambda_{l})\\\
\overline{d(\lambda)}:=([\lambda_{C(1)}]_{\xi_{\mathcal{N}}})\\\ \end{cases}$
(6.4)

We then consider the specialisation of coefficients (which is associated to
the multi-indices $\bar{\alpha}=\bar{\lambda}$ and
$\eta=\overline{d(\lambda)}$):

$\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}:\mathbb{C}[x_{1}^{\pm
1},...,x_{l}^{\pm 1},y^{\pm 1},d^{\pm
1}]\rightarrow\mathbb{C}[\xi_{\mathcal{N}}^{\pm\frac{1}{2}},\xi_{\mathcal{N}}^{\pm\frac{\lambda_{1}}{2}},...,\xi_{\mathcal{N}}^{\pm\frac{\lambda_{l}}{2}}]$
$\begin{cases}&\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}(x_{i})=\xi_{\mathcal{N}}^{\lambda_{i}},\
i\in\\{1,...,l\\}\\\
&\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}(y)=([\lambda_{C(1)}]_{\xi_{\mathcal{N}}}),\\\
&\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}(d)=\xi_{\mathcal{N}}^{-\frac{1}{2}}.\end{cases}$
(6.5)

###  6.2 Intersection model

In this section we will prove the topological model for the coloured Alexander
polynomials of a link coloured with the colours $\lambda_{1},...,\lambda_{l}$.
We show that this multivariable invariant can be read off from an intersection
pairing via the homology classes

$\\{\mathscr{F}_{\bar{i},\mathcal{N}},\mathscr{L}_{\bar{i},\mathcal{N}}\\}\text{
for all indices }\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{N}-1}\\}.$

We recall the model presented in Theorem 1.9.

######  Theorem 6.7 (Topological model for coloured Alexander polynomials for
coloured links via ovals)

We have the following topological model for a link coloured with colours
$\bar{\lambda}$:

$\displaystyle\Phi^{\mathcal{N}}_{\bar{\lambda}}(L)$
$\displaystyle=~{}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)\Bigm{|}_{\psi^{\bar{\lambda}}_{\mathcal{N}}}.$
(6.6)

Here, the intersection is given by:

$\begin{cases}&A\Gamma^{\mathcal{N}}(\beta_{n})\in\mathbb{Z}[u_{1}^{\pm
1},...,u_{l}^{\pm 1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y,d^{\pm 1}]\\\
&A\Gamma^{\mathcal{N}}(\beta_{n}):=\prod_{i=1}^{l}u_{i}^{\left(f_{i}-\sum_{j\neq{i}}lk_{i,j}\right)}\cdot\prod_{i=2}^{n}u^{-1}_{C(i)}\cdot\\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{N}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle.\end{cases}$
(6.7)

###### Proof.

First of all, we look at the change of coefficients described in Section 2,
and using this we want to show the following:

$\displaystyle\Phi^{\mathcal{N}}_{\bar{\lambda}}(L)$
$\displaystyle=~{}{\xi_{\mathcal{N}}}^{\sum_{i=1}^{l}\left(f_{i}-\sum_{j\neq
i}lk_{i,j}\right)\lambda_{i}(1-N_{i})}\cdot$ (6.8)
$\displaystyle\cdot\left(\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{N}-1}\\}}\left(\prod_{i=1}^{n}u^{-1}_{C(i)}\right)\cdot\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle\right)\Bigm{|}_{\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}}.$

We will prove this model in $6$ main parts. The first two steps are purely
topological and for the rest we use the connection to representation theory.
The strategy has some similarities to the one used for the model for coloured
Jones polynomials from the previous section, but there are certain important
topological differences, which we will emphasise in the next part.

A feature of this topological perspective is that it will be a key part in the
globalisation of all of these invariants with colours less than the level
$\mathcal{N}$, as we will present in the next section. This globalisation, in
turn, will be the main tool for constructing the universal ADO invariant.

Step 1

We consider the homology classes obtained from the geometric supports of the
classes $\mathscr{F}_{\bar{i},\bar{N}}$ and $\mathscr{L}_{\bar{i},\bar{N}}$,
but where we remove the 1-dimensional part supported around the puncture of
disc labeled by $0$ (this means that we remove the purple segment and the blue
central circle), as in Figure 6.4:

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

Figure 6.4: Removing the middle circle from the geometric support

Counting the contribution of the intersection point between the circle and the
middle arc, this gives the following relation between the intersection
pairings:

$\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+2})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle=d^{-4\sum_{k=1}^{n-1}i_{k}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime}_{\bar{i},\mathcal{N}}}\right\rangle$

This means that we want to prove the following:

$\displaystyle\Phi^{\mathcal{N}}_{\bar{\lambda}}(L)$
$\displaystyle=~{}{\xi_{\mathcal{N}}}^{\sum_{i=1}^{l}\left(f_{i}-\sum_{j\neq
i}lk_{i,j}\right)\lambda_{i}(1-N_{i})}\cdot$ (6.9)
$\displaystyle\cdot\left(\sum_{\bar{i}\in
C(\bar{N})}\left(\prod_{i=1}^{n}u^{-1}_{C(i)}d^{-4\sum_{k=1}^{n}i_{k}}\right)\cdot\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime}_{\bar{i},\mathcal{N}}}\right\rangle\right)\Bigm{|}_{\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}}.$

Step 2

Next, we consider the homology classes given by the support of
$\mathscr{F}^{\prime}_{\bar{i},\bar{N}}$ and
$\mathscr{L}^{\prime}_{\bar{i},\bar{N}}$ where we remove the arc and circle
that end or go around the last puncture, as in Figure 6.5:

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

$\hskip 14.22636pt\downarrow\text{ lifts }$

Figure 6.5: Removing the middle circle and the extremal circle from the
geometric support

In this step we want to compare the graded intersection from Step I, given by

$\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+2})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime}_{\bar{i},\mathcal{N}}}\right\rangle$

and the new pairing between the classes from Figure 6.5, given by:

$\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+2})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime\prime}_{\bar{i},\mathcal{N}}}\right\rangle.$

The unique difference between the intersections of the underlying pairs of
submanifolds comes from the fact that the first pair of submanifolds has an
extra intersection point between the extremal circle and the arc that ends in
the puncture $2n$, as in Figure 6.6:

$\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+2})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime}_{\bar{i},\mathcal{N}}}\right\rangle\
\ \ \ \ \ \ \ \ \ \ \ \ \ \text{ and }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+2})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime\prime}_{\bar{i},\mathcal{N}}}\right\rangle$

Figure 6.6: Removing the middle circle and the extremal circle for the
coloured Alexander polynomials

That means that the associated intersection pairing has an extra coefficient
coming from the contribution of this intersection point.

This is a computation which is very similar to the analog one presented in
Step 2 for the case of the Coloured Jones polynomials (which we saw in the
previous section).

This coefficient comes from the monodromy of the loop around the $s$-puncture,
presented in the right hand side of Figure 6.6, which gets evaluated to the
variable $y$.

Then, we have the following relation between the two intersection pairings:

$\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+2})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime}_{\bar{i},\mathcal{N}}}\right\rangle=y\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime\prime}_{\bar{i},\mathcal{N}}}\right\rangle$

Then, let us specialise these intersections via the change of variables
$\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}$. We recall
that:

$\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}(y)=d(\lambda_{C(1)})=[\lambda_{C(1)}]_{\xi_{\mathcal{N}}}.$

So, we obtain the following relation:

$\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+2})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime}_{\bar{i},\mathcal{N}}}\right\rangle|_{\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}}=[\lambda_{C(1)}]_{\xi_{\mathcal{N}}}\cdot\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime\prime}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}^{\prime\prime}_{\bar{i},\mathcal{N}}}\right\rangle|_{\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}}$

Step 3

Now we turn our attention to the definition of the coloured Alexander
invariants, coming from representation theory. We will use the version of the
quantum group $U_{q}(sl(2))$ over two variables, which has the divided powers
of one of the generators. We refer to [3, Section 3] for the definition of
this quantum group, its Verma modules and representation theory.

We showed in [3, Section 3] that we can specialise this representation theory
at roots of unity and obtain the coloured Alexander invariant for a link whose
components are coloured with the same colour. We did this by first considering
the subspace generated by the first $\mathcal{N}$ weight vectors in the Verma
module. Then, we noticed that when we specialise at the root of unity
$\xi_{\mathcal{N}}$, this subspace does not lead to a well-defined sub-
representation over the quantum group $U_{\xi_{\mathcal{N}}}(sl(2))$. However,
the key point is that the tensor product of this specialised module is
preserved by the specialised braid group action.

This was shown in [3, Lemma 3.1.7], and it relies on the computation of the
R-matrix action onto a tensor product of two vectors $v_{i}\otimes v_{j}$ from
the Verma module together with the fact that when specialised at roots of
unity we have a vanishing of certain quantum factorials. That ensures that if
$i,j\leqslant\mathcal{N}-1$, then the action of the specialised braiding on
$v_{i}\otimes v_{j}$ will be a linear combination of $v_{i-n}\otimes v_{j+n}$
where all the indices remain bounded by $\mathcal{N}$.

For our situation, namely for the coloured Alexander invariants for links, we
need to work with different colours. This means that we have to consider
tensor products of modules associated to different highest weights
$\lambda_{1}$ and $\lambda_{2}$.

In a nutshell, the main property that changes and needs to be checked concerns
the action of the braiding on such two Verma modules that have weights
$\lambda_{1}$ and $\lambda_{2}$.

We notice that also for this situation we can generalise the argument from [3,
Lemma 3.1.7]. A direct computation of the coefficients coming from the
R-matrix shows that we have the property that the tensor power of finite
dimensional subspaces of dimension $\mathcal{N}$ associated to Verma modules
of different weights are preserved by the braid group action, when specialised
to the root of unity $\xi_{\mathcal{N}}$.

This means that we can obtain the coloured Alexander polynomial of the link
coloured with different colours via the Reshetikhin-Turaev construction
applied to this version of the quantum group. More precisely, looking on the
bottom of the diagram, we have that the cups evaluated on the algebraic side
correspond exactly to the sum classes
$\mathscr{F}^{\prime\prime}_{\bar{i},\mathcal{N}}$ over all choices of indices
$\bar{i}\in\\{\bar{0},...,\overline{\mathcal{N}-1}\\}$. After that, we use the
property that the braid action on the quantum side and on the homological side
are isomorphic, via the identification showed by Martel in [19].

Step 4

The upper part of the diagram, given by the set of caps is associated on the
algebraic side to the evaluations. This means that we evaluate non-trivially
just the tensor powers of vectors from the specialised Verma module which
satisfy the following two requirements:

  * •

components are all strictly less than $\mathcal{N}$

  * •

the indices of the components are symmetric with respect to the colour
$\mathcal{N}$, meaning that they have the sum $\mathcal{N}-1$ (if we remove
the first component, because this is associated to the open strand).

Now, we translate the meaning of these evaluation conditions on the
homological side.

We notice that this condition gives us the requirement that we have to
evaluate non-trivially just the homology classes which:

  * •

have all multiplicities of the multi-indices strictly less than $\mathcal{N}$

  * •

the number of arcs that end in symmetric punctures with respect to $0$ (which
is in the middle of the punctured disc) adds up to $\mathcal{N}-1$ – excluding
the component that ends in $1$ (this is associated to the open strand).

Step 5

Looking more closely at this requirement, we see that it enforces that we
evaluate non-trivially just the classes whose indices associated to the
p-punctures $k$ and $2n+1-k$ have the sum $\mathcal{N}-1$. The main point is
that we are able to obtain this condition in a geometric way, by intersecting
with the dual class $\mathscr{L}^{\prime\prime}_{\bar{i},\mathcal{N}}$.

Step 6

Algebraically, we have to encode an additional piece of data, which is a
coefficient corresponding to the caps of the diagram. On the representation
theory side, this is provided by the quantum trace. The details of the
argument for the case where we have just a single colour are discussed in [3,
(Section 5, Section 8)]. We will proceed with an analog argument, so we
present below the main differences that appears in this new situation, given
by the choice of different colours for our link.

The quantum trace is a trace that is twisted by certain coefficients that come
from the pivotal structure. In this case, the pivotal structure is provided by
the action of the element $K^{\mathcal{N}-1}$ of the quantum group (in [3] we
present in details the precise definition of this quantum trace).

Now, we fix a set of indices $i_{1},...,i_{n-1}$ which are strictly bounded by
$\mathcal{N}$. On the algebraic side, the action of the element
$K^{\mathcal{N}-1}$ on the associated monomial has the following coefficient:

${\xi_{\mathcal{N}}}^{(\mathcal{N}-1)\sum_{k=2}^{n}\left((\lambda_{C(k)})-2i_{k-1}\right)}=\left(\prod_{k=2}^{n}{\xi_{\mathcal{N}}}^{(\mathcal{N}-1)\lambda_{C(k)}}\right)\cdot{\xi_{\mathcal{N}}}^{\
\sum_{k=1}^{n-1}2i_{k}}.$ (6.10)

In the following part, we encode this coefficient using the specialisation of
variables which are used for the local system.

We recall the following property:

$\displaystyle\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}(u_{C(i)})={\xi_{\mathcal{N}}}^{(1-\mathcal{N})\lambda_{C(i)}},\forall
i\in\\{1,...,n\\}\ (\text{following }\eqref{eq:N})$ (6.11)
$\displaystyle\psi_{q,\bar{N},\overline{[N]}}(d^{-4})=q^{2}.$

Using this, we obtain that the coefficient from (6.10) is precisely the
following specialisation:

$\psi_{\xi_{\mathcal{N}},\bar{\lambda},\overline{d(\lambda)}}\left(\left(\prod_{i=2}^{n}u^{-1}_{C(i)}\right)\cdot
d^{-4\sum_{k=1}^{n-1}i_{k}}\right).$ (6.12)

It remains one last coefficient in the formula presented in (6.8). This comes
from the framing contribution of the components of the link $L$.

Putting all of these together, we conclude the intersection model for the
coloured Alexander polynomial of the link $L$ coloured with colours
$\lambda_{1},...,\lambda_{l}$. ∎

##  7 Unifying all Coloured Alexander polynomials for links with colours
bounded by $\mathcal{N}$

In this part we put together the provious models and we will show that if we
fix $\mathcal{N}$, then the intersection at level $\mathcal{N}$ recovers all
coloured Alexander polynomials of levels less than $\mathcal{N}$, as presented
in Theorem 1.12 which we remind below.

######  Theorem 7.1 (Unifying coloured Alexander polynomials of bounded level)

Let us fix a level $\mathcal{N}\in\mathbb{N}$. Then the state sum of
Lagrangian intersection in the configuration space of $(n-1)(\mathcal{N}-1)+2$
points in the punctured disc:

$A\Gamma^{\mathcal{N}}(\beta_{n})\in\mathbb{Z}[u_{1}^{\pm 1},...,u_{l}^{\pm
1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y,d^{\pm 1}].$ (7.1)

(given by the set of Lagrangian intersections
$\\{\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\rangle\\}_{\bar{i}=(i_{1},...,i_{n})\in\\{\bar{0},\dots,\overline{\mathcal{N}-1}}$)
recovers all coloured Alexander polynomials of colours
$\mathcal{M}\leqslant\mathcal{N}$:

$\displaystyle\Phi^{\mathcal{M}}_{\bar{\lambda}}(L)$
$\displaystyle=~{}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)\Bigm{|}_{\psi^{\bar{\lambda}}_{\mathcal{M}}}$
(7.2)

###### Proof.

Let us start by recalling the definition of the intersection form at level
$\mathcal{N}$:

$\displaystyle
A\Gamma^{\mathcal{N}}(\beta_{n}):=\prod_{i=1}^{l}u_{i}^{\left(f_{i}-\sum_{j\neq{i}}lk_{i,j}\right)}\cdot\prod_{i=2}^{n}u^{-1}_{C(i)}\cdot$
(7.3) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{N}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle.$

Similarly, the intersection form at the inferior level, $\mathcal{M}$, has the
formula:

$\displaystyle
A\Gamma^{\mathcal{M}}(\beta_{n}):=\prod_{i=1}^{l}u_{i}^{\left(f_{i}-\sum_{j\neq{i}}lk_{i,j}\right)}\cdot\prod_{i=2}^{n}u^{-1}_{C(i)}\cdot$
(7.4) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\right\rangle.$

Following the topological model presented in Theorem 1.9, we have that
$A\Gamma^{\mathcal{M}}(\beta_{n})$ recovers the $\mathcal{M}^{th}$ coloured
Alexander invariant, through the specialisation of coefficients
$\psi^{\bar{\lambda}}_{\mathcal{M}}$, as below:

$\displaystyle\Phi^{\mathcal{M}}_{\bar{\lambda}}(L)$
$\displaystyle=~{}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)\Bigm{|}_{\psi^{\bar{\lambda}}_{\mathcal{M}}}.$
(7.5)

Putting together these equations, we see that it is enough to prove that the
state sums of Lagrangian intersections $A\Gamma^{\mathcal{N}}(\beta_{n})$ and
$A\Gamma^{\mathcal{M}}(\beta_{n})$ become equal when specialised through
$\psi^{\bar{\lambda}}_{\mathcal{M}}$:

$\displaystyle\left(A\Gamma^{\mathcal{M}}(\beta_{n})\right)\Bigm{|}_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=~{}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)\Bigm{|}_{\psi^{\bar{\lambda}}_{\mathcal{M}}}.$
(7.6)

Using the formulas from equation (7.3) and (7.4), we want to prove the
following relation:

$\displaystyle\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\right\rangle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=$
(7.7)
$\displaystyle\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{N}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}.$

Looking at the second sum we notice that we have two type of terms, namely the
ones associated to indices less than $\overline{\mathcal{M}}$ and the ones
which are in between $\overline{\mathcal{M}}$ and $\overline{\mathcal{N}-1}$,
so we want to show the following relation:

$\displaystyle\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\right\rangle$
$\displaystyle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=$ (7.8) $\displaystyle=$
$\displaystyle\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}+$
$\displaystyle+\sum_{\bar{i}\in\\{\bar{\mathcal{M}},\dots,\overline{\mathcal{N}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}.$

We will do this in two parts. First we show that the terms from the left and
right hand side of the above equation that are associated to an index
$\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}$ become equal when
specialised at level $\mathcal{M}$. More specifically we will prove the
following relation:

$\displaystyle\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\right\rangle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle$
$\displaystyle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}},$ (7.9)
$\displaystyle\forall\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}.$

Secondly, the terms associated to higher indices
$\bar{i}\in\\{\bar{\mathcal{M}},\dots,\overline{\mathcal{N}-1}\\}$ vanish when
specialised at level $\mathcal{M}$. More specifically we will prove the
following relation:

$\displaystyle\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle$
$\displaystyle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=0,\forall\bar{i}\in\\{\overline{\mathcal{M}},\dots,\overline{\mathcal{N}-1}\\}.$
(7.10)

Step I - Classes that are associated to a multi-index bounded by $\mathcal{M}$

In order to prove the first part, namely relation (7.9), we will actually show
a slightly stronger property, namely that the classes corresponding to the
indices $\mathcal{M}$ and $\mathcal{N}$ give the same intersection as long as
the index is bounded by $\bar{\mathcal{M}}$, as below.

######  Lemma 7.2 (Equality of intersections associated to indices less than
$\mathcal{M}$)

The following intersection pairings, between classes associated to indices
$\bar{i}$ bounded by $\bar{\mathcal{M}}$ give the same result, even before
specialisation:

$\displaystyle\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\right\rangle=\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle,\forall\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}.$
(7.11)

###### Proof.

Looking at the classes $\mathscr{F}_{\bar{i},\mathcal{M}}$ and
$\mathscr{F}_{\bar{i},\mathcal{N}}$ we remark that they have the same
geometric support in the left hand side of the puncture disc. The only
difference is that the class $\mathscr{F}_{\bar{i},\mathcal{N}}$comes from a
support that has more segments ending in the right hand side of the disc.

Now, we have to act with the braid $\beta_{n}\cup{\mathbb{I}}_{n+1}$. This
action does not change the support associated to the right hand side of the
disc.

So, we remark that the classes
$\langle(\beta_{n}\cup{\mathbb{I}}_{n+1}\rangle)\mathscr{F}_{\bar{i},\mathcal{M}}$
and
$\langle(\beta_{n}\cup{\mathbb{I}}_{n+1}\rangle)\mathscr{F}_{\bar{i},\mathcal{N}}$
have the same geometric support in the left hand side of the disc, the only
difference coming from the number of arcs ending in the right hand side of the
disc.

Now, we have to intersect with the dual classes, which are
$\mathscr{L}_{\bar{i},\mathcal{M}}$ and $\mathscr{L}_{\bar{i},\mathcal{N}}$.
Looking at the geometric supports that give the submanifolds in the base
configuration space, we remark that we have a bijective correspondence between
the associated sets of intersection points, as below:

$(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i},\mathcal{M}}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}L_{\bar{i},\mathcal{M}}}\longleftrightarrow^{f}(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i},\mathcal{N}}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}L_{\bar{i},\mathcal{N}}}$
$\bar{x}\ \ \ \longleftrightarrow\ \ \ f(\bar{x}).$

In order to construct this bijection, let us consider an intersection point
$\bar{x}\in(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i},\mathcal{M}}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}L_{\bar{i},\mathcal{M}}}$.
We define $f(\bar{x})$ to be the point in the configuration space coming from
$\bar{x}$ where we add $(\mathcal{N}-M)(n-1)$ particles. More specifically, we
add to $\bar{x}$ the set of points situated in the right hand side of the
disc, at the intersection between the extra arcs ending in a puncture labeled
by $\\{n+1,...,2n-1\\}$ (from the support of
$(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i},\mathcal{N}}}$)
and the circles that give $L_{\bar{i},\mathcal{N}}$.

In order to compute the intersections we use the formula from (3.17). Having
in mind that our submanifolds come from products of arcs, we can compute the
intersections by looking at the product of signs at the local orientations in
the punctured disc, associated to $\bar{x}$ and $f(\bar{x})$.

$\displaystyle\ll(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\gg$
$\displaystyle=\sum_{\bar{x}\in(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i},\mathcal{M}}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}L_{\bar{i},\mathcal{M}}}}\alpha_{\bar{x}}\cdot\bar{\Phi}^{\bar{\mathcal{M}}}(l_{\bar{x}})$
(7.12)
$\displaystyle\ll(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\gg$
$\displaystyle=\sum_{\bar{y}\in(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i},\mathcal{N}}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}}\alpha_{\bar{y}}\cdot\bar{\Phi}^{\bar{\mathcal{N}}}(l_{\bar{y}})=$
$\displaystyle=\sum_{\bar{x}\in(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i},\mathcal{M}}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}L_{\bar{i},\mathcal{M}}}}\alpha_{f(\bar{x})}\cdot\bar{\Phi}^{\bar{\mathcal{N}}}(l_{f(\bar{x}})).$

Now, looking at the components of the intersection points $\bar{x}$ and
$f(\bar{x})$, we see that the points in the left hand side of the disc
coincide, so they have the same local signs and all the components from the
right hand side have positive orientation, so they do not contribute to the
sign cunt. This shows that:

$\alpha_{\bar{x}}=\alpha_{f(\bar{x})}.$ (7.13)

Next, let us look at the monomials that are associated to these intersection
points. The loop in the configuration space which is associated to
$f(\bar{x})$, $l_{f(\bar{x})}$, can be obtained from the loop associated to
$\bar{x}$ union with another set of loops that pass through the extra
intersection points in the right hand side of the disc (which we used in order
to construct the bijection $f$).

However, we remark that these extra loops are evaluated trivially by the local
system $\bar{\phi}^{\bar{\mathcal{N}}}$. This is due to the fact that they do
not twist and do not go around any puncture, and so they contribute with
coefficients which are all 1.

The last remark is that the two local systems $\bar{\Phi}^{\bar{\mathcal{M}}}$
and $\bar{\Phi}^{\bar{\mathcal{N}}}$ evaluate in the same way the loops
associated to the punctures in the left hand side of the disc, and also the
relative winding. So, the contribution of the components of the intersection
points from the left hand side of the disc is the same, when evaluated through
these two local systems.

Putting everything together, we conclude that:

$\bar{\Phi}^{\bar{\mathcal{M}}}(\bar{x})=\bar{\Phi}^{\bar{\mathcal{N}}}(f(\bar{x})),\
\ \ \
\forall\bar{x}\in(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}F_{\bar{i},\mathcal{M}}}\cap{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}L_{\bar{i},\mathcal{M}}}.$
(7.14)

This shows that the two intersection pairings lead to the same result:

$\ll(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\gg=\ll(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\gg.$
(7.15)

In order to get to the pairings that we have, we still have to change the
coefficients via the colouring $f_{C}$, following relation (3.19).
Specialising the previous relation we get:

$\displaystyle\langle(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\rangle=$
(7.16)
$\displaystyle=\ll(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\gg|_{f_{C}}=$
$\displaystyle=\ll(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\gg|_{f_{C}}=$
$\displaystyle=\langle(\beta_{n}\cup{\mathbb{I}}_{n+1}){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\rangle.$

∎

Step II- Classes associated to higher indices, specialised at level
$\mathcal{M}$

Now, we will investigate what happens with the classes associated to multi-
indices biger than $\mathcal{M}$, when specialised via the change of
coefficients at level $\mathcal{M}$. We will prove that in this situation the
specialisation vanishes, which is what we wanted in relation (7.10):

###### Lemma 7.3

$\mathscr{F}_{\bar{i},\mathcal{N}}\mid_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=0,\
\ \
\forall\bar{i}\in\\{\overline{\mathcal{M}},\dots,\overline{\mathcal{N}-1}\\}.$
(7.17)

###### Proof.

We will prove this relation in a topological way, showing that something
special happens with the geometry of the above class when we specialise it to
a root of unity. For this, we will relate it to another homological class,
which has a slightly different geometric support.

Let $\mathscr{F}^{c}_{\bar{i},\mathcal{N}}\in
H^{\bar{\mathcal{N}}}_{n,1+(n-1)(\mathcal{N}-1)}$ be the homology class which
is given by the lift of the geometric support obtained from the product of
configuration spaces on the collection of arcs presented in the Figure 7.1.

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}\in
H^{\bar{\mathcal{N}}}_{n,1+(n-1)(\mathcal{N}-1)}}\ \ \ \ \ \ \ \ \ \ \text{
and }\ \ \ \ \ \ \ \ \ \ \ \ \
{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{F}^{c}_{\bar{i},\mathcal{N}}\in
H^{\bar{\mathcal{N}}}_{n,1+(n-1)(\mathcal{N}-1)}}.$

$\hskip 14.22636pt\downarrow\text{ lifts }$

Figure 7.1: Classes at parameters which are roots of unity

∎

We remark that the difference between the above geometric supports consists in
the fact that the one from the left hand side of Figure 7.1 is based on arcs
with multiplicities, and the one from the right hand side is given by
configurations on arcs, of prescribed number of particles. The main point is
that these two classes are related, one being a multiple of the other, as
below.

###### Notation 7.4

We denote the following quantum integers:

$(n)_{y}=\frac{1-y^{n}}{1-y}\ \ \ \ \ \ (n)_{y}!=(1)_{y}\cdot...\cdot(n)_{y}.$
(7.18)

###### Proposition 7.5

Following [3, Proposition 4.8], we have the relation:

$\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}=\prod_{k=1}^{n-1}(i_{k})_{d^{4}}!\cdot(\mathcal{N}-1-i_{k})_{d^{4}}!\
\mathscr{F}^{c}_{\bar{i},\mathcal{N}}.$ (7.19)

Now, we want to see what happens when we specialise at level $\mathcal{M}$,
through the function $\psi^{\bar{\lambda}}_{\mathcal{M}}$. We remind that:

$\psi^{\bar{\lambda}}_{\mathcal{M}}(d)=\xi_{\mathcal{M}}^{-\frac{1}{2}}.$

We notice that this formula tells us the following property:

$\psi^{\bar{\lambda}}_{\mathcal{M}}\left((\mathcal{M})_{d^{4}}\right)=0.$
(7.20)

This implies that:

$\psi^{\bar{\lambda}}_{\mathcal{M}}\left((i)_{d^{4}}!\right)=0,\text{ if
}i\geqslant\mathcal{M}.$ (7.21)

So, we have that:

$\psi^{\bar{\lambda}}_{\mathcal{M}}\left(\prod_{k=1}^{n-1}(i_{k})_{d^{4}}!\cdot(\mathcal{N}-1-i_{k})_{d^{4}}!\right)=0,\forall\bar{i}\in\\{\overline{\mathcal{M}},\dots,\overline{\mathcal{N}-1}\\}.$
(7.22)

Now following relations (7.19) and (7.22) we obtain:

$\displaystyle\mathscr{F}_{\bar{i},\mathcal{N}}\mid_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=$
(7.23)
$\displaystyle=\psi^{\bar{\lambda}}_{\mathcal{M}}\left(\prod_{k=1}^{n-1}(i_{k})_{d^{4}}!\cdot(\mathcal{N}-1-i_{k})_{d^{4}}!\right)\
\mathscr{F}^{c}_{\bar{i},\mathcal{N}}\mid_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=$
$\displaystyle=0,\ \
\forall\bar{i}\in\\{\overline{\mathcal{M}},\dots,\overline{\mathcal{N}-1}\\},$

which concludes the relation presented in the statement of Lemma 7.3.

Now, we put together these properties in order to finish the proof of the
globalisation property. Following Lemma 7.2 and 7.3, we have:

$\displaystyle\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}+$
(7.24)
$\displaystyle+\sum_{\bar{i}\in\\{\bar{\mathcal{M}},\dots,\overline{\mathcal{N}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}=$
$\displaystyle=\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{M}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{M}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{M}}}\right\rangle|_{\psi^{\bar{\lambda}}_{\mathcal{M}}}.$

This shows the relation that we wanted, presented in equation (7.8) and
concludes the proof of the Theorem 7.1.

∎

##  8 Universal ADO Invariant

In this part our aim is to unify and extract an invariant out of the sequence
of graded intersections

$\\{A\Gamma^{\mathcal{N}}(\beta_{n})\mid\mathcal{N}\in\mathbb{N}\\}.$

The first part for this construction is dedicated to the definition of an
appropiate ring of coefficients where this universal invariant will be
defined. Secondly, we will show that in such a universal ring, we have indeed
a well defined link invariant which is a universal coloured Alexander
invariant, recovering all $\Phi^{\mathcal{N}}(L)$ for all colours
$\mathcal{N}\in\mathbb{N}$.

###  8.1 Set-up and notations

######  Definition 8.1 (Rings for the universal invariant)

Let us define the following rings:

$\begin{cases}\mathbb{L}:=\mathbb{Z}[u_{1}^{\pm 1},...,u_{l}^{\pm
1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y^{\pm 1},d^{\pm
1}]=\mathbb{Z}[(\bar{u})^{\pm 1},(\bar{x})^{\pm 1},y^{\pm 1},d^{\pm 1}]\\\
\Lambda_{\mathcal{N}}:=\mathbb{Q}({\xi_{\mathcal{N}}})(x^{2\mathcal{N}}_{1}-1,...,x^{2\mathcal{N}}_{l}-1)[x_{1}^{\pm
1},...,x_{l}^{\pm 1}]\end{cases}$ (8.1)

where we compressed the multi-indices as $(\bar{u})^{\pm 1}=(u_{1}^{\pm
1},...,u_{l}^{\pm 1})$ and $(\bar{x})^{\pm 1}=(x_{1}^{\pm 1},...,x_{l}^{\pm
1})$.

###### Remark 8.2

(Immage of the coloured Alexander invariant) In the literature the ADO
invariant is seen also as a polynomial if one replaces
$\xi_{\mathcal{N}}^{\lambda_{i}}$ by variables $x_{i}$.

For the next part, we change the ring of coefficients, and see the image of
the coloured Alexander invariants in the cyclotomic field
$\mathbb{Q}({\xi_{\mathcal{N}}})$ in variables $x_{1}^{\pm 1},...,x_{l}^{\pm
1}$ (which are associated to the strands of the link), as below:

$\Phi^{\mathcal{N}}(L)\in\Lambda_{\mathcal{N}}=\mathbb{Q}({\xi_{\mathcal{N}}})(x^{2\mathcal{N}}_{1}-1,...,x^{2\mathcal{N}}_{l}-1)[x_{1}^{\pm
1},...,x_{l}^{\pm 1}].$ (8.2)

######  Definition 8.3 (Specialisation for the universal invariant)

We change the ring of coefficients as above, and let us define the associated
globalised specialisation of coefficients as below:

$\psi_{\mathcal{N}}:\mathbb{L}=\mathbb{Z}[u_{1}^{\pm 1},...,u_{l}^{\pm
1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y^{\pm 1},d^{\pm
1}]\rightarrow\Lambda_{\mathcal{N}}$
$\begin{cases}&\psi_{\mathcal{N}}(u_{i})=x_{i}^{1-\mathcal{N}},\forall
i\in\\{1,...,l\\}\\\ &\psi_{\mathcal{N}}(d)=\xi_{2\mathcal{N}}^{-1}\\\
&\psi_{\mathcal{N}}(y)={\text{qd}}_{\mathcal{N}}(x_{C(1)}):=\frac{x_{C(1)}-x_{C(1)}^{-1}}{x^{\mathcal{N}}_{C(1)}-x_{C(1)}^{-\mathcal{N}}}.\end{cases}$
(8.3)

###  8.2 Graded intersection in the ring with integer coefficients

###### Lemma 8.4

The graded intersection $A\Gamma^{\mathcal{N}}(\beta_{n})$ takes values in the
Laurent polynomial ring with integer coefficients: $\mathbb{Z}[u_{1}^{\pm
1},...,u_{l}^{\pm 1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y,d^{\pm 1}]$.

###### Proof.

We remind the formula for this intersection form:

$\displaystyle A\Gamma^{\mathcal{N}}(\beta_{n})$
$\displaystyle:=\prod_{i=1}^{l}u_{i}^{\left(f_{i}-\sum_{j\neq{i}}lk_{i,j}\right)}\cdot\prod_{i=2}^{n}u^{-1}_{C(i)}\cdot$
(8.4) $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\sum_{\bar{i}\in\\{\bar{0},\dots,\overline{\mathcal{N}-1}\\}}\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle.$

Since the homology classes $\mathscr{F}_{\bar{i},\mathcal{N}}$ and
$\mathscr{L}_{\bar{i},\mathcal{N}}$ belong to the homology groups
$H^{\bar{\mathcal{N}}}_{n,2+(n-1)(\mathcal{N}-1)}$ and
$H^{\bar{\mathcal{N}},\partial}_{n,2+(n-1)(\mathcal{N}-1)}$ that are modules
over $\mathbb{C}[x_{1}^{\pm 1},...,x_{l}^{\pm 1},y,d^{\pm 1}]$, we have that a
priori

$A\Gamma^{\mathcal{N}}(\beta_{n})\in\mathbb{C}[u_{1}^{\pm 1},...,u_{l}^{\pm
1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y,d^{\pm 1}].$

We will show that for our particular homology classes, we have the property
that actually their intersection pairing has all coefficients that are
integers:

$\left\langle(\beta_{n}\cup{\mathbb{I}}_{n+1})\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathscr{F}_{\bar{i},\mathcal{N}}},{\color[rgb]{0,0.58984375,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.58984375,0}\mathscr{L}_{\bar{i},\mathcal{N}}}\right\rangle\in\mathbb{Z}[x_{1}^{\pm
1},...,x_{l}^{\pm 1},y,d^{\pm 1}].$ (8.5)

As we have seen, this intersection is computed from the set of intersection
points between the geometric supports, graded by the local system. Now, the
only contribution of the local system that could give complex coefficients
would come from loops that have non-trivial winding number around the
$p$-punctures from the right hand side of the disc. However, the intersection
points between our geometric supports for the classes from above have
associated loops that do not wind around punctures from the right hand side of
the disc. This shows that (8.5) holds and so our intersection form has integer
coefficients:

$A\Gamma^{\mathcal{N}}(\beta_{n})\in\mathbb{Z}[u_{1}^{\pm 1},...,u_{l}^{\pm
1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y,d^{\pm 1}].$

The same argument applies also for the intersection form:

$J{\Gamma}^{\bar{N}}(\beta_{n})\in\mathbb{Z}[u_{1}^{\pm 1},...,u_{l}^{\pm
1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y,d^{\pm 1}].$

∎

###  8.3 Product up to a finite level

######  Definition 8.5 (Product up to level $\mathcal{N}$)

For a fixed level $\mathcal{N}$, let us define the product of the rings for
smaller levels, as:

$\tilde{\Lambda}_{\mathcal{N}}:=\prod_{\mathcal{M}\leqslant\mathcal{N}}\Lambda_{\mathcal{M}}$
(8.6)

and
$p^{\mathcal{N}}_{\mathcal{M}}:\tilde{\Lambda}_{\mathcal{N}}\rightarrow\Lambda_{\mathcal{M}}$
the projection onto the $\mathcal{M}^{th}$ component.

Also, let us denote by
$\tilde{\psi}_{\mathcal{N}}:\mathbb{L}\rightarrow\tilde{\Lambda}_{\mathcal{N}}$
the product of specialisations:

$\tilde{\psi}_{\mathcal{N}}:=\prod_{\mathcal{M}\leqslant\mathcal{N}}\psi_{\mathcal{M}}.$
(8.7)

######  Definition 8.6 (Coloured Alexander invariants up to a fixed level)

We denote the product of the invariants up to level $\mathcal{N}$ as below:

$\tilde{\Phi}^{\mathcal{N}}(L):=\prod_{\mathcal{M}\leqslant\mathcal{N}}\Phi^{\mathcal{M}}(L)\in\tilde{\Lambda}_{\mathcal{N}}.$
(8.8)

We will construct our universal ring using this sequence of morphisms
$\\{\tilde{\psi}_{\mathcal{N}}\mid\mathcal{N}\in\mathbb{N}\\}$. More
specifically, we will use the sequence of kernels associated to these maps, as
follows.

######  Definition 8.7 (Kernel and quotient rings)

Let us denote by:

$\tilde{I}_{\mathcal{N}}:=\text{Ker}\left(\tilde{\psi}_{\mathcal{N}}\right)\subseteq\mathbb{L}.$
(8.9)

Associated to it, we denote the quotient ring by:

$\bar{\mathbb{L}}_{\mathcal{N}}:=\mathbb{L}/\tilde{I}_{\mathcal{N}}.$ (8.10)

Also, we define the product of the rings for smaller levels, as:

$\tilde{\Lambda}_{\mathcal{N}}:=\prod_{\mathcal{M}\leqslant\mathcal{N}}\Lambda_{\mathcal{M}}$
(8.11)

This means that we have the following diagram with quotient maps, which we
denote as below:

$\mathbb{L}$$\tilde{\Lambda}_{\mathcal{N}}$$\bar{\mathbb{L}}_{\mathcal{N}}$$\hat{\mathbb{L}}$$A\Gamma^{\mathcal{N}}(\beta_{n})\in$${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\overline{A\Gamma}^{\mathcal{N}}(L)}\in$$\Lambda_{\mathcal{N}}$$g_{\mathcal{N}}$$\tilde{\psi}_{\mathcal{N}}$$\bar{\bar{\psi}}_{\mathcal{N}}$$\bar{\psi}_{\mathcal{N}}$$p^{\mathcal{N}}_{\mathcal{N}}$$\psi^{u}_{\mathcal{N}}$
Figure 8.1: Quotient rings

###### Remark 8.8

By the definition of the product rings $\tilde{\Lambda}_{\mathcal{N}}$, we see
that we have a sequence of nested ideals:

$\cdots\supseteq\tilde{I}_{\mathcal{N}}\supseteq\tilde{I}_{\mathcal{N}+1}\supseteq\cdots$
(8.12)

######  Definition 8.9 (Sequence of quotient rings)

In this manner, we obtain a sequence of associated quotient rings, with maps
between them:

$\displaystyle l_{\mathcal{N}}\hskip 28.45274ptl_{\mathcal{N}+1}$ (8.13)
$\displaystyle\cdots\bar{\mathbb{L}}_{\mathcal{N}}\leftarrow\bar{\mathbb{L}}_{\mathcal{N}+1}\leftarrow\cdots$

######  Definition 8.10 (Limit ring)

We define the projective limit of this sequence of rings and denote it as
follows:

$\hat{\mathbb{L}}:=\underset{\longleftarrow}{\mathrm{lim}}\
\bar{\mathbb{L}}_{\mathcal{N}}.$ (8.14)

###### Remark 8.11

Then, we have induced maps:

$\bar{\psi}_{\mathcal{N}}:\hat{\mathbb{L}}\rightarrow\tilde{\Lambda}_{\mathcal{N}}.$
(8.15)

Further, composing with the projection onto the $\mathcal{N}^{th}$ component
we obtain a map, which we denote:

$\psi^{u}_{\mathcal{N}}:\hat{\mathbb{L}}\rightarrow\Lambda_{\mathcal{N}},$
(8.16)

defined as

$\psi^{u}_{\mathcal{N}}:=p^{\mathcal{N}}_{\mathcal{N}}\circ\bar{\psi}_{\mathcal{N}}.$

######  Definition 8.12 (Projection maps for the ADO coefficient rings)

We have projection maps which go from the coefficient rings at consecutive
levels, as follows:

$\text{pr}_{\mathcal{N}}:\tilde{\Lambda}_{\mathcal{N}+1}\rightarrow\tilde{\Lambda}_{\mathcal{N}}.$
(8.17)

Now we are going to define step by step the universal invariant coming from
our intersection pairings.

###  8.4 Link invariants at level $\mathcal{N}$

First, we define a set of link invariants that will be used to build the
globalised coloured Alexander invariant. We recall that we have the graded
intersection:

$A\Gamma^{\mathcal{N}}(\beta_{n})\in\mathbb{L}$

and it recovers the coloured Alexander invariants, as below:

$\psi_{\mathcal{N}}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)=\Phi^{\mathcal{N}}(L).$
(8.18)

Taking the product component-wise, we conclude that:

$\tilde{\psi}_{\mathcal{N}}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)=\tilde{\Phi}^{\mathcal{N}}(L).$
(8.19)

######  Definition 8.13 (Level $\mathcal{N}$ link invariant)

Let us consider the intersection form obtained from
$A\Gamma^{\mathcal{N}}(\beta_{n})$ by taking the quotient through the
projection $g_{\mathcal{N}}$ (as in Figure 8.1):

$\overline{A\Gamma}^{\mathcal{N}}(L)\in\bar{\mathbb{L}}_{\mathcal{N}}.$ (8.20)

######  Lemma 8.14 (Invariant)

The intersection $\overline{A\Gamma}^{\mathcal{N}}(L)$ is a well-defined link
invariant.

###### Proof.

Following relation (8.19), we have that:

$\tilde{\psi}_{\mathcal{N}}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)=\tilde{\Phi}^{\mathcal{N}}(L)$

for $L$ the link obtained as the closure of the braid $\beta_{n}$. On the
other hand, the projection $g_{\mathcal{N}}$ is defined via the kernel of the
map $\tilde{\psi}_{\mathcal{N}}$ and we have that

$\overline{A\Gamma}^{\mathcal{N}}(L)=g_{\mathcal{N}}(A\Gamma^{\mathcal{N}}(\beta_{n})).$
(8.21)

Putting these properties together, it follows that
$\overline{A\Gamma}^{\mathcal{N}}(L)$ is a link invariant.

∎

###  8.5 Definition of the universal ring and invariants

Now we are ready to define our universal invariant, which will be build from
the sequence of intersections up to level $\mathcal{N}$.

######  Definition 8.15 (Unification of all coloured Alexander link
invariants)

We define the projective limit of the graded intersection pairings
$\overline{A\Gamma}^{\mathcal{N}}(L)$ over all levels, and denote it as
follows:

${\large\hat{{\Huge{\Gamma}}}}(L):=\underset{\longleftarrow}{\mathrm{lim}}\
\overline{A\Gamma}^{\mathcal{N}}(L)\in\hat{\mathbb{L}}.$ (8.22)

######  Definition 8.16 (Limit ring of coefficients and limit invariant)

Now, let us consider the product of the rings where the coloured Alexander
invariants belong to, where we put no condition about the level:

$\hat{\Lambda}:=\prod_{\mathcal{M}\in\mathbb{N}}\Lambda_{\mathcal{M}}$ (8.23)

$\hat{\Phi}(L):=\prod_{\mathcal{M}\leqslant\mathcal{N}}\Phi^{\mathcal{M}}(L)\in\hat{\Lambda}.$
(8.24)

Using the definition from above, let us denote the projection map:

$\hat{p}_{\mathcal{M}}:\hat{\Lambda}\rightarrow\Lambda_{\mathcal{M}}.$ (8.25)

We remark that this means that projecting onto the $\mathcal{M}^{th}$
component we obtain the $\mathcal{M}^{th}$ coloured Alexander invariant:

$\hat{p}_{\mathcal{M}}(\hat{\Phi}(L))=\Phi^{\mathcal{M}}(L).$ (8.26)

######  Theorem 8.17 (Universal ADO link Invariant)

The limit of the graded intersections ${\large\hat{{\Huge{\Gamma}}}}(L)$ is a
well-defined link invariant, that recovers all coloured Alexander invariants:

$\Phi^{\mathcal{N}}(L)={\large\hat{{\Huge{\Gamma}}}}(L)|_{\psi^{u}_{\mathcal{N}}}.$
(8.27)

###### Proof.

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(8.28) Figure 8.2: The universal ring and universal invariant as projective
limits

In order to have a well-defined limit, we should prove that:

$l_{\mathcal{N}}\left(\overline{A\Gamma}^{\mathcal{N}+1}(L)\right)=\overline{A\Gamma}^{\mathcal{N}}(L).$
(8.29)

###### Remark 8.18

Following the definition of the projection maps for the coefficient rings from
relation (8.17), we have the property:

$\text{pr}_{\mathcal{N}}\left(\tilde{\Phi}^{\mathcal{N}+1}(L)\right)=\tilde{\Phi}^{\mathcal{N}}(L).$
(8.30)

###### Lemma 8.19

The link invariant at level $\mathcal{N}$ recovers all coloured Alexander
invariants with levels smaller than $\mathcal{N}$, which means that we have
the following relation:

$\bar{\bar{\psi}}_{\mathcal{N}}\left(\overline{A\Gamma}^{\mathcal{N}}(L)\right)=\tilde{\Phi}^{\mathcal{N}}(L).$
(8.31)

###### Proof.

This property will follow from Theorem 1.12 together with the definition of
the quotient maps, as we will see below. We notice that it is enough to prove
that this relation holds when composed with the set of projections
$p^{\mathcal{N}}_{\mathcal{M}}$ for all $\mathcal{M}\leqslant\mathcal{N}$. So,
we want to show:

$p^{\mathcal{N}}_{\mathcal{M}}\circ\bar{\bar{\psi}}_{\mathcal{N}}\left(\overline{A\Gamma}^{\mathcal{N}}(L)\right)=p^{\mathcal{N}}_{\mathcal{M}}\circ\tilde{\Phi}^{\mathcal{N}}(L)\left(=\Phi^{\mathcal{N}}(L)\right),\
\ \ \forall\mathcal{M}\leqslant\mathcal{N}.$ (8.32)

By construction we have that:

$\overline{A\Gamma}^{\mathcal{N}}(L)=g_{\mathcal{N}}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right).$
(8.33)

This means that we want to show:

$p^{\mathcal{N}}_{\mathcal{M}}\circ\bar{\bar{\psi}}_{\mathcal{N}}\circ
g_{\mathcal{N}}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)=\Phi^{\mathcal{N}}(L),\
\ \ \forall\mathcal{M}\leqslant\mathcal{N}.$ (8.34)

On the other hand, we have: $\bar{\bar{\psi}}_{\mathcal{N}}\circ
g_{\mathcal{N}}=\tilde{\psi}_{\mathcal{N}}$ and
$p^{\mathcal{N}}_{\mathcal{M}}\circ\tilde{\psi}_{\mathcal{N}}=\psi_{\mathcal{M}}$.
This means that relation (8.34) is equivalent to:

$\psi_{\mathcal{M}}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)=\Phi^{\mathcal{N}}(L),\
\ \ \forall\mathcal{M}\leqslant\mathcal{N}.$ (8.35)

This is precisely the statement of the unification result up to level
$\mathcal{N}$ from the Unification Theorem 1.12, which concludes the proof of
this Lemma. ∎

######  Lemma 8.20 (Well behaviour of the intersection pairings when changing
the level)

When changing the level from $\mathcal{N}$ to $\mathcal{N}+1$, the
intersection pairings recover one another through the induced map
$l_{\mathcal{N}}$ as below:

$l_{\mathcal{N}}\left(\overline{A\Gamma}^{\mathcal{N}+1}(L)\right)=\overline{A\Gamma}^{\mathcal{N}}(L).$
(8.36)

###### Proof.

From the construction of quotienting by the kernel, we have that the maps
$\bar{\bar{\psi}}_{\mathcal{N}}$ and $\bar{\bar{\psi}}_{\mathcal{N}+1}$ are
injective. Following Lemma 8.19, we have the property that:

$\bar{\bar{\psi}}_{\mathcal{N}}\left(\overline{A\Gamma}^{\mathcal{N}}(L)\right)=\tilde{\Phi}^{\mathcal{N}}(L).$

This means that our statement is equivalent to:

$\text{pr}_{\mathcal{N}}\left(\tilde{\Phi}^{\mathcal{N}+1}(L)\right)=\tilde{\Phi}^{\mathcal{N}}(L).$

This is precisely the property from Remark 8.18, and so the statement holds. ∎

As a conclusion after this discussion, we have that:

$l_{\mathcal{N}}\left(\overline{A\Gamma}^{\mathcal{N}+1}(L)\right)=\overline{A\Gamma}^{\mathcal{N}}(L).$
(8.37)

######  Lemma 8.21 (Commutation of the squares from the diagram)

All squares associated to consecutive levels $\mathcal{N}$ and $\mathcal{N}+1$
commute.

###### Proof.

This comes from the commutativity when we project onto each factor and by the
definition of the quotient maps. ∎

This shows that there exists a well-defined ring homomorphism between the
limits, which we denote as below:

$\hat{\psi}:\hat{\mathbb{L}}\rightarrow\hat{\Lambda}.$ (8.38)

This shows that there exists a well-defined element in the projective limit,
which is a link invariant:

${\large\hat{{\Huge{\Gamma}}}}(L)\in\hat{\mathbb{L}}.$

The last part that we need to prove is that this universal invariant recoveres
all ADO invariants, for any level, as stated in relation (9.27):

$\Phi^{\mathcal{N}}(L)={\large\hat{{\Huge{\Gamma}}}}(L)|_{\psi^{u}_{\mathcal{N}}}.$
(8.39)

Following the commutativity of the squares at all levels, we have that:

$\hat{\psi}\left({\large\hat{{\Huge{\Gamma}}}}(L)\right)=\hat{\Phi}(L).$
(8.40)

Also, using the commutativity of the diagrams together with Definition 8.10 we
obtain that:

$\hat{p}_{\mathcal{N}}\circ\hat{\psi}=p^{\mathcal{N}}_{\mathcal{N}}\circ\bar{\psi}_{\mathcal{N}}=\psi^{u}_{\mathcal{N}}.$
(8.41)

Now, following relation (8.26), we have that:

$\hat{p}_{\mathcal{N}}(\hat{\Phi}(L))=\Phi^{\mathcal{N}}(L).$ (8.42)

Using the previous three relations, we conclude that:

$\displaystyle{\large\hat{{\Huge{\Gamma}}}}(L)|_{\psi^{u}_{\mathcal{N}}}$
$\displaystyle=\psi^{u}_{\mathcal{N}}\left({\large\hat{{\Huge{\Gamma}}}}(L)\right)=$
(8.43)
$\displaystyle=\hat{p}_{\mathcal{N}}\circ\hat{\psi}\left({\large\hat{{\Huge{\Gamma}}}}(L)\right)=$
$\displaystyle=\hat{p}_{\mathcal{N}}(\hat{\Phi}(L))=$
$\displaystyle=\Phi^{\mathcal{N}}(L),\ \ \ \
\forall\mathcal{N}\in\mathbb{N},\mathcal{N}\geqslant 2.$

This concludes our construction and shows that we have a universal ADO
invariant for links ${\large\hat{{\Huge{\Gamma}}}}(L)$, constructed from
graded intersections in configuration spaces, which recovers the coloured
Alexander invariants at all levels, through specialisation of coefficients. ∎

##  9 Universal refined intersection in a module over the Habiro ring

Up to this moment, we have our universal invariant
${\large\hat{{\Huge{\Gamma}}}}(L)$ in the ring $\hat{\mathbb{L}}$ which is a
projective limit of quotient rings of

$\mathbb{L}=\mathbb{Z}[u_{1}^{\pm 1},...,u_{l}^{\pm 1},x_{1}^{\pm
1},...,x_{l}^{\pm 1},y^{\pm 1},d^{\pm 1}].$

More precisely, we have defined the sequence of nested ideals in $\mathbb{L}$:
$\cdots\supseteq\tilde{I}_{\mathcal{N}}\supseteq\tilde{I}_{\mathcal{N}+1}\supseteq\cdots$
leading to a sequence of associated quotient rings:

$\displaystyle\phantom{A}\ l_{\mathcal{N}}\hskip 28.45274ptl_{\mathcal{N}+1}$
(9.1)
$\displaystyle\cdots\bar{\mathbb{L}}_{\mathcal{N}}\leftarrow\bar{\mathbb{L}}_{\mathcal{N}+1}\leftarrow\cdots\
\ \ \ \ \ \ \ \ \ \ \
\hat{\mathbb{L}}:=\underset{\longleftarrow}{\mathrm{lim}}\
\bar{\mathbb{L}}_{\mathcal{N}}$

Then, we defined the projective limit of this sequence of rings, and denoted
it as below:

$\hat{\mathbb{L}}:=\underset{\longleftarrow}{\mathrm{lim}}\
\bar{\mathbb{L}}_{\mathcal{N}}.$ (9.2)

Now, the question that we pose concerns the relation between our rings and the
ring introduced by Habiro in [15], [16]. Habiro’s ring is a projective limit
of quotients of a Laurent polynomial ring in two variables by a sequence of
ideals generated by cyclotomic polynomials in one of these variables.

In the next part we aim to do a similar construction in our situation,
considering quotients by cyclotomic polynomials in one of our variables. We
recall that we have the specialisation of coefficients from Definition 2.19:

$\psi_{\mathcal{N}}:\mathbb{L}=\mathbb{Z}[u_{1}^{\pm 1},...,u_{l}^{\pm
1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y^{\pm 1},d^{\pm
1}]\rightarrow\Lambda_{\mathcal{N}}$
$\begin{cases}&\psi_{\mathcal{N}}(u_{i})=x_{i}^{1-\mathcal{N}},\forall
i\in\\{1,...,l\\}\\\ &\psi_{\mathcal{N}}(d)=\xi_{2\mathcal{N}}^{-1}\\\
&\psi_{\mathcal{N}}(y)={\text{qd}}_{\mathcal{N}}(x_{C(1)}):=\frac{x_{C(1)}-x_{C(1)}^{-1}}{x^{\mathcal{N}}_{C(1)}-x_{C(1)}^{-\mathcal{N}}}.\end{cases}$
(9.3)

In this setting, we have three main parameters, as below:

  * •

$x$-which remains as a variable

  * •

$d$-which gets specialised to a root of unity

  * •

$u$-which controls the pivotal part of the construction and gets evaluated to
a power of the variable $x$.

Question Can we construct a ring suitable for our setting– where we have our
graded intersection pairings– which is related to Habiro’s ring?

Idea Our plan is to start from $\hat{\mathbb{L}}$ and consider quotients by
ideals generated by cyclotomic polynomials in $d$. This will be an “extended
Habiro ring”, having two extra free variables $u$ and $y$. Further, we will
quotient by relations suggested by the specialisation $\psi_{\mathcal{N}}$,
which provide relations involving $u_{i}$ and $x_{i}$ on one hand and $y$ and
$x_{i}$ on the other. We will make this precise in the following part.

######  Definition 9.1 (Cyclotomic quotients in powers of $d$)

Let us consider the following ideals:

$J^{H}_{\mathcal{N}}:=\langle(j)_{d^{4}}!\mid
j\geqslant\mathcal{N}\rangle\subseteq\mathbb{L}.$ (9.4)

We remark that this ideal is generated by cyclotomic polynomials (up to a
change of variables):

$J^{H}_{\mathcal{N}}=\left((\mathcal{N})_{d^{4}}!\right)\subseteq\mathbb{L}.$
(9.5)

These form a nested sequence of ideals, which give us a sequence of quotient
rings:

$\mathbb{L}/J^{H}_{\mathcal{N}}=\mathbb{Z}[u_{1}^{\pm 1},...,u_{l}^{\pm
1},x_{1}^{\pm 1},...,x_{l}^{\pm 1},y^{\pm 1},d^{\pm
1}]/\left((\mathcal{N})_{d^{4}}!\right).$ (9.6)

More precisely, we have the following sequence:

$\displaystyle l^{H}_{\mathcal{N}}\hskip 28.45274ptl^{H}_{\mathcal{N}+1}$
(9.7)
$\displaystyle\cdots\mathbb{L}/J^{H}_{\mathcal{N}}\leftarrow\mathbb{L}/J^{H}_{\mathcal{N}+1}\leftarrow\cdots.$

######  Definition 9.2 (Extended Habiro ring)

We consider the projective limit of this sequence of rings and call it
“extended Habiro ring”:

$\mathbb{L}^{H}:=\underset{\longleftarrow}{\mathrm{lim}}\
\mathbb{L}/J^{H}_{\mathcal{N}}.$ (9.8)

This ring is (up to a change of variables) the Habiro ring with respect to the
variables $x_{i}$ and $d$, where we add $\mathbb{Z}[u_{1}^{\pm
1},...,u_{l}^{\pm 1},y^{\pm 1}]$.

This ring is too large to have an invariant coming from our intersections
$A\Gamma^{\mathcal{N}}(\beta_{n})$. For this reason, we will quotient further,
by relations involving the variables $u_{i}$ and $y$, as follows.

######  Definition 9.3 (Refined cyclotomic ideals)

Let us consider the ideals $J_{\mathcal{N}}^{R}$ obtained from
$J_{\mathcal{N}}^{H}$ where we add extra relations, as follows:

$J_{\mathcal{N}}^{R}:=J^{H}_{\mathcal{N}}+\langle
u_{i}-x_{i}^{1-\mathcal{N}},y\left(x^{j}_{C(1)}-x^{-j}_{C(1)}\right)-(x_{C(1)}-x^{-1}_{C(1)})\mid
j\geqslant\mathcal{N},1\leqslant i\leqslant l\rangle\subseteq\mathbb{L}.$
(9.9)

These give also a sequence of nested ideals, which in turn lead to a sequence
of quotient rings:

$\displaystyle l^{R}_{\mathcal{N}}\hskip 28.45274ptl^{R}_{\mathcal{N}+1}$
(9.10)
$\displaystyle\cdots\mathbb{L}/J^{R}_{\mathcal{N}}\leftarrow\mathbb{L}/J^{R}_{\mathcal{N}+1}\leftarrow\cdots.$

###### Definition 9.4

We consider the projective limit of this sequence of rings:

$\mathbb{L}^{R}:=\underset{\longleftarrow}{\mathrm{lim}}\
\mathbb{L}/J^{R}_{\mathcal{N}}.$ (9.11)

We will refer to this ring as “refined universal ring”.

We notice that these ideals are included in the kernel ideals that we used for
the construction of our universal ADO invariant, as follows:

$J^{H}_{\mathcal{N}}\subset
J^{R}_{\mathcal{N}}\subset\tilde{I}_{\mathcal{N}}.$ (9.12)

This induces maps at the level of quotient rings, which we denote as below:

$\displaystyle p_{\mathcal{N}}^{H,R}\ \ \ \ \ \ \ \ \ p_{\mathcal{N}}^{R,U}$
(9.13) $\displaystyle\mathbb{L}/J^{H}_{\mathcal{N}}\ \ \rightarrow\ \
\mathbb{L}/J^{R}_{\mathcal{N}}\ \ \rightarrow\ \
\mathbb{L}/\tilde{I}_{\mathcal{N}}.$

###  9.1 Refined graded intersections invariants at level $\mathcal{N}$

Now, let us define a set of graded intersections in these refined rings. We
recall that for a fixed $\mathcal{N}$, we have constructed our intersection:

$A\Gamma^{\mathcal{N}}(\beta_{n})\in\mathbb{L}.$

Then we proved that it recovers all the coloured Alexander invariants of
colours $\mathcal{M}$ bounded by $\mathcal{N}$, as below:

$\psi_{\mathcal{M}}\left(A\Gamma^{\mathcal{N}}(\beta_{n})\right)=\Phi^{\mathcal{M}}(L).$
(9.14)

######  Definition 9.5 (Level $\mathcal{N}$ extended intersection)

We define $\overline{A\Gamma}^{H,\mathcal{N}}(\beta_{n})$ to be the image of
$A\Gamma^{\mathcal{N}}(\beta_{n})$ in the quotient of $\mathbb{L}$ by the
ideal $J^{H}_{\mathcal{N}}$:

$\overline{A\Gamma}^{H,\mathcal{N}}(\beta_{n})\in\mathbb{L}/J^{H}_{\mathcal{N}}.$
(9.15)

######  Definition 9.6 (Level $\mathcal{N}$ refined intersection)

We also consider the intersection given by $A\Gamma^{\mathcal{N}}(\beta_{n})$
via the quotient of $\mathbb{L}$ associated to the ideal
$J^{R}_{\mathcal{N}}$:

$\overline{A\Gamma}^{R,\mathcal{N}}(\beta_{n})\in\mathbb{L}/J^{R}_{\mathcal{N}}.$
(9.16)

###### Lemma 9.7

The sequences of quotients of graded intersections
$\\{\overline{A\Gamma}^{R,\mathcal{N}}(\beta_{n})\\}$ and
$\\{\overline{A\Gamma}^{H,\mathcal{N}}(\beta_{n})\\}$ behave well with respect
to the projection maps, and we have:

$l^{H}_{\mathcal{N}}(\overline{A\Gamma}^{H,\mathcal{N}+1}(\beta_{n}))=\overline{A\Gamma}^{H,\mathcal{N}}(\beta_{n}).$
(9.17)
$l^{R}_{\mathcal{N}}(\overline{A\Gamma}^{R,\mathcal{N}+1}(\beta_{n}))=\overline{A\Gamma}^{R,\mathcal{N}}(\beta_{n}).$
(9.18)

###### Proof.

We will use the fact that both intersections
$\overline{A\Gamma}^{H,\mathcal{N}+1}(\beta_{n})$ and
$\overline{A\Gamma}^{R,\mathcal{N}+1}(\beta_{n})$ are quotients of
$A\Gamma^{\mathcal{N}+1}(\beta_{n})$, for all $\mathcal{N}\geqslant 2$.

In order to prove the good behaviour with respect to the projection maps, it
is enough to prove that:

$A\Gamma^{\mathcal{N}+1}(\beta_{n})-A\Gamma^{\mathcal{N}}(\beta_{n})$

belongs to the ideals $J^{H}_{\mathcal{N}}$ and $J^{R}_{\mathcal{N}}$. Now,
since

$J^{H}_{\mathcal{N}}\subset J^{R}_{\mathcal{N}}$

it is enough to prove the following relation:

$A\Gamma^{\mathcal{N}+1}(\beta_{n})-A\Gamma^{\mathcal{N}}(\beta_{n})\in
J^{H}_{\mathcal{N}}.$ (9.19)

Using the definition of these graded intersections, we remark that the
difference between these pairings comes from the extra intersections that
involve homology classes $\mathscr{F}_{\bar{i},\mathcal{N}+1}$ associated to
an index $\bar{i}$ which has at least one component
$i_{k}\geqslant\mathcal{N}$. The next part of the proof will follow in a
similar manner as the proof of Lemma 7.3.

There the subtle part of the argument was related to the choice of our
homology classes and the fact that we have the relation provided by
Proposition 7.19:

$\mathscr{F}^{\prime}_{\bar{i},\mathcal{N}}=\prod_{j=1}^{n-1}(i_{j})_{d^{4}}!\cdot(\mathcal{N}-1-i_{j})_{d^{4}}!\
\mathscr{F}^{c}_{\bar{i},\mathcal{N}}.$ (9.20)

The key point is that the coefficient
$\prod_{j=1}^{n-1}(i_{j})_{d^{4}}!\cdot(\mathcal{N}-1-i_{j})_{d^{4}}!$ from
above contains as factor:

$(i_{k})_{d^{4}}!.$

Now, since $i_{k}\geqslant\mathcal{N}$, it means that

$(i_{k})_{d^{4}}!\in J^{H}_{\mathcal{N}}.$

This shows, in turn, that every element associated to a multi-index $\bar{i}$
that contributes to the difference
$A\Gamma^{\mathcal{N}+1}(\beta_{n})-A\Gamma^{\mathcal{N}}(\beta_{n})$ belongs
to $J^{H}_{\mathcal{N}}$. So, we have that:

$A\Gamma^{\mathcal{N}+1}(\beta_{n})-A\Gamma^{\mathcal{N}}(\beta_{n})\in
J^{H}_{\mathcal{N}}$

and this concludes the proof of this globalisation with respect to the level.
∎

######  Definition 9.8 (Universal refined invariants)

We define the projective limits of the graded extended / refined intersection
pairings $\overline{A\Gamma}^{H,\mathcal{N}}(\beta_{n})$ and
$\overline{A\Gamma}^{R,\mathcal{N}}(\beta_{n})$ over all levels, and denote
them as follows:

${\large{\Huge{\Gamma^{H}}}}(\beta_{n}):=\underset{\longleftarrow}{\mathrm{lim}}\