# A diagrammatic computation of abelian link invariants

David Cimasoni  David Cimasoni – Section de mathématiques, Université de
Genève, Suisse<EMAIL_ADDRESS>,  Livio Ferretti  Livio Ferretti –
Section de mathématiques, Université de Genève, Suisse
<EMAIL_ADDRESS>and  Jessica Liu  Jessica Liu – Department of
Mathematics, University of Toronto, Canada<EMAIL_ADDRESS>
###### Abstract.

We show how the multivariable signature and Alexander polynomial of a colored
link can be computed from a single symmetric matrix naturally defined from a
colored link diagram. In the case of a single variable, it coincides with the
matrix introduced by Kashaev [13], which was recently proven to compute the
Levine-Tristram signature and the Alexander polynomial of oriented links [16,
3]. As a corollary, we obtain a multivariable extension of Kauffman’s
determinantal model of the Alexander polynomial [15], recovering a result of
Zibrowius [18].

###### Key words and phrases:

Link diagrams, multivariable signature, multivariable Alexander polynomial

###### 1991 Mathematics Subject Classification:

57K10

##  1\. Introduction

As its title suggests, the aim of the present article is to give a way of
computing several classical link invariants directly from a diagram. Before
specifying these invariants, let us mention that this story is best told in
the context of colored links, that we now recall.

Given an integer $\mu>0$, a $\mu$-colored link is an oriented link $L\subset
S^{3}$ each of whose components is endowed with a color in $\\{1,\dots,\mu\\}$
in such a way that all these colors are used. Two colored links are isotopic
if they are related by an ambient isotopy which respects the orientation and
color of all components. Clearly, a $1$-colored link is just an oriented link,
while a $\mu$-component $\mu$-colored link is an oriented ordered link. A
$\mu$-colored link can be described by an oriented link diagram $D$ with
colored components, an object which we will refer to as a $\mu$-colored
diagram (see Figure 1 for an example). As usual, a crossing $v$ of $D$ is
naturally endowed with a sign that we denote by $\operatorname{sgn}(v)=\pm 1$,
see Figure 1. Finally, a crossing will be called monochromatic if the two
corresponding strands are of the same color, and bichromatic otherwise.

$L_{1}$$L_{2}$

$\bullet$

$\scriptstyle{1}$$\scriptstyle 2$$\scriptstyle 3$$\scriptstyle 4$$\scriptstyle
5$$a$$b$$c$$d$$e$$f$$g$ Figure 1. A $2$-colored diagram $D$ for a $2$-colored
link $L=L_{1}\cup L_{2}$. The crossings are labelled 1 through 5 and the
regions are labelled $a$ through $g$. Crossings $1$ to $4$ are positive and
bichromatic, while crossing $5$ is negative and monochromatic. The marked
point on $L_{1}$ will serve a further purpose.

The invariants we are interested in computing are the classical abelian
invariants of a $\mu$-colored link $L$, namely its multivariable signature and
nullity $\sigma_{L},\eta_{L}\colon(S^{1}\setminus\\{1\\})^{\mu}\to\mathbb{Z}$,
and its multivariable Alexander polynomial $\Delta_{L}$ in the normalized form
given by the Conway function $\nabla_{L}$. In the case $\mu=1$, these are the
well-known Levine-Tristram signature and nullity and Alexander-Conway
polynomial, without doubt among the most studied of link invariants. We refer
to Section 2 for the definition of these classical objects.

$a$$b$$c$$d$$j$$k$$\begin{array}[]{c|cccc}&a&b&c&d\\\ \hline\cr
a&x_{jk}&x_{j}&1&x_{k}\\\ b&x_{j}&2x_{j}x_{k}-x_{jk}&x_{k}&1\\\
c&1&x_{k}&x_{jk}&x_{j}\\\ d&x_{k}&1&x_{j}&2x_{j}x_{k}-x_{jk}\end{array}$
Figure 2. A crossing $v$ together with the corresponding $4\times 4$ minor of
$\tau_{v}(x)$. The incoming left strand is of color $j$, the incoming right
strand of color $k$, and the four adjacent regions are $a,b,c,$ and $d$.

As our main result will show, these invariants can all be computed from a
single symmetric matrix $\tau_{D}(x)$, whose coefficients are functions of
formal variables $x=\\{x_{j},x_{jk}\mid 1\leq j,k\leq\mu\\}$ indexed by
(unordered pairs of) colors. We now give its definition.

######  Definition 1.

Given a $\mu$-colored diagram $D$, let $\tau_{D}(x)$ be the symmetric matrix
with rows and columns indexed by the regions of $D$ defined by

$\tau_{D}(x)=\sum_{v}\frac{\operatorname{sgn}(v)}{\sqrt{1-x_{j}^{2}}\sqrt{1-x_{k}^{2}}}\tau_{v}(x)\,,$

where the sum is over all crossings of $D$, the indices
$j,k\in\\{1,\dots,\mu\\}$ are the (possibly identical) colors of the two
strands crossing at $v$, and the only non-vanishing coefficients of the matrix
$\tau_{v}(x)$ are given in Figure 2.

Also, we shall denote by $\widetilde{\tau}_{D}(x)$ the matrix obtained by
removing the two rows and columns corresponding to two adjacent regions of $D$
determined by a marked point on $D$. We will assume without loss of generality
that this point is on a strand of color $1$.

Note that if the regions $a,b,c,d$ around a crossing $v$ are not all distinct,
then one should add the corresponding rows and columns of $\tau_{v}(x)$. This
happens in the following example.

######  Example 1.

Consider the $2$-colored diagram $D$ illustrated in Figure 1. Ordering the
regions alphabetically, the corresponding matrix is given by

$\displaystyle\tau_{D}(x)=\frac{1}{\sqrt{1-x_{1}^{2}}\sqrt{1-x_{2}^{2}}}\left[\begin{array}[]{ccccccc}4(2x_{1}x_{2}-x_{12})&2x_{2}&2x_{1}&2x_{2}&4&0&2x_{1}\\\\[2.0pt]
2x_{2}&2x_{12}&1&0&2x_{1}&0&1\\\\[2.0pt]
2x_{1}&1&2x_{12}&1&2x_{2}&0&0\\\\[2.0pt]
2x_{2}&0&1&2x_{12}&2x_{1}&0&1\\\\[2.0pt]
4&2x_{1}&2x_{2}&2x_{1}&4(2x_{1}x_{2}-x_{12})&0&2x_{2}\\\\[2.0pt]
0&0&0&0&0&0&0\\\\[2.0pt] 2x_{1}&1&0&1&2x_{2}&0&2x_{12}\end{array}\right]$
$\displaystyle-\frac{1}{1-x_{1}^{2}}\left[\begin{array}[]{ccccccc}0&0&0&0&0&0&0\\\\[2.0pt]
0&0&0&0&0&0&0\\\\[2.0pt] 0&0&0&0&0&0&0\\\\[2.0pt] 0&0&0&0&0&0&0\\\\[2.0pt]
0&0&0&0&1&1&2x_{1}\\\\[2.0pt] 0&0&0&0&1&1&2x_{1}\\\\[2.0pt]
0&0&0&0&2x_{1}&2x_{1}&4x_{1}^{2}\end{array}\right]\,,$

where the first summand contains the contributions from the four (positive)
bichromatic crossings and the second summand is the contribution from the
(negative) monochromatic crossing.

Here is our main result.

######  Theorem 1.

Let $D$ be an arbitrary $\mu$-colored diagram for a $\mu$-colored link $L$.

  1. (i)

For any
$\omega=(\omega_{1},\dots,\omega_{\mu})\in(S^{1}\setminus\\{1\\})^{\mu}$, the
signature and nullity of $L$ are given by

$\sigma_{L}(\omega)=\textstyle{\frac{1}{2}}(\operatorname{sign}(\widetilde{\tau}_{D}(\omega))-w_{\mathrm{m}}(D))\quad\text{and}\quad\eta_{L}(\omega)=\textstyle{\frac{1}{2}}\operatorname{null}(\widetilde{\tau}_{D}(\omega))\,,$

where $w_{\mathrm{m}}(D)$ is the sum of the signs of all monochromatic
crossings of $D$, and $\tau_{D}(\omega)$ stands for the evaluation of
$\tau_{D}(x)$ at

$x_{j}=\operatorname{Re}(\omega_{j}^{1/2})\,,\quad
x_{jk}=\operatorname{Re}(\omega_{j}^{1/2}\omega_{k}^{1/2})\,.$

  2. (ii)

If $D$ is connected, then the Conway function of $L$ satisfies

$\nabla^{2}_{L}(t_{1},\dots,t_{\mu})=\frac{1}{(t_{1}-t_{1}^{-1})^{2}}\prod_{v}(-\operatorname{sgn}(v))\frac{t_{j}-t_{j}^{-1}}{2}\frac{t_{k}-t_{k}^{-1}}{2}\cdot\det(\widetilde{\tau}_{D}(t^{2}))\,,$

where the product is over all crossings of $D$, the indices $j,k$ are the
(possibly identical) colors of the two strands crossing at $v$, and
$\tau_{D}(t^{2})$ stands for the evaluation of $\tau_{D}(x)$ at

$x_{j}=\frac{t_{j}+t_{j}^{-1}}{2}\,,\quad
x_{jk}=\frac{t_{j}t_{k}+t_{j}^{-1}t_{k}^{-1}}{2}\,.$

Several remarks are in order.

######  Remarks.

  1. (i)

We need to fix one square root of each coordinate $\omega_{j}\in
S^{1}\setminus\\{1\\}$ of $\omega$: our choice is to take
$\omega_{j}=e^{i\theta_{j}}$ with $\theta_{j}\in(0,2\pi)$, and
$\omega_{j}^{1/2}=e^{i\theta_{j}/2}$. In other words, $\omega_{j}^{1/2}$
denotes the unique square root such that $\operatorname{Im}(\omega_{j}^{1/2})$
lies in $[0,1)$. In particular, we have
$(\bar{\omega}_{j})^{1/2}=-(\overline{\omega_{j}^{1/2}})$. Note that
$x_{j}^{2}\neq 1$, so $\tau_{D}(\omega)$ is a well-defined symmetric real
matrix.

  2. (ii)

In both points of Theorem 1, the evaluations of the formal variables satisfy
$x_{jj}=2x_{j}^{2}-1$ for all $j$. Therefore, if a crossing $v$ is
monochromatic, then the matrix $\tau_{v}(x)$ can be written in a simple form
which only depends on the single variable $x_{j}$.

  3. (iii)

In particular, if $\mu=1$, then $\tau_{D}(x)$ depends on a single variable.
This matrix was first introduced by Kashaev in [13], see discussion below.

######  Example 2.

Consider once again the $2$-colored link illustrated in Figure 1. From the
corresponding matrix $\tau_{D}(x)$ given in Example 1, we compute

$\footnotesize\widetilde{\tau}_{D}(t^{2})=\frac{-4}{(t_{1}-t_{1}^{-1})(t_{2}-t_{2}^{-1})}\left[\begin{array}[]{ccccc}2(t_{1}t_{2}^{-1}+t_{1}^{-1}t_{2})&t_{2}+t_{2}^{-1}&t_{1}+t_{1}^{-1}&t_{2}+t_{2}^{-1}&4\\\\[2.0pt]
t_{2}+t_{2}^{-1}&t_{1}t_{2}+t_{1}^{-1}t_{2}^{-1}&1&0&t_{1}+t_{1}^{-1}\\\\[2.0pt]
t_{1}+t_{1}^{-1}&1&t_{1}t_{2}+t_{1}^{-1}t_{2}^{-1}&1&t_{2}+t_{2}^{-1}\\\\[2.0pt]
t_{2}+t_{2}^{-1}&0&1&t_{1}t_{2}+t_{1}^{-1}t_{2}^{-1}&t_{1}+t_{1}^{-1}\\\\[2.0pt]
4&t_{1}+t_{1}^{-1}&t_{2}+t_{2}^{-1}&t_{1}+t_{1}^{-1}&2(t_{1}t_{2}^{-1}+t_{1}^{-1}t_{2})-\frac{t_{2}-t_{2}^{-1}}{t_{1}-t_{1}^{-1}}\end{array}\right]$

and obtain

$\det(\widetilde{\tau}_{D}(t^{2}))=-\left(\frac{-4}{(t_{1}-t_{1}^{-1})(t_{2}-t_{2}^{-1})}\right)^{5}(t_{1}-t^{-1}_{1})(t_{2}-t_{2}^{-1})(t_{1}t_{2}+t_{1}^{-1}t_{2}^{-1})^{2}\,.$

Evaluating at $t_{i}=\omega^{1/2}_{i}$, and recalling that the signature can
change value only when the determinant vanishes, we compute
$\sigma_{L}(\omega_{1},\omega_{2})=-\operatorname{sign}\left[\operatorname{Re}((1-\omega_{1})(1-\omega_{2}))\right]$,
$\eta_{L}(\omega_{1},\omega_{2})=1$ if $\omega_{1}\omega_{2}=-1$ and
$\eta_{L}(\omega_{1},\omega_{2})=0$ otherwise. Finally, we also get
$\pm\nabla_{L}(t_{1},t_{2})=t_{1}t_{2}+t_{1}^{-1}t_{2}^{-1}$.

The first point of this theorem provides a practical new way of computing
multivariable signatures, but it also yields much simpler proofs of known
properties of this invariant. For example, consider the following situation:
let $L^{\prime}$ be the $\mu$-colored link obtained from a $(\mu+1)$-colored
link $L$ by identifying the colors $\mu$ and $\mu+1$; then, for all
$(\omega_{1},\dots,\omega_{\mu})\in(S^{1}\setminus\\{1\\})^{\mu}$, we have the
equality

$\sigma_{L^{\prime}}(\omega_{1},\dots,\omega_{\mu})=\sigma_{L}(\omega_{1},\dots,\omega_{\mu},\omega_{\mu})-\sum\operatorname{lk}(K_{\mu},K_{\mu+1})\,,$

the sum being over all components $K_{\mu}\subset L$ of color $\mu$ and all
components $K_{\mu+1}\subset L$ of color $\mu+1$. The original proof of this
fact is rather tedious, see [4, Proposition 2.5]. It is an amusing exercise to
check that this fact immediately follows from Theorem 1.

The second point of this theorem implies a corollary that we now present.
Given a connected colored diagram $D$, let $K_{D}$ be the matrix whose rows
are indexed by the crossings of $D$, whose columns are indexed by the regions
of $D$, and whose coefficients are defined by the label of the corners in
Figure 3. (If a region abuts a corner from two sides, then the corresponding
labels should be added.) Finally, let $\widetilde{K}_{D}$ denote the square
matrix obtained from $K_{D}$ by removing two columns corresponding to two
adjacent regions (separated by a strand of color $1$).

$\left(t_{j}^{1/2}t_{k}^{1/2}\right)^{s}$$\left(t_{j}^{1/2}t_{k}^{-1/2}\right)^{s}$$\left(t_{j}^{-1/2}t_{k}^{-1/2}\right)^{s}$$\left(t_{j}^{-1/2}t_{k}^{1/2}\right)^{s}$$j$$k$
Figure 3. The labels in the definition of $K_{D}$ around a vertex $v$ with
$s=\operatorname{sgn}(v)$.

######  Corollary 1.

If $D$ is a connected diagram for a colored link $L$, then

$\det\widetilde{K}_{D}=\pm(t_{1}-t_{1}^{-1})\nabla_{L}(t_{1},\dots,t_{\mu})\,.$

######  Example 3.

Consider one last time the $2$-colored link illustrated in Figure 1. The
matrix $K_{D}$ is given by

$\left[\begin{array}[]{ccccccc}t_{1}^{-1/2}t_{2}^{1/2}&t_{1}^{1/2}t_{2}^{1/2}&0&0&t_{1}^{1/2}t_{2}^{-1/2}&0&t_{1}^{-1/2}t_{2}^{-1/2}\\\\[3.0pt]
t_{1}^{1/2}t_{2}^{-1/2}&t_{1}^{-1/2}t_{2}^{-1/2}&t_{1}^{1/2}t_{2}^{1/2}&0&t_{1}^{-1/2}t_{2}^{1/2}&0&0\\\\[3.0pt]
t_{1}^{-1/2}t_{2}^{1/2}&0&t_{1}^{-1/2}t_{2}^{-1/2}&t_{1}^{1/2}t_{2}^{1/2}&t_{1}^{1/2}t_{2}^{-1/2}&0&0\\\\[3.0pt]
t_{1}^{1/2}t_{2}^{-1/2}&0&0&t_{1}^{-1/2}t_{2}^{-1/2}&t_{1}^{-1/2}t_{2}^{1/2}&0&t_{1}^{1/2}t_{2}^{1/2}\\\\[3.0pt]
0&0&0&0&1&1&t_{1}+t_{1}^{-1}\end{array}\right]\,,$

from which we compute once again
$\pm\nabla_{L}(t_{1},t_{2})=t_{1}t_{2}+t_{1}^{-1}t_{2}^{-1}$.

Let us now put our results in the context of the preexisting literature.

In 2018, Kashaev [13] defined the matrix $\tau_{D}(x)$ in the case $\mu=1$,
and conjectured Theorem 1 in this special case. Recently, the first two named-
authors [3] provided a proof of the second part of this conjecture by
establishing a connection with Kauffman’s determinantal model of the Alexander
polynomial [15]; they also proved the first part of the Kashaev conjecture in
a very restrictive case and indirect way. Immediately afterwards, the third
named-author [16] gave a complete proof of the conjecture. Joining our
efforts, we now extend the approach of [16] to the general multivariable case
in Theorem 1 and in its proof.

As for Corollary 1, in the case $\mu=1$ it is nothing but Kauffman’s
aforementioned model for the Alexander polynomial [15]. Interestingly,
Kauffman did state a multivariable version of his model (a detailed proof was
only given many years later by Sato [17]), but it is different from our model.
However, Zibrowius [19] gave a state sum model for the multivariable Conway
function which uses the same labels as the ones of Figure 3 up to a sign;
using an extension of Kauffman’s Clock Theorem [15], this state sum model can
be turned into a determinantal model which coincides with the one of Corollary
1. This later fact can be found in the first version of Zibrowius’s article on
the ArXiv [18] (but not in the published version [19]). Therefore, and even
though our proof is completely different, Corollary 1 is not a new result in
the strict sense.

Let us finally mention that Friedl-Kausik-Quintanilha [11] recently provided
an algorithm for the computation of generalized Seifert matrices (see Section
2.1) for colored links given as closures of colored braids. Since such
matrices can be used to define $\sigma_{L},\eta_{L}$ and $\nabla_{L}$, this
method yields an algorithmic computation of these invariants. However, the
remarkable feature of Theorem 1 remains: a new way of computing these
invariants from a single symmetric matrix obtained directly from a colored
diagram.

This paper is organised as follows. In Section 2, we recall the necessary
background on generalized Seifert matrices (Section 2.1), multivariate
signatures of colored links (Section 2.2), and the Conway function (Section
2.3). Section 3 contains the proof of our results, namely the first and second
points of Theorem 1 in Section 3.1 and 3.2, respectively, and of Corollary 1
in Section 3.3. A slightly informal last Section 3.4 contains results on the
Alexander module.

### Acknowledgments

We thank Claudius Zibrowius for informing us of his work on the Kauffman
model. DC and LF are supported by the Swiss NSF grant 200021-212085. JL is
partially supported by NSERC CGS-D.

##  2\. Background

The aim of this section is to briefly recall the necessary background for our
work: we start in Section 2.1 with the definition of C-complexes and
generalized Seifert forms, then move on to multivariate signatures in Section
2.2, before dealing with the Conway function in Section 2.3.

###  2.1. Generalized Seifert surfaces and matrices

Seifert surfaces and matrices are well known tools in the construction and
study of (single-variable) abelian link invariants, such as the Levine-
Tristram signature and the Alexander polynomial. Less well-known is the fact
that multivariate invariants can be defined and studied via generalized
Seifert surfaces, known as C-complexes. We now introduce these objects,
following [8, 2].

To do so, we will use the notation $L=L_{1}\cup\dots\cup L_{\mu}$ for a
$\mu$-colored link, where $L_{i}$ is the sublink of $L$ consisting of all the
components of color $i$.

Figure 4. A positive clasp intersection (left), and a negative one (right).

######  Definition 2.

A C-complex for a $\mu$-colored link $L=L_{1}\cup\dots\cup L_{\mu}$ is a union
$S=S_{1}\cup\dots\cup S_{\mu}$ of surfaces embedded in $S^{3}$ satisfying the
following conditions:

  1. (i)

for all $i$, the surface $S_{i}$ is a (possibly disconnected) Seifert surface
for $L_{i}$;

  2. (ii)

for all $i\neq j$, the surfaces $S_{i}$ and $S_{j}$ are either disjoint or
intersect in a finite number of clasps, see Figure 4;

  3. (iii)

for all $i,j,k$ pairwise distinct, the intersection $S_{i}\cap S_{j}\cap
S_{k}$ is empty.

Note that a C-complex for a $1$-colored link $L$ is nothing but a (possibly
disconnected) Seifert surface for the oriented link $L$. The existence of a
C-complex for any given colored link is easy to establish [2]. On the other
hand, the corresponding notion of S-equivalence is more difficult to prove,
see [10] for the recently corrected statement.

These C-complexes allow us to define generalized Seifert forms as follows. For
any choice of signs
$\varepsilon=(\varepsilon_{1},\dots,\varepsilon_{\mu})\in\\{\pm 1\\}^{\mu}$,
let

$\alpha^{\varepsilon}\colon H_{1}(S)\times H_{1}(S)\longrightarrow\mathbb{Z}$

be the bilinear form given by
$\alpha^{\varepsilon}(x,y)=\operatorname{lk}(x^{\varepsilon},y)$, where
$x^{\varepsilon}$ denotes a well-chosen representative of the homology class
$x\in H_{1}(S)$ pushed-off $S_{i}$ in the $\varepsilon_{i}$-normal direction
(see [4] for a more formal definition). We denote by $A^{\varepsilon}$ the
corresponding generalized Seifert matrices, defined with respect to a fixed
basis of $H_{1}(S)$. One easily checks the equality

(1) $A^{-\varepsilon}=(A^{\varepsilon})^{\mathrm{T}}$

for all $\varepsilon\in\\{\pm 1\\}^{\mu}$. Note that the two generalized
Seifert matrices $A^{-},A^{+}$ of a $1$-colored link $L$ coincide with the
usual Seifert matrix $A$ of the oriented link $L$ and its transposed matrix
$A^{\mathrm{T}}$.

The general principle regarding these matrices is the following: what Seifert
matrices can do in one variable for oriented links, generalized Seifert
surfaces can do in $\mu$-variables for $\mu$-colored links. In Sections 2.2
and 2.3 we illustrate this principle with two examples of invariants.

###  2.2. Signatures and nullities of colored links

Fix a C-complex $S$ for a $\mu$-colored link $L$ and a basis of $H_{1}(S)$.
Consider an element $\omega=(\omega_{1},\dots,\omega_{\mu})$ of
$\mathbb{T}^{\mu}_{*}\coloneqq(S^{1}\setminus\\{1\\})^{\mu}$, and set

(2) $H(\omega)\coloneqq\sum_{\varepsilon\in\\{\pm
1\\}^{\mu}}\prod_{i=1}^{\mu}(1-\overline{\omega}_{i}^{\varepsilon_{i}})A^{\varepsilon}\,.$

Using (1), one easily checks that $H(\omega)$ is a Hermitian matrix and hence
admits a well-defined signature $\operatorname{sign}(H(\omega))\in\mathbb{Z}$
and nullity $\operatorname{null}(H(\omega))\in\mathbb{Z}_{\geq 0}$.

######  Definition 3 ([4]).

The signature and nullity of the $\mu$-colored link $L$ are functions

$\sigma_{L},\eta_{L}\colon\mathbb{T}^{\mu}_{*}\longrightarrow\mathbb{Z}$

defined by $\sigma_{L}(\omega)\coloneqq\operatorname{sign}(H(\omega))$ and
$\eta_{L}(\omega)\coloneqq\operatorname{null}(H(\omega))$, respectively.

The fact that these functions are well-defined invariants, i.e. do not depend
on the choice of the C-complex $S$ for $L$, relies on the aforementioned
notion of S-equivalence [4, 10]. Note that in the case $\mu=1$, the functions
$\sigma_{L},\eta_{L}\colon S^{1}\setminus\\{1\\}\to\mathbb{Z}$ are the
signature and nullity of the Hermitian matrix
$(1-\omega)A+(1-\overline{\omega})A^{\mathrm{T}}$, i.e. they coincide with the
Levine-Tristram signature and nullity of the oriented link $L$. We refer to
the recent survey [5] for background on this classical invariant.

In a nutshell, all the remarkable properties of the Levine-Tristram signature
extend to the multivariable setting. For example, the function $\sigma_{L}$ is
constant on the connected components of the complement in
$\mathbb{T}^{\mu}_{*}$ of the zeros of the multivariable Alexander polynomial
$\Delta_{L}(t_{1},\dots,t_{\mu})$ [4] (see Section 2.3 below). Also, if
$(\omega_{1},\dots,\omega_{\mu})\in\mathbb{T}^{\mu}_{*}$ is not the root of
any Laurent polynomial $p(t_{1},\dots,t_{\mu})$ with $p(1,\dots,1)=\pm 1$,
then $\sigma_{L}(\omega_{1},\dots,\omega_{\mu})$ and
$\eta_{L}(\omega_{1},\dots,\omega_{\mu})$ are invariant under topological
concordance of colored links [6].

###  2.3. The Conway function of a colored link

The one-variable Alexander polynomial $\Delta_{L}(t)$ of an oriented link $L$
can be generalized to a $\mu$-variable polynomial invariant
$\Delta_{L}(t_{1},\dots,t_{\mu})$ of a $\mu$-colored link $L$, a fact known to
Alexander himself [1]. To do so, consider the exterior
$X_{L}:=S^{3}\setminus\nu(L)$ of $L$ and the surjective group homomorphism

$\pi_{1}(X_{L})\to\mathbb{Z}^{\mu}\,,\quad[\gamma]\mapsto(\operatorname{lk}(\gamma,L_{1}),\dots,\operatorname{lk}(\gamma,L_{\mu}))\,.$

This defines a regular $\mathbb{Z}^{\mu}$-cover $\widehat{X}_{L}$ of $X_{L}$
whose homology groups are hence equipped with the structure of a module over
the group ring $\mathbb{Z}[\mathbb{Z}^{\mu}]=\mathbb{Z}[t_{1}^{\pm
1},\dots,t_{\mu}^{\pm 1}]$. In particular, the module
$\mathscr{A}_{L}:=H_{1}(\widehat{X}_{L})$ is called the (multivariable)
Alexander module of $L$ (see Section 3.4), and a greatest common divisor of
the elements of its order ideal is the Alexander polynomial of $L$.

Note however that this Laurent polynomial in $\mu$-variables is only well-
defined up to multiplication by units of the ring $\mathbb{Z}[t_{1}^{\pm
1},\dots,t_{\mu}^{\pm 1}]$, i.e. up to a sign and powers of the variables.
This later indeterminacy can be easily overcome by harnessing the symmetry of
$\Delta$ and requiring it to satisfy
$\Delta_{L}(t_{1}^{-1},\dots,t_{\mu}^{-1})=\pm\Delta_{L}(t_{1},\dots,t_{\mu})$,
but the sign issue is a non-trivial one.

The solution was suggested by Conway in his landmark paper [7]. He claimed the
existence of a well-defined rational function $\nabla_{L}$ satisfying

(3)
$\nabla_{L}(t_{1},\dots,t_{\mu})\stackrel{{\scriptstyle\boldsymbol{\cdot}}}{{=}}\begin{cases}\frac{1}{t_{1}-t_{1}^{-1}}\Delta_{L}(t^{2}_{1})&\text{if\leavevmode\nobreak\
$\mu=1$;}\\\ \Delta(t_{1}^{2},\dots,t_{\mu}^{2})&\text{if\leavevmode\nobreak\
$\mu>1$,}\end{cases}$

where $\stackrel{{\scriptstyle\boldsymbol{\cdot}}}{{=}}$ stands for the
equality up to multiplication by $\pm t_{1}^{\nu_{1}}\cdots
t_{\mu}^{\nu_{\mu}}$ with $\nu_{1},\dots,\nu_{\mu}\in\mathbb{Z}$. The first
explicit construction of this Conway function was given by Hartley [12] using
free differential calculus, but we will make use of the following geometric
construction [2]. Given any _connected_ C-complex $S=S_{1}\cup\dots\cup
S_{\mu}$ for a $L$, consider the matrix

(4) $A_{S}\coloneqq\sum_{\varepsilon\in\\{\pm
1\\}^{\mu}}\prod_{i=1}^{\mu}\varepsilon_{i}t_{i}^{\varepsilon_{i}}A^{\varepsilon}\,.$

Then, the Conway function of $L$ is given by

(5)
$\nabla_{L}(t_{1},\dots,t_{\mu})=\operatorname{sgn}(S)\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{\chi(S\setminus
S_{i})-1}\det(-A_{S})\,,$

where $\operatorname{sgn}(S)$ denotes the product of the signs of the clasps
of $S$ (recall Figure 4). Note that in the case $\mu=1$, equations (3) and (5)
lead to the formula $\Delta_{L}(t)=\det(t^{-1/2}A-t^{1/2}A^{\mathrm{T}})$, the
classical definition of the Alexander-Conway polynomial of the oriented link
$L$ [14].

This geometric construction of the Conway function yields straightforward
proofs of the various properties of this invariant. In particular, it yields a
“geometric explanation” of the local relations that can be used to compute it
from a link diagram, see [2] for more details.

##  3\. Proofs of the main results

As evident from its title, this section contains the proofs of our results.
More precisely, we start in Section 3.1 with the demonstration of the first
part of Theorem 1 on signatures and nullities. Section 3.2 deals with the
second part on the Conway function, while Section 3.3 contains the proof of
Corollary 1 on the multivariable Kauffman model. Finally, Section 3.4 consists
in a slightly informal discussion on the Alexander module.

###  3.1. Signatures and nullities

In this section we discuss how to compute the multivariable signature, proving
part (i) of Theorem 1 which we now restate for convenience.

######  Proposition 1.

Let $D$ be a diagram for a $\mu$-colored link $L$. For any
$\omega=(\omega_{1},\dots,\omega_{\mu})\in(S^{1}\setminus\\{1\\})^{\mu}$, the
signature and nullity of $L$ are given by

$\sigma_{L}(\omega)=\textstyle{\frac{1}{2}}(\operatorname{sign}(\widetilde{\tau}_{D}(\omega))-w_{\mathrm{m}}(D))\quad\text{and}\quad\eta_{L}(\omega)=\textstyle{\frac{1}{2}}\operatorname{null}(\widetilde{\tau}_{D}(\omega))\,,$

where $w_{\mathrm{m}}(D)$ is the sum of the signs of all monochromatic
crossings of $D$, and $\tau_{D}(\omega)$ stands for the evaluation of
$\tau_{D}(x)$ at $x_{j}=\operatorname{Re}(\omega_{j}^{1/2})$ and
$x_{jk}=\operatorname{Re}(\omega_{j}^{1/2}\omega_{k}^{1/2})$.

###### Proof.

Fix an arbitrary $\mu$-colored link $L$, and let $-L$ denote the $\mu$-colored
link $L$ with reverse orientation but same coloring as $L$. Let $L\\#_{1}-L$
denote a connected sum of $L$ and $-L$ along two components of color 1. Unlike
for knots, the isotopy type of the connected sum of (colored) links is not
well-defined. However, any two such connected sums have the same signature and
nullity, as these invariants behave additively under this ill-defined
operation [4, Proposition 2.12]. Since the signature and nullity are unchanged
by reversing the orientation (see [4, Corollary 2.9]), the relations

(6) $\sigma_{L\\#_{1}-L}(\omega)=2\sigma_{L}(\omega)\hskip
28.45274pt\text{and}\hskip
28.45274pt\eta_{L\\#_{1}-L}(\omega)=2\eta_{L}(\omega)$

hold for any such connected sum.

The idea of the proof is to use a diagram $D$ for $L$ to construct a C-complex
$S$ for $L\\#_{1}-L$ whose first homology has a basis given by classes of
loops corresponding to the regions and the monochromatric crossings of $D$ –
minus the two regions near the connected sum. By taking generalized Seifert
matrices with respect to this basis, we show that the matrix $H(\omega)$ used
in Definition 3 is congruent to a block-diagonal matrix of the form
$\widetilde{\tau}_{D}(\omega)\oplus Z$ with $\sigma(Z)=-w_{\mathrm{m}}(D)$ and
$\operatorname{null}(Z)=0$. Combined with Equation (6), this completes the
proof.

We now give the details. To construct a C-complex for $L\\#_{1}-L$ from $D$,
we use the following procedure (see Figure 5 for the construction near
crossings, and Figure 6 for an example).

  1. (i)

At each crossing of $D$, draw a copy of the corresponding crossing for $-L$ “a
bit above and behind” the crossing of $D$.

Figure 5. The construction of a C-complex $S$ for $L\\#_{1}-L$ near a
monochromatic crossing (left) and a bichromatic crossing (right).

  2. (ii)

Connect the remaining strands of $-L$ to each other following along the edges
of $D$, possibly creating an additional crossing along each edge (with $-L$
passing under $L$). This yields a diagram for the disjoint sum of $L$ and
$-L$.

  3. (iii)

Create a clasp intersection near each bichromatic crossing of $D$ and apply
the usual Seifert algorithm near each monochromatic crossing of $D$. This
yields a C-complex for the disjoint sum of $L$ and $-L$.

  4. (iv)

Finally, pick a point on a strand of color 1 in $D$ and cut the corresponding
surface at that place. The result is a C-complex $S$ for (some version of)
$L\\#_{1}-L$.

Figure 6. Left: the diagram for the disjoint sum of $L$ and $-L$ obtained from
$D$ of Figure 1. Right: the corresponding C-complex $S$ for $L\\#_{1}-L$.

Note that $S$ deformation retracts onto a graph defined as follows: take the
4-regular graph underlying the diagram $D$, add a loop at each vertex
corresponding to a monochromatic vertex, and remove the edge along which the
connected sum was performed. As a consequence, a natural basis for $H_{1}(S)$
is given by classes of cycles corresponding to the regions and monochromatic
crossings of $D$ – minus the two regions adjacent to where the connected sum
happens. We use the convention that the cycles representing our basis of
$H_{1}(S)$ are oriented counterclockwise in the plane of $D$ where $S$ is
drawn, and denote by the same letter a region or monochromatic crossing and
its corresponding cycle of $H_{1}(S)$. Using this explicit basis, we now study
the local contribution to generalized Seifert matrices near crossings of $D$.

As illustrated in Figure 7, there are five homology classes in $H_{1}(S)$
coming into play near a monochromatic crossing of $D$, and four near a
bichromatic crossing. The local contribution to the matrices $A^{\varepsilon}$
near different types of crossings of $D$ are given in Figure 8. By the
symmetry (1) and the fact that the C-complex near a negative bichromatic
crossing is the mirror image of the C-complex near a positive bichromatic
crossing, the local linking of all possible cases can be computed from those
in Figure 8.

$a$$b$$c$$d$$v$$a$$b$$c$$d$ Figure 7. The five cycles of $S$ near a
monochromatic crossing (left, one image) and the four cycles of $S$ near a
bichromatic crossing (right, four images). The labels $a,b,c,d,v$ for the
regions are used for the local linking matrices in Figure 8.

$\begin{array}[]{c||cccc|c}\operatorname{lk}&a&b&c&d&v\\\
\hline\cr\hline\cr&&&&&\\\\[-8.5359pt] a^{(1)}&-1/2&-1/2&0&0&1\\\
b^{(1)}&0&0&0&0&0\\\ c^{(1)}&0&-1/2&-1/2&0&1\\\ d^{(1)}&1/2&0&1/2&0&-1\\\
\hline\cr&&&&&\\\\[-8.5359pt] v^{(1)}&0&1&0&0&-1\end{array}\hskip
28.45274pt\begin{array}[]{c||cccc|c}\operatorname{lk}&a&b&c&d&v\\\
\hline\cr\hline\cr&&&&&\\\\[-8.5359pt] a^{(1)}&1/2&-1/2&0&0&0\\\
b^{(1)}&0&0&0&0&0\\\ c^{(1)}&0&-1/2&1/2&0&0\\\ d^{(1)}&1/2&0&1/2&0&-1\\\
\hline\cr&&&&&\\\\[-8.5359pt] v^{(1)}&-1&1&-1&0&1\end{array}$ (a)
Contributions near a positive (left) and negative (right) monochromatic
crossing of color $j$, where $x^{(1)}$ stands for $x^{\varepsilon}$ with
$\varepsilon_{j}=1$.

$\begin{array}[]{c||cccc}\operatorname{lk}&a&b&c&d\\\
\hline\cr\hline\cr&&&&\\\\[-8.5359pt] a^{(1,1)}&-1/2&-1/2&0&0\\\
b^{(1,1)}&0&0&0&0\\\ c^{(1,1)}&0&1/2&-1/2&0\\\ d^{(1,1)}&1/2&-1&1/2&0\\\
\end{array}\hskip
28.45274pt\begin{array}[]{c||cccc}\operatorname{lk}&a&b&c&d\\\
\hline\cr\hline\cr&&&&\\\\[-8.5359pt] a^{(1,-1)}&0&0&0&0\\\
b^{(1,-1)}&-1/2&1/2&0&0\\\ c^{(1,-1)}&1&-1/2&0&-1/2\\\
d^{(1,-1)}&-1/2&0&0&1/2\\\ \end{array}$ (b) Contributions near a positive
bichromatic crossing of colors $(j,k)$, with $x^{\varepsilon}$ denoted by
$x^{(\varepsilon_{j},\varepsilon_{k})}$.

Figure 8. The local contributions to linking numbers near crossings of $D$,
where the curves are labeled as in Figure 7.

For the remainder of the proof, we adopt the following convention: we say a
crossing $v$ has color $(j,k)=(j_{v},k_{v})$ if its incoming left strand is
color $j$, and incoming right strand has color $k$, as in Figure 2. If $j=k$,
we may simply say it has color $j$.

We now write $H(\omega)$ as a sum over crossings of $D$. For a crossing $v$,
let $A_{v}^{\varepsilon}$ denote the square matrix (of size equal to the first
Betti number of $S$) given by the contribution to $A^{\varepsilon}$ from the
linking near $v$; in other words, $A^{\varepsilon}_{v}$ is zero everywhere
except in the $5\times 5$ or $4\times 4$ minor corresponding to the homology
classes coming into play near $v$, where its values are given by the local
contributions to linking numbers given by the matrices from Figure 8. We have:

(7) $H(\omega)=\sum_{\varepsilon\in\\{\pm
1\\}^{\mu}}\prod_{i=1}^{\mu}(1-\overline{\omega}_{i}^{\varepsilon_{i}})A^{\varepsilon}=\sum_{v}\sum_{\varepsilon\in\\{\pm
1\\}^{\mu}}\prod_{i=1}^{\mu}(1-\overline{\omega}_{i}^{\varepsilon_{i}})A_{v}^{\varepsilon}=:\sum_{v}H_{v}\,,$

where
$H_{v}=\sum_{\varepsilon}\prod_{i}(1-\overline{\omega}_{i}^{\varepsilon_{i}})A_{v}^{\varepsilon}$,
and the sums indexed by $v$ always refer to the sum over all crossing of $D$.
Note that $A_{v}^{\varepsilon}$ is entirely specified by $\varepsilon_{j_{v}}$
and $\varepsilon_{k_{v}}$. Thus we use $A_{v}^{(\alpha,\beta)}$ to denote
$A_{v}^{\varepsilon}$ for any $\varepsilon$ with $\varepsilon_{j_{v}}=\alpha$
and $\varepsilon_{k_{v}}=\beta$. If $v$ is monochromatic with
$\varepsilon_{j_{v}}=\varepsilon_{k_{v}}=\alpha$, we simply write
$A_{v}^{(\alpha)}$.

When $v$ is monochromatic of color $j=j_{v}$, the matrix $H_{v}$ can be
rewritten as

$\begin{split}H_{v}&=\sum_{(\varepsilon_{1},\cdots,\widehat{\varepsilon}_{j},\cdots,\varepsilon_{\mu})\in\\{\pm
1\\}^{\mu-1}}\prod_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{\mu}(1-\overline{\omega}_{i}^{\varepsilon_{i}})\left((1-\overline{\omega}_{j})A^{(1)}_{v}+(1-\omega_{j})A^{(-1)}_{v}\right)\\\
&=\Bigg{(}\prod_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{\mu}(1-\overline{\omega}_{i})(1-\omega_{i})\Bigg{)}\left((1-\overline{\omega}_{j})A^{(1)}_{v}+(1-\omega_{j})A^{(-1)}_{v}\right)\,,\end{split}$

where the second equality uses the relation
$(1-\overline{\omega}_{i})+(1-\omega_{i})=(1-\overline{\omega}_{i})(1-\omega_{i})$.
Hence, using the notation $s_{i}:=(1-\omega_{i})$, the matrix $H_{v}$ for a
monochromatic crossing $v$ of color $j$ is given by

(8) $H_{v}=\prod_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}^{\mu}|s_{i}|^{2}\left(\bar{s}_{j}A^{(1)}_{v}+s_{j}A^{(-1)}_{v}\right)=:\prod_{\begin{subarray}{c}i=1\\\
i\neq j\end{subarray}}^{\mu}|s_{i}|^{2}\,A_{v}\,,$

while for a bichromatic crossing of colors $(j,k)$, it is given by

(9) $H_{v}=\prod_{\begin{subarray}{c}i=1\\\ i\neq
j,k\end{subarray}}^{\mu}|s_{i}|^{2}\left(\bar{s}_{j}\bar{s_{k}}A^{(1,1)}_{v}+\bar{s}_{j}s_{k}A^{(1,-1)}_{v}+s_{j}\bar{s}_{k}A^{(-1,1)}_{v}+s_{j}s_{k}A^{(-1,-1)}_{v}\right)=:\prod_{\begin{subarray}{c}i=1\\\
i\neq j,k\end{subarray}}^{\mu}|s_{i}|^{2}\,A_{v}\,,$

where we use the notation

$A_{v}:=\begin{cases}\bar{s}_{j}A^{(1)}_{v}+s_{j}A^{(-1)}_{v}&\text{if\leavevmode\nobreak\
$v$ is monochromatic of color\leavevmode\nobreak\ $j$};\\\
\bar{s}_{j}\bar{s_{k}}A^{(1,1)}_{v}+\bar{s}_{j}s_{k}A^{(1,-1)}_{v}+s_{j}\bar{s}_{k}A^{(-1,1)}_{v}+s_{j}s_{k}A^{(-1,-1)}_{v}&\text{if\leavevmode\nobreak\
$v$ is bichromatic of colors\leavevmode\nobreak\ $j,k$}\,.\end{cases}$

Plugging Equations (8) and (9) into Equation (7), we get

(10) $H(\omega)=\sum_{v}H_{v}=\sum_{v}\prod_{i\neq
j_{v},k_{v}}^{\mu}|s_{i}|^{2}A_{v}\,.$

Writing the Hermitian matrix $H(\omega)$ as a block matrix of the form

$H(\omega)=\begin{array}[]{c||c|c}&\text{regions}&\text{mono crossings}\\\
\hline\cr\hline\cr\text{regions}&X&Y\\\ \hline\cr\text{mono
crossings}&Y^{*}&Z\\\ \end{array}$

we see that $Z$ is a diagonal matrix with coefficient corresponding to the
monochromatic crossing $v$ given by
$-\prod_{i}|s_{i}|^{2}\operatorname{sgn}(v)$. In particular, $Z$ is invertible
(and hence has nullity $\operatorname{null}(Z)=0$), while its signature is
equal to $\sigma(Z)=-w_{\mathrm{m}}(D)$. Furthermore, $H(\omega)$ is congruent
to the block diagonal matrix

(11) $MH(\omega)M^{*}=\begin{pmatrix}X-YZ^{-1}Y^{*}&0\\\
0&Z^{-1}\end{pmatrix}$

via $M=\begin{pmatrix}I&-YZ^{-1}\\\ 0&Z^{-1}\end{pmatrix}$. Since
$\sigma(Z^{-1})=\sigma(Z)=-w_{\mathrm{m}}(D)$ and
$\operatorname{null}(Z^{-1})=0$, it remains to show that the matrix
$X-YZ^{-1}Y^{*}$ coincides with $\widetilde{\tau}_{D}(\omega)$ up to
transformations that do not affect the signature and nullity.

To determine $X-YZ^{-1}Y^{*}$, let us fix two regions $a$ and $b$. Note that

$(YZ^{-1}Y^{*})_{a,b}=\sum_{v}Y_{a,v}Z_{v,v}^{-1}\overline{Y_{b,v}}\,,$

and that it is only possible for both $Y_{a,v}$ and $Y_{b,v}$ to be nonzero if
the regions $a$ and $b$ are both adjacent to the crossing $v$. Therefore, the
matrix $YZ^{-1}Y^{*}$ is a sum over crossings, where the contribution at each
crossing is a matrix that is zero everywhere except in the $4\times 4$ minor
corresponding to the four adjacent regions of the crossing. The same is then
true for $X-YZ^{-1}Y^{*}$. For a bichromatic crossing $v$, there is no column
of $Y$ that corresponds to $v$, so this $4\times 4$ minor is nothing but the
nonzero $4\times 4$ minor of $A_{v}$. For a monochromatic crossing $v$, the
$4\times 4$ minor of $X-YZ^{-1}Y^{*}$ is obtained by performing the
corresponding matrix operations to the $5\times 5$ minor of $A_{v}$. The
computation is similar to the single variable case, see [16] for more details.
The explicit values for these local contributions to $X-YZ^{-1}Y^{*}$ are
given in Figure 9. Note that we only provide these minors in the case of
positive crossings, as the minor for a negative crossing is obtained by the
negative of the minor for a positive one.

$\left[\begin{array}[]{c||cccc}&a&b&c&d\\\
\hline\cr\hline\cr&&&&\\\\[-8.5359pt]
a&\frac{\omega_{j}+\overline{\omega}_{j}}{2}&-\frac{1+\overline{\omega}_{j}}{2}&1&-\frac{1+\omega_{j}}{2}\\\
&&&&\\\\[-8.5359pt]
b&-\frac{1+\omega_{j}}{2}&1&-\frac{1+\omega_{j}}{2}&\omega_{j}\\\
&&&&\\\\[-8.5359pt]
c&1&-\frac{1+\overline{\omega}_{j}}{2}&\frac{\omega_{j}+\overline{\omega}_{j}}{2}&-\frac{1+\omega_{j}}{2}\\\
&&&&\\\\[-8.5359pt]
d&-\frac{1+\overline{\omega}_{j}}{2}&\overline{\omega}_{j}&-\frac{1+\overline{\omega}_{j}}{2}&1\end{array}\right]$
(a) $v$ positive monochromatic crossing of color $j$

$\left[\begin{array}[]{c||cccc}&a&b&c&d\\\
\hline\cr\hline\cr&&&&\\\\[-8.5359pt]
a&-\frac{\overline{s_{j}}\overline{s_{k}}+s_{j}s_{k}}{2}&\frac{\overline{s_{k}}(\omega_{j}-\overline{\omega}_{j})}{2}&\overline{s_{k}}\overline{s_{j}}&\frac{s_{j}(\overline{\omega}_{k}-\omega_{k})}{2}\\\
&&&&\\\\[-8.5359pt]
b&\frac{(\overline{\omega}_{j}-\omega_{j})s_{k}}{2}&\frac{s_{j}\overline{s_{k}}+\overline{s_{j}}s_{k}}{2}&\frac{s_{j}(\overline{\omega}_{k}-\omega_{k})}{2}&-s_{j}s_{k}\\\
&&&&\\\\[-8.5359pt]
c&\overline{s_{j}}s_{k}&\frac{(\omega_{k}-\overline{\omega}_{k})\overline{s_{j}}}{2}&-\frac{\overline{s_{j}}\overline{s_{k}}+s_{j}s_{k}}{2}&\frac{(\overline{\omega}_{j}-\omega_{j})s_{k}}{2}\\\
&&&&\\\\[-8.5359pt]
d&\frac{\overline{s_{j}}(\omega_{k}-\overline{\omega}_{k})}{2}&-\overline{s_{j}}\overline{s_{k}}&\frac{(\omega_{j}-\overline{\omega}_{j})\overline{s_{k}}}{2}&\frac{s_{j}\overline{s_{k}}+\overline{s_{j}}s_{k}}{2}\end{array}\right]$
(b) $v$ positive bichromatic crossing of colors $(j,k)$

Figure 9. The local contribution to $X-YZ^{-1}Y^{*}$ for a positive
monochromatic and bichromatic crossing $v$. If $v$ is a negative crossing, the
matrix is the negative of the corresponding matrix for a positive crossing.

We need to perform one last change of basis, which we now describe. If the
sublink $L_{i}$ winds around the region $a$ a total of $\alpha_{i}$ times,
then multiply the basis element corresponding to $a$ with
$\prod_{i=1}^{\mu}(-\omega^{-1/2}_{i})^{\alpha_{i}}$. This change of basis
alters the matrices in Figure 9 in the following way. If $v$ is a
monochromatic crossing of color $j$, then $L_{j}$ winds around $b$ once more
that it does around $a$ and $c$, and it winds around $d$ once fewer. Thus the
rows corresponding to $b$ and $d$ are multiplied by $-\omega^{-1/2}_{i}$ and
$-\omega^{1/2}_{i}$ respectively, and the columns are multiplied by the
conjugates $-\omega^{1/2}_{j}$ and $-\omega^{-1/2}_{j}$. If $v$ is bichromatic
crossing of colors $(j,k)$, then $L_{j}$ winds once more around $a$ and $b$
than it does around $c$ and $d$, and $L_{k}$ winds once more around $b$ and
$c$ than it does around $a$ and $d$. Thus we multiply the row for $a$ by
$-\omega^{-1/2}_{j}$, the row for $c$ by $-\omega^{-1/2}_{k}$, and the row for
$b$ by $\omega^{-1/2}_{j}\omega^{-1/2}_{k}$, and we multiply the corresponding
columns by the conjugates.

Remarkably, the sum of local contributions to $X-YZ^{-1}Y^{*}$ from Figure 9
can now be written in terms of the single matrix $\tau_{v}(\omega)$, the
evaluation of $\tau_{v}(x)$ at $x_{j}=\operatorname{Re}(\omega_{j}^{1/2})$ and
$x_{jk}=\operatorname{Re}(\omega_{j}^{1/2}\omega_{k}^{1/2})$, in both the
monochromatic and bichromatic cases. Indeed, for a monochromatic crossing $v$,
it coincides with $\operatorname{sgn}(v)\tau_{v}(\omega)$, while for a
bichromatic crossing, it yields

$4\operatorname{sgn}(v)\sqrt{1-x_{j_{v}}^{2}}\sqrt{1-x_{k_{v}}^{2}}\tau_{v}(\omega)\,.$

The result of the matrix $X-YZ^{-1}Y^{*}$ after this change of basis thus
gives

$\begin{split}\sum_{v\text{ mono}}\prod_{i\neq
j_{v}}|s_{i}|^{2}\operatorname{sgn}(v)\tau_{v}(\omega)+&\sum_{v\text{
bi}}\prod_{i\neq
j_{v},k_{v}}|s_{i}|^{2}4\operatorname{sgn}(v)\sqrt{1-x_{j_{v}}^{2}}\sqrt{1-x_{k_{v}}^{2}}\tau_{v}(\omega)\\\
=&\frac{1}{4}\prod_{i=1}^{\mu}|s_{i}|^{2}\;\sum_{v}\frac{\operatorname{sgn}(v)}{\sqrt{1-x_{j_{v}}^{2}}\sqrt{1-x_{k_{v}}^{2}}}\tau_{v}(\omega)=\frac{1}{4}\prod_{i=1}^{\mu}|s_{i}|^{2}\;\widetilde{\tau}_{D}(\omega)\,.\end{split}$

The positive constant $\frac{1}{4}\prod_{i=1}^{\mu}|s_{i}|^{2}$ affects
neither the signature nor the nullity, so we have

$\displaystyle 2\sigma_{L}(\omega)$
$\displaystyle=\sigma(X-YZ^{-1}Y^{*})-\sigma(Z^{-1})=\sigma(\widetilde{\tau}_{D}(\omega))-w_{\mathrm{m}}(D)\,,$
$\displaystyle 2\eta_{L}(\omega)$
$\displaystyle=\eta(X-YZ^{-1}Y^{*}-\eta(Z^{-1})=\eta(\widetilde{\tau}_{D}(\omega))\,,$

and the proof is complete.

∎

###  3.2. The Conway function

In this section we discuss how to compute the Conway function of a colored
link from the matrix $\widetilde{\tau}_{D}$ and prove the second point of
Theorem 1, which we now restate for convenience.

######  Proposition 2.

If $D$ is a connected $\mu$-colored diagram for a $\mu$-colored link $L$, we
have the equality

$\nabla^{2}_{L}(t_{1},\dots,t_{\mu})=\frac{1}{(t_{1}-t_{1}^{-1})^{2}}\prod_{v}(-\operatorname{sgn}(v))\frac{t_{j}-t_{j}^{-1}}{2}\frac{t_{k}-t_{k}^{-1}}{2}\cdot\det(\widetilde{\tau}_{D}(t^{2}))\,,$

where the product is over all crossings of $D$, the indices $j,k$ are the
(possibly identical) colors of the two strands crossing at $v$, and
$\tau_{D}(t^{2})$ stands for the evaluation of $\tau_{D}(x)$ at

$x_{j}=\frac{t_{j}+t_{j}^{-1}}{2}\,,\quad
x_{jk}=\frac{t_{j}t_{k}+t_{j}^{-1}t_{k}^{-1}}{2}\,.$

First recall that, for any
$\omega=(\omega_{1},\dots,\omega_{\mu})\in(S^{1}\setminus\\{1\\})^{\mu}$,
$\tau_{D}(\omega)$ stands for the evaluation of $\tau_{D}$ at

$x_{j}=\operatorname{Re}(\omega_{j}^{1/2})=\frac{\omega_{j}^{1/2}+\omega_{j}^{-1/2}}{2}\,,\quad
x_{jk}=\operatorname{Re}(\omega_{j}^{1/2}\omega_{k}^{1/2})=\frac{\omega_{j}^{1/2}\omega_{k}^{1/2}+\omega_{j}^{-1/2}\omega_{k}^{-1/2}}{2}\,.$

Hence, $\tau_{D}(t^{2})$ can be understood as $\tau_{D}(\omega)$ evaluated at
$\omega_{i}=t_{i}^{2}$ (or equivalently, at $\omega_{i}=t_{i}^{-2}$),
justifying the notation. We frequently make use of this change of variables in
what follows.

Our starting point is Equation (5), which expresses the Conway function
$\nabla_{L}$ in terms of the matrix $A_{S}$ associated to a connected
C-complex $S$ for $L$, recall (4). First note that if $H(\omega)$ denotes the
matrix defined in (2), which is used for computing the multivariable
signature, we have

(12)
$\begin{split}H(t^{-2})&:=H(t_{1}^{-2},\cdots,t_{\mu}^{-2})=\sum_{\varepsilon\in\\{\pm
1\\}^{\mu}}\prod_{i=1}^{\mu}(1-t_{i}^{2\varepsilon_{i}})A^{\varepsilon}=\sum_{\varepsilon\in\\{\pm
1\\}^{\mu}}\prod_{i=1}^{\mu}\varepsilon_{i}t_{i}^{\varepsilon_{i}}\varepsilon_{i}(t_{i}^{-\varepsilon_{i}}-t_{i}^{\varepsilon_{i}})A^{\varepsilon}\\\
&=\prod_{i=1}^{\mu}-(t_{i}-t_{i}^{-1})\sum_{\varepsilon\in\\{\pm
1\\}^{\mu}}\prod_{i=1}^{\mu}\varepsilon_{i}t_{i}^{\varepsilon_{i}}A^{\varepsilon}=(-1)^{\mu}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})A_{S}\,.\end{split}$

Hence, the Conway function can in fact be computed from the matrix $H$.

Now, in order to prove Proposition 2, we adopt the same strategy as in the
computation of the signature: starting from a _connected_ diagram $D$, we use
the C-complex $S$ for $L\\#_{1}-L$ constructed in the previous section to
compute the Conway function of $L\\#_{1}-L$, and conclude by applying well-
known formulas relating the Conway function of a connected sum to the Conway
functions of the summands. Note that, by construction, requiring $D$ to be
connected precisely means that $S$ is connected.

So, let $D=D_{1}\cup\dots\cup D_{\mu}$ be a connected, $\mu$-colored diagram
of a $\mu$-colored link $L$ and let $S=S_{1}\cup\dots\cup S_{\mu}$ be the
C-complex for $L\\#_{1}-L$ defined in the proof of Proposition 1. As before,
the notation $\\#_{1}$ stands for the connected sum performed along a
component of color $1$. Let $n_{\mathrm{m}}$ and $n_{\mathrm{b}}$ denote the
number of monochromatic and bichromatic crossings of $D$, respectively, and
$n=n_{\mathrm{m}}+n_{\mathrm{b}}$ the total number of crossings. Similarly,
let $n_{{\mathrm{m}},i}$ (resp. $n_{{\mathrm{b}},i}$) denote the number of
monochromatic (resp. bichromatic) crossings of $D$ _without_ the color $i$.

######  Lemma 1.

With the notations above, the C-complex $S$ satisfies:

  1. (i)

$\operatorname{sgn}{S}=\prod\limits_{v\text{ bichr.}}\operatorname{sgn}{v}$,
where the product is taken over all bichromatic crossings.

  2. (ii)

Its first Betti number is equal to $b_{1}(S)=n+n_{\mathrm{m}}$ and is even.

  3. (iii)

$\chi(S\setminus S_{1})=-n_{{\mathrm{b}},1}-2n_{{\mathrm{m}},1}$ and
$\chi(S\setminus S_{i})=1-n_{{\mathrm{b}},i}-2n_{{\mathrm{m}},i}$ for all
$i\neq 1$.

###### Proof.

The first equality is clear by construction, since $S$ has one clasp for each
bichromatic crossing of $D$, and the sign of the clasp is equal to the sign of
the corresponding crossing. To check the second point let $r$ denote the
number of regions of $D$. By construction, we have
$b_{1}(S)=(r-2)+n_{\mathrm{m}}$, while an Euler characteristic computations
yields the equality $r-2=n$. Since $n_{\mathrm{b}}$ is always even, it follows
that $b_{1}(S)=n+n_{\mathrm{m}}=n_{\mathrm{b}}+2n_{\mathrm{m}}$ is also even.
As for the third point, one just needs to notice that $S\setminus S_{i}$
deformation retracts onto a graph $\Gamma_{i}$ constructed from (the planar
projection of) the diagram $D\setminus D_{i}$ by adding one loop to each
monochromatic crossing and, if $i\neq 1$, by removing one edge of color $1$
(which corresponds to performing the connected sum). The number of vertices of
$\Gamma_{i}$ minus the number of its edges yields the result. ∎

To shorten our formulas, we denote $s_{\mathrm{b}}:=\prod\limits_{v\text{
bichr.}}\operatorname{sgn}{v}$ and $s_{\mathrm{m}}:=\prod\limits_{v\text{
mono.}}\operatorname{sgn}{v}$, where the product is taken over all bichromatic
(resp. monochromatic) crossings of $D$. We are now ready to prove Proposition
2.

###### Proof of Proposition 2.

Recall that, by the proof of Proposition 1 (in particular (11)), there exists
a matrix $M$ such that

$MH(\omega)M^{*}=\widetilde{\tau}^{\prime}_{D}(\omega)\oplus Z^{-1}\,,$

where the matrix

$Z=\prod_{i=1}^{\mu}(1-\omega_{i})(1-\omega_{i}^{-1})\operatorname{diag}(-\operatorname{sgn}(v))$

is indexed by the monochromatic crossings of $D$, and
$\det(M)=\det(M^{*})=\det(Z^{-1})$. Therefore,

$\det(H(\omega))=\det(Z)\det(\widetilde{\tau}^{\prime}_{D}(\omega))=(-1)^{n_{\mathrm{m}}}s_{\mathrm{m}}\Big{(}\prod_{i=1}^{\mu}(1-\omega_{i})(1-\omega_{i}^{-1})\Big{)}^{n_{\mathrm{m}}}\det(\widetilde{\tau}^{\prime}_{D}(\omega))\,.$

Evaluating $H$ at $\omega_{i}=t_{i}^{-2}$, and using the equality
$(1-t_{i}^{2})(1-t_{i}^{-2})=-(t_{i}-t_{i}^{-1})^{2}$, we obtain

(13) $\det(H(t^{-2}))=(-1)^{n_{\mathrm{m}}+\mu
n_{\mathrm{m}}}s_{\mathrm{m}}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{2n_{\mathrm{m}}}\det(\widetilde{\tau}^{\prime}_{D}(t^{2}))\,.$

Furthermore, since $\widetilde{\tau}^{\prime}_{D}$ is a matrix of size $n$
and, by definition,

$\widetilde{\tau}^{\prime}_{D}(\omega)=\frac{1}{4}\prod_{i=1}^{\mu}(1-\omega_{i})(1-\omega_{i}^{-1})\widetilde{\tau}_{D}(\omega)\,,$

we have

(14) $\det(\widetilde{\tau}^{\prime}_{D}(t^{2}))=(-1)^{\mu
n}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{2n}\det({\textstyle\frac{1}{4}}\widetilde{\tau}_{D}(t^{2}))\,.$

Putting everything together, and writing $t$ for $(t_{1},\dots,t_{\mu})$, we
obtain

$\begin{split}\nabla^{2}_{L}(t)=&\nabla_{L}(t)\nabla_{-L}(t)=\frac{1}{t_{1}-t_{1}^{-1}}\nabla_{L\\#_{1}-L}(t)\\\
\overset{\eqref{eq:Conway}}{=}&\frac{1}{t_{1}-t_{1}^{-1}}(-1)^{b_{1}(S)}\operatorname{sgn}(S)\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{\chi(S\setminus
S_{i})-1}\det(A_{S})\\\
\overset{\eqref{eq:H-Conway}}{=}&\frac{s_{\mathrm{b}}}{t_{1}-t_{1}^{-1}}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{\chi(S\setminus
S_{i})-1}(-1)^{\mu
b_{1}(S)}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{-b_{1}(S)}\det(H(t^{-2}))\\\
\overset{\eqref{eq:H-tau}}{=}&\frac{s_{\mathrm{b}}}{t_{1}-t_{1}^{-1}}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{\chi(S\setminus
S_{i})-1-b_{1}(S)}(-1)^{n_{\mathrm{m}}+\mu
n_{\mathrm{m}}}s_{\mathrm{m}}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{2n_{\mathrm{m}}}\det(\widetilde{\tau}^{\prime}_{D}(t^{2}))\\\
\overset{\eqref{eq:tau}}{=}&\frac{(-1)^{n_{\mathrm{m}}+\mu
n_{\mathrm{m}}}s_{\mathrm{b}}s_{\mathrm{m}}}{t_{1}-t_{1}^{-1}}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{\chi(S\setminus
S_{i})-1-b_{1}(S)+2n_{\mathrm{m}}}(-1)^{\mu
n}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{2n}\det({\textstyle\frac{1}{4}}\widetilde{\tau}_{D}(t^{2}))\\\
\overset{}{=}&\frac{(-1)^{n_{\mathrm{m}}}}{(t_{1}-t_{1}^{-1})^{2}}\frac{s_{\mathrm{b}}s_{\mathrm{m}}}{4^{n}}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{-n_{{\mathrm{b}},i}-2n_{{\mathrm{m}},i}+n+n_{\mathrm{m}}}\det(\widetilde{\tau}_{D}(t^{2}))\,,\end{split}$

where in the first line we used Corollary $2$ and Proposition $5$ of [2]. The
following equalities derive, as indicated, from Equations (5), (12), (13) and
(14), together with the first point of Lemma 1 in the third line (with (12)),
the second point in the fourth line (with (13)), and the second and third
points in the last line.

To conclude, we note that the exponent
$-n_{{\mathrm{b}},i}-2n_{{\mathrm{m}},i}+n+n_{\mathrm{m}}$ appearing in the
last line is equal to the number of bichromatic crossings involving a strand
of color $i$ plus twice the number of monochromatic crossings of color $i$.
Therefore,

$\frac{s_{\mathrm{b}}s_{\mathrm{m}}}{4^{n}}\prod_{i=1}^{\mu}(t_{i}-t_{i}^{-1})^{-n_{{\mathrm{b}},i}-2n_{{\mathrm{m}},i}+n+n_{\mathrm{m}}}=\prod_{v}\operatorname{sgn}(v)\frac{t_{j}-t_{j}^{-1}}{2}\frac{t_{k}-t_{k}^{-1}}{2}\,,$

where the product on the right-hand side is over all crossings of $D$ and the
indices $j,k$ are the two (possibly identical) colors of strands crossing at
$v$. The proposition now follows from observing that
$(-1)^{n_{\mathrm{m}}}=(-1)^{n}$ since $n=n_{\mathrm{m}}+n_{\mathrm{b}}$ and
$n_{\mathrm{b}}$ is even. ∎

###  3.3. The multivariate Kauffman model

Having finished the proof of Theorem 1, we now turn our attention to Corollary
1.

Starting from a _connected_ diagram $D$ of a $\mu$-colored link, let $K_{D}$
(or simply $K$) be the matrix defined by the labels in Figure 3, and
$\widetilde{K}$ the square matrix obtained from $K$ by removing two columns
corresponding to two adjacent regions of $D$ separated by a strand of color
$1$. Corollary 1 is a direct consequence of Proposition 2 together with the
following lemma.

######  Lemma 2.

Let $S=(S_{v,v})$ be the diagonal matrix indexed by the crossings of $D$ with
coefficients

$S_{v,v}=\frac{-4\operatorname{sgn}(v)}{(t_{j}-t_{j}^{-1})(t_{k}-t_{k}^{-1})}\,,$

where $j$ and $k$ are the colors of the two strands meeting at $v$. Then, we
have

$\tau_{D}(t^{2})=K^{\mathrm{T}}SK\,.$

We start by proving Corollary 1, before addressing the proof of Lemma 2.

###### Proof of Corollary 1.

By Lemma 2, we have
$\widetilde{\tau}_{D}(t^{2})=\widetilde{K}^{\mathrm{T}}S\widetilde{K}$,
yielding the equality

$(\det\widetilde{K})^{2}=\det(S)^{-1}\det(\widetilde{\tau}_{D}(t^{2}))\,.$

Since moreover

$\det(S)^{-1}=\prod_{v}(-\operatorname{sgn}(v))\frac{t_{j}-t_{j}^{-1}}{2}\frac{t_{k}-t_{k}^{-1}}{2}\,,$

Proposition 2 implies
$(\det\widetilde{K})^{2}=(t_{1}-t_{1}^{-1})^{2}\nabla^{2}_{L}(t_{1},\dots,t_{\mu})$,
and Corollary 1 follows. ∎

###### Proof of Lemma 2.

Recall that $K=(K_{v,a})$ is a matrix with rows indexed by the crossings of
$D$ and columns indexed by the regions of $D$. Let us fix two regions $a$ and
$b$ of $D$. By definition, the corresponding coefficient of $K^{\mathrm{T}}SK$
is

(15)
$(K^{\mathrm{T}}SK)_{a,b}=\sum_{v}\frac{-4\operatorname{sgn}(v)}{(t_{j}-t_{j}^{-1})(t_{k}-t_{k}^{-1})}K_{v,a}K_{v,b}\,,$

while the corresponding coefficient of $\tau_{D}(t^{2})$ is

(16)
$(\tau_{D}(t^{2}))_{a,b}=\sum_{v}\frac{-4\operatorname{sgn}(v)}{(t_{j}-t_{j}^{-1})(t_{k}-t_{k}^{-1})}(\tau_{v}(t^{2}))_{a,b}\,.$

In both cases, the sum is over all crossings of $D$ (and $j,k$ denote the
colors of the strands crossing at $v$), but the only non-zero contributions
come from the crossings adjacent to both $a$ and $b$.

Comparing the labels of Figure 2 evaluated at
$x_{j}=\frac{t_{j}+t_{j}^{-1}}{2}$ and
$x_{jk}=\frac{t_{j}t_{k}+t_{j}^{-1}t_{k}^{-1}}{2}$ with the labels of Figure
3, one notices an interesting relation. To state it precisely, let
$\mathbb{Q}(t)$ denote the field of fractions of $\mathbb{Z}[t_{1}^{\pm
1},\cdots,t_{\mu}^{\pm 1}]$, and let
$\varphi:\mathbb{Q}(t)\rightarrow\mathbb{Q}(t)$ be the involution induced by
$t_{i}\mapsto t_{i}^{-1}$ for all $i$. We claim that the following equality
holds:

(17)
$(\tau_{v}(t^{2}))_{a,b}=\frac{K_{v,a}K_{v,b}+\varphi(K_{v,a}K_{v,b})}{2}\,.$

The proof of this claim is divided into three cases, depending on the relative
positions of the regions $a$ and $b$. Let us first assume that $a$ and $b$ are
two different regions of the same checkerboard color. In such a case, we have
$K_{v,a}K_{v,b}=1=(\tau_{v}(t^{2}))_{a,b}$ for each crossing $v$ incident to
both $a$ and $b$, so (17) holds. Let us now assume that $a$ and $b$ are
regions with different checkerboard colors, meeting at a crossing $v$ with
strands of colors $j$ and $k$. If $a$ and $b$ are adjacent to the strand of
color $j$ (resp. $k$), we get $K_{v,a}K_{v,b}=t_{k}^{\pm 1}$ (resp.
$t_{j}^{\pm 1}$). Since the coefficient of $\tau_{v}(t^{2})$ is
$x_{k}=\frac{t_{k}+t_{k}^{-1}}{2}$ (resp. $x_{j}=\frac{t_{j}+t_{j}^{-1}}{2}$),
Equation (17) holds in this case as well. Finally, let us assume that $a=b$.
For a crossing $v$ incident to $a$, we get either
$K_{v,a}^{2}=(t_{j}t_{k})^{\pm 1}$ or $K_{v,a}^{2}=(t_{j}^{-1}t_{k})^{\pm 1}$,
depending to the position of $a$ around $v$. Similarly, the corresponding
coefficient of $\tau_{v}(t^{2})$ is either
$x_{jk}=\frac{t_{j}t_{k}+t_{j}^{-1}t_{k}^{-1}}{2}$ or
$2x_{j}x_{k}-x_{jk}=\frac{t_{j}t_{k}^{-1}+t_{j}^{-1}t_{k}}{2}$, respectively.
This concludes the proof of (17).

Equations (15), (16) and (17) immediately imply the equality

$\tau_{D}(t^{2})=\frac{K^{\mathrm{T}}SK+\varphi(K^{\mathrm{T}}SK)}{2}\,,$

where $\varphi$ is applied to matrices coefficientwise. To conclude the proof
of Lemma 2, it remains to check that
$\varphi(K^{\mathrm{T}}SK)=K^{\mathrm{T}}SK$. This fact being suprisingly
technical, we make it the object of a final separate lemma. ∎

######  Lemma 3.

Let $\mathbb{Q}(t)$ denote the field of fractions of $\mathbb{Z}[t_{1}^{\pm
1},\cdots,t_{\mu}^{\pm 1}]$, and let
$\varphi:\mathbb{Q}(t)\rightarrow\mathbb{Q}(t)$ be the involution induced by
$t_{i}\mapsto t_{i}^{-1}$ for all $i$. Then, we have the equality
$\varphi(K^{\mathrm{T}}SK)=K^{\mathrm{T}}SK$.

###### Proof.

We have to show that, for any two regions $a$ and $b$ of $D$, the coefficient

$(K^{\mathrm{T}}SK)_{a,b}=\sum_{v}\frac{-4\operatorname{sgn}(v)}{(t_{j}-t_{j}^{-1})(t_{k}-t_{k}^{-1})}K_{v,a}K_{v,b}$

is invariant under $\varphi$, where the sum is taken over all crossings
adjacent to both $a$ and $b$. We will consider several cases, according to the
relative positions of $a$ and $b$ with respect to the crossings.

First of all, if $a$ and $b$ are two regions of the same checkerboard color,
each common crossing $v$ contributes a term
$\frac{-4\operatorname{sgn}(v)}{(t_{j}-t_{j}^{-1})(t_{k}-t_{k}^{-1})}$ to the
coefficient $(K^{\mathrm{T}}SK)_{a,b}$. Since all these terms are invariant
under $\varphi$, this case is checked.

Next, suppose that $a$ and $b$ have different checkerboard colors. Each edge
of $D$ adjacent to both regions gives two contributions to the sum, one for
each crossing adjacent to the edge. So, let us consider a common edge with
endpoints $v$ and $v^{\prime}$, and suppose without loss of generality that
the colors and orientations of the strands are as in Figure 10, left. The two
contributions sum up to

$\frac{-4st_{j}^{-s}}{(t_{i}-t_{i}^{-1})(t_{j}-t_{j}^{-1})}+\frac{-4s^{\prime}t_{k}^{s^{\prime}}}{(t_{i}-t_{i}^{-1})(t_{k}-t_{k}^{-1})}\,,$

where $s=\operatorname{sgn}(v)$ and
$s^{\prime}=\operatorname{sgn}(v^{\prime})$. Proving that the term displayed
above is invariant under $\varphi$ is clearly equivalent to showing that

$G:=st_{j}^{-s}(t_{k}-t_{k}^{-1})+s^{\prime}t_{k}^{s^{\prime}}(t_{j}-t_{j}^{-1})$

satisfies $\varphi(G)=-G$. Expanding the products and denoting by $\chi$ the
characteristic function, one checks that $G$ is equal to

$(t_{j}t_{k}-t_{j}^{-1}t_{k}^{-1})(\chi_{s^{\prime}=1}-\chi_{s=-1})+(t_{j}t_{k}^{-1}-t_{j}^{-1}t_{k})(\chi_{s=-1}-\chi_{s^{\prime}=-1})\,,$

which is clearly antisymmetric, thus finishing this case.

Finally, let us consider the diagonal coefficient corresponding to a region
$a$. Suppose that, when moving around the boundary of $a$ counterclockwise,
one encounters $n$ crossings $v_{1},\dots,v_{n}$ of respective signs
$s_{1},\dots,s_{n}$. (It can happen that $a$ abuts the same crossing from two
sides, but since the corresponding labels are added, our computations remain
valid in this case.) Let us also number the edges of the boundary from $1$ to
$n$ as in Figure 10, right. To each edge, we assign a sign
$\varepsilon_{i}\in\\{\pm 1\\}$, where $\varepsilon_{i}=1$ if the edge $i$ is
oriented coherently with the counterclockwise orientation of the boundary of
$a$, and $\varepsilon_{i}=-1$ otherwise. Without loss of generality, we can
assume that all the edges have different colors, that we also denote by
$1,\dots,n$; in the general case, if two colors coincide, one simply needs to
identify the corresponding variables in the following computations, a
transformation which does not affect the symmetry.

$j$$k$$i$$\scriptstyle v$$\scriptstyle
v^{\prime}$$a$$b$$\scriptstyle{v_{1}}$$\scriptstyle v_{2}$$\scriptstyle
v_{3}$$\scriptstyle v_{n}$$1$$2$$3$$4$$n$$a$ Figure 10. The conventions in the
proof of Lemma 3.

With these notations and the help of Figure 3, one computes

$(K^{\mathrm{T}}SK)_{a,a}=\sum_{i=1}^{n}\frac{-4s_{i}(t_{i}^{\varepsilon_{i+1}}t_{i+1}^{-\varepsilon_{i}})^{s_{i}}}{(t_{i}-t_{i}^{-1})(t_{i+1}-t_{i+1}^{-1})}\,.$

As before, proving that this coefficient is invariant under $\varphi$ is
equivalent to showing that

$G:=\prod_{j=1}^{n}(t_{j}-t_{j}^{-1})\sum_{i=1}^{n}\frac{s_{i}(t_{i}^{\varepsilon_{i+1}}t_{i+1}^{-\varepsilon_{i}})^{s_{i}}}{(t_{i}-t_{i}^{-1})(t_{i+1}-t_{i+1}^{-1})}$

satisfies $\varphi(G)=(-1)^{n}G$. Expanding as a sum of monomials, we obtain

$\displaystyle G$ $\displaystyle=\sum_{i=1}^{n}\prod_{j\neq
i,i+1}(t_{j}-t_{j}^{-1})s_{i}(t_{i}^{\varepsilon_{i+1}}t_{i+1}^{-\varepsilon_{i}})^{s_{i}}=\sum_{i=1}^{n}\sum_{\alpha\in\\{\pm
1\\}^{n-2}}\prod_{j\neq
i,i+1}\alpha_{j}t_{j}^{\alpha_{j}}s_{i}(t_{i}^{\varepsilon_{i+1}}t_{i+1}^{-\varepsilon_{i}})^{s_{i}}$
$\displaystyle=\sum_{\beta\in\\{\pm 1\\}^{n}}c_{\beta}t_{1}^{\beta_{1}}\cdots
t_{n}^{\beta_{n}}$

for some coefficient $c_{\beta}$. To compute these coefficients explicitly,
let us define for each $\beta\in\\{\pm 1\\}^{n}$ the (possibly empty) set of
indices
$I_{\beta}=\\{i\in\\{1,\dots,n\\}\mid\beta_{i}=\varepsilon_{i+1}s_{i},\beta_{i+1}=-\varepsilon_{i}s_{i}\\}$.
We then have

$c_{\beta}=\sum_{i\in
I_{\beta}}\beta_{1}\dots\beta_{i-1}s_{i}\beta_{i+2}\dots\beta_{n}=\sum_{i\in
I_{\beta}}s_{i}\beta_{i}\beta_{i+1}(\beta_{1}\dots\beta_{n})=\beta_{1}\dots\beta_{n}\sum_{i\in
I_{\beta}}s_{i}\beta_{i}\beta_{i+1}=\beta_{1}\dots\beta_{n}d_{\beta}\,,$

with $d_{\beta}=\sum_{i\in I_{\beta}}s_{i}\beta_{i}\beta_{i+1}$. The desired
equality $\varphi(G)=(-1)^{n}G$ is equivalent to
$c_{-\beta}=(-1)^{n}c_{\beta}$ for all $\beta\in\\{\pm 1\\}^{n}$, which in
turns is equivalent to $d_{-\beta}=d_{\beta}$.

Therefore, we are left with the proof of the equality $d_{-\beta}=d_{\beta}$
for all $\beta\in\\{\pm 1\\}^{n}$. Given any such $\beta$ and any index
$i\in\\{1,\dots,n\\}$, define $\tilde{\beta}$ by
$\tilde{\beta}_{i}=-\beta_{i}$ and $\tilde{\beta}_{j}=\beta_{j}$ for $j\neq
i$. A straightforward but slightly cumbersome computation yields

$d_{\beta}-d_{\tilde{\beta}}=\begin{cases}-\varepsilon_{i}\beta_{i},&\text{ if
$(\beta_{i-1},\beta_{i+1})=(-\varepsilon_{i}s_{i-1},-\varepsilon_{i}s_{i})$}\\\
\varepsilon_{i}\beta_{i},&\text{ if
$(\beta_{i-1},\beta_{i+1})=(\varepsilon_{i}s_{i-1},\varepsilon_{i}s_{i})$}\\\
0,&\text{ otherwise. }\end{cases}$

This expression is invariant if we replace $\beta$ by $-\beta$. It thus
follows that, for any two $\beta,\beta^{\prime}\in\\{\pm 1\\}^{n}$, we have
$d_{\beta}-d_{\beta^{\prime}}=d_{-\beta}-d_{-\beta^{\prime}}$. To prove the
equality $d_{-\beta}=d_{\beta}$ for all $\beta$, we therefore only need to
check it for a single $\beta$. Taking $\beta=(1,\dots,1)$, we get

$d_{\beta}-d_{-\beta}=\sum_{i:(\varepsilon_{i},\varepsilon_{i+1})=(-s_{i},s_{i})}s_{i}-\sum_{i:(\varepsilon_{i},\varepsilon_{i+1})=(s_{i},-s_{i})}s_{i}=\sum_{i:\varepsilon_{i}\neq\varepsilon_{i+1}}\varepsilon_{i+1}=0\,,$

since there is an even number of crossings at which $\varepsilon_{i}$ changes
sign, going from $1$ to $-1$ in exactly half of the cases and form $-1$ to $1$
in the others. ∎

###  3.4. The Alexander module

We conclude this article with a slighlty informal discussion on yet another
abelian link invariant, namely the Alexander module (recall its definition
from Section 2.3).

The question we address is whether it is possible to obtain a presentation of
(the square of) the Alexander module $\mathscr{A}_{L}$ of a $\mu$-colored link
$L$ from the matrix $\widetilde{\tau}_{D}(x)$ associated to a colored diagram
$D$ for $L$. As we will see, the answer is positive in the case $\mu=1$, but
not in general.

First, recall that $\mathscr{A}_{L}$ does not admit a square presentation
matrix over $\Lambda=\mathbb{Z}[t_{1}^{\pm 1},\dots,t_{\mu}^{\pm 1}]$ if
$\Delta_{L}\neq 0$ and $\mu\geq 4$ [9]. For this reason alone, there is no
hope of answering the above question positively in general. However, the
module $\mathscr{A}_{L}$ does admit a square presentation matrix over the
localised ring

$\Lambda_{S}:=\mathbb{Z}[t_{1}^{\pm 1},\dots,t_{\mu}^{\pm
1},(t_{1}-1)^{-1},\dots,(t_{\mu}-1)^{-1}]\,.$

Moreover, by Corollary 3.6 of [4], the generalized Seifert matrices can be
used to compute such a square presentation matrix. Therefore, it is natural to
hope that the strategy developed in this work could be applied to this
invariant as well. However, since the change of variables
$2x_{j}=t_{j}^{1/2}+t_{j}^{-1/2}$ makes use of fractional powers of the
variables, we will need to work over the slightly larger ring

$\Lambda^{\prime}_{S}:=\mathbb{Z}[t_{1}^{\pm 1/2},\dots,t_{\mu}^{\pm
1/2},(t_{1}-1)^{-1},\dots,(t_{\mu}-1)^{-1}]\,.$

In the case $\mu=1$, this program can be carried out, yielding the following
result. Let $D$ be a connected diagram for an oriented link $L$. As one easily
checks, the coefficients of the matrix $\tau_{D}(x)$ are polynomials in
$2x=:y$. Let $\mathscr{M}_{D}$ denote the $\mathbb{Z}[y]$-module presented by
the matrix $\widetilde{\tau}_{D}(y)$. Then, we have an isomorphism of
$\Lambda^{\prime}_{S}$-modules

$\mathscr{M}_{D}\otimes_{\mathbb{Z}[y]}\Lambda^{\prime}_{S}\simeq\mathscr{A}_{L}^{\oplus
2}\otimes_{\Lambda}\Lambda^{\prime}_{S}\,,$

where $\Lambda^{\prime}_{S}=\mathbb{Z}[t^{\pm 1/2},(t-1)^{-1}]$ is a
$\mathbb{Z}[y]$-module via the ring homomorphism
$\mathbb{Z}[y]\to\Lambda^{\prime}_{S}$ mapping $y$ to $t^{1/2}+t^{-1/2}$, and
a $\Lambda$-module via the natural inclusion $\Lambda\to\Lambda^{\prime}_{S}$.
Less formally, one can say that the matrix $\widetilde{\tau}_{D}(y)$ is a
presentation matrix of $\mathscr{A}_{L}^{\oplus 2}$ over
$\Lambda^{\prime}_{S}$ via the substitution $y=t^{1/2}+t^{-1/2}$. Moreover, if
$L=K$ is a knot, then the multiplication by $(t-1)$ is invertible in
$\mathscr{A}_{K}$. As a consequence, the matrix $\widetilde{\tau}_{D}(y)$
presents $\mathscr{A}_{K}^{\oplus 2}$ over the ring $\mathbb{Z}[t^{\pm 1/2}]$.

Unfortunately, these results do not carry over to the case $\mu>1$. Indeed,
let $D$ be a $\mu$-colored diagram for a $\mu$-colored link $L$, and assume
that each pair of colors meet in $D$. Then, using Corollary 3.6 of [4], it is
possible to prove that the matrix $\widetilde{\tau}_{D}(x)$ presents the
Alexander module of $L\\#_{1}-L$ over the ring

$\mathbb{Z}[{\textstyle{\frac{1}{2}}},t_{1}^{\pm 1/2},\dots,t_{\mu}^{\pm
1/2},(t_{1}-1)^{-1},\dots,(t_{\mu}-1)^{-1}]$

under the substitutions $x_{j}=\frac{t_{j}+t_{j}^{-1}}{2}$ and
$x_{jk}=\frac{t_{j}t_{k}+t_{j}^{-1}t_{k}^{-1}}{2}$. However, the isomorphism

$\mathscr{A}_{L\\#_{1}L^{\prime}}\simeq\mathscr{A}_{L}\oplus\mathscr{A}_{L^{\prime}}$

that we used in the case $\mu=1$ is no longer valid in general for $\mu>1$. In
other words, the addivity under connected sum enjoyed by the other abelian
invariants considered in this work does not extend to the Alexander module in
general.

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