# A Slope invariant and the A-polynomial of knots

Leo Benard  Mathematisches Institut, Georg-August Unversität, Göttingen,
Deutschland<EMAIL_ADDRESS>,  Vincent Florens
Université de Pau, Pau, France<EMAIL_ADDRESS>and  Adrien Rodau
Université de Pau, France<EMAIL_ADDRESS>
###### Abstract.

The A-polynomial is a knot invariant related to the space of
$\operatorname{SL}_{2}(\mathbb{C})$ representations of the knot group. In this
paper our interests lies in the _logarithmic Gauss map_ of the $A$-polynomial.
We develop a homological point of view on this function by extending the
constructions of Degtyarev, the second author and Lecuona to the setting of
non-abelian representations. It defines a rational function on the character
variety, which unifies various known invariants such as the _change of curves_
in the Reidemeister function, the _modulus_ of boundary-parabolic
representations, the _boundary slope_ of some incompressible surfaces embedded
in the exterior of the knot $K$ or equivalently the slopes of the sides of the
Newton polygon of the A-polynomial $A_{K}$. We also present a method to
compute this invariant in terms of Alexander matrices and Fox calculus.

###### 2000 Mathematics Subject Classification:

Primary: 57K10, Secondary: 57K14, Tertiary: 57K31

##  1\. Introduction

The set of all representations of a knot group in
$\operatorname{SL}_{2}(\mathbb{C})$ carries naturally the structure of an
algebraic set. This holds also for the characters of these representations,
whose set is called the $\operatorname{SL}_{2}(\mathbb{C})$-character variety
of the knot. Given a peripheral structure of the knot, the character variety
is a plane curve in $\mathbb{C}^{*}\times\mathbb{C}^{*}$, whose coordinates
$M$ and $L$ correspond to the eigenvalues of the meridian $m$ and the prefered
longitude $\ell$. The polynomial $A_{K}(L,M)$ defining this curve is an
invariant of the knot, called the A-polynomial. This invariant contains many
interesting informations on the knot; in particular, Boyer and Zhang [4] and
Dunfield and Garoufalidis [13] showed that $A_{K}=L-1$ if and only if $K$ is
trivial.

In this paper, our motivations come, among others, from the following result
of Boden:

######  Theorem 1.1 ([3]).

If the $M$-degree $\deg_{M}A_{K}(L,M)$ of the $A$-polynomial is zero, then $K$
is the trivial knot.

This result motivates the systematic study of the _logarithmic Gauss map_ of
the A-polynomial

(1) $\frac{M}{L}\cdot\frac{\partial_{M}A_{K}(L,M)}{\partial_{L}A_{K}(L,M)},$

where $\partial_{M}$ and $\partial_{L}$ denote the partial derivatives. By
Theorem 1.1, this rational function vanishes identically on $\\{A_{K}=0\\}$ if
and only if $K$ is trivial.

The logarithmic Gauss map has introduced in [14] by Guelfand, Kapranov and
Zelevinsky in order to study some determinantial varieties. Then it has been
used for instance by Mikhalkin in [17] for studying the topology of
arrangements of real plane curves. In [15], Marché and Guilloux showed it is
related with the volume function of the A-polynomial of knots, or more
generally of exact polynomials.

Our proposal is to develop a homological point of view on this function, by
extending the constructions of Degtyarev, the second author and Lecuona [10,
11] to the setting of non-abelian representations. Let $K$ be an oriented knot
in the 3-sphere $S^{3}$ with exterior $M_{K}$. Denote by $R(M_{K})$ and
$X(M_{K})$ the $\operatorname{SL}_{2}(\mathbb{C})$-representation and
character varieties of the knot $K$. We consider representations
$\rho\colon\pi_{1}(M_{K})\to\operatorname{SL}_{2}(\mathbb{C})$ composed with
the adjoint action of $\operatorname{SL}_{2}(\mathbb{C})$ on the Lie algebra
$\operatorname{Ad}\colon\operatorname{SL}_{2}(\mathbb{C})\to\operatorname{Aut}(\mathfrak{sl}_{2}(\mathbb{C}))$,
and show that there is a non-empty Zariski open subset of $X(M_{K})$ such that
for all $\rho$ in this subset

  * -

there is an element $v_{\rho}\in\mathfrak{sl}_{2}(\mathbb{C})$ such that
$(v_{\rho}\otimes\ell,v_{\rho}\otimes m)$ is a basis of the homology group
$H_{1}(\partial M_{K},\operatorname{Ad}\circ\rho)\simeq\mathbb{C}^{2}$ with
coefficients twisted by $\operatorname{Ad}\circ\rho$, and

  * -

the kernel of the homomorphism induced by the inclusion:

$\mathcal{Z}(K,\operatorname{Ad}\circ\rho)=\ker\left(H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}H_{1}(M_{K},\operatorname{Ad}\circ\rho)\right)$

is generated by a single vector of the form
$a\,v_{\rho}\otimes\ell+b\,v_{\rho}\otimes m$ for some
$[b:a]\in\mathbb{C}\mathbb{P}^{1}$.

The representations which verify these conditions are called _admissible_. We
define the slope of $K$ at the admissible representation $\rho$ by

$s_{K}(\rho)=-\frac{b}{a}\in\mathbb{C}\mathbb{P}^{1}.$

We prove that representations which restrict to _non-parabolic_
representations of the boundary $\partial M_{K}$ of $M_{K}$ are admissible,
see Lemma 3.4. If $\rho$ is a boundary-parabolic representation, we define the
slope $s_{K}(\rho)$ as the modulus of the euclidean structure induced by the
restricted representation on $\pi_{1}(\partial M_{K})$, see Section 3.3. It
turns out that these two different definitions fit well and that the following
holds.

######  Proposition 1.2.

The slope depends only on the conjugacy classes of the representations and
induces a rational function

$s_{K}:X\subset X(M_{K})\longrightarrow\mathbb{C}\mathbb{P}^{1}$

on each irreducible component $X$ of the character variety.

Note that if the representation is real or unitary, then $s_{K}$ takes values
in $\mathbb{R}\mathbb{P}^{1}$ (see Proposition 3.13). For any knot, the
function $s_{K}$ can be computed by Fox calculus, see Section 3.5. We
illustrate the method in the case of the trefoil knot, and further compute the
slope of the figure-eight knot.

The following theorem relates $s_{K}$ to the original motivation; a precise
statement is given in Theorem 4.1.

######  Theorem 1.3.

The slope function $s_{K}$ equals minus the logarithmic Gauss map of the
A-polynomial defined in Eq. 1.

We also relate $s_{K}$ to the change of curve factor for the Reidemeister
torsion. Let $\mathbb{T}_{M_{K},\ell}(\rho)$ and $\mathbb{T}_{M_{K},m}(\rho)$
be the Reidemeister torsions according to homology bases induced by the
choices of the curves $\ell$ and $m$ in $\partial M_{K}$, see Section 3.4.

######  Proposition 1.4.

The slope coincides with the quotient of Reidemeister torsion:

$s_{K}(\rho)=\frac{\mathbb{T}_{M_{K},\ell}(\rho)}{\mathbb{T}_{M_{K},m}(\rho)}$

for all $\rho$ such that this formula is well-defined.

Porti had already observed ([18, Corollary 4.9]) that the logarithmic Gauss
map of the A-polynomial could be expressed as a ratio of torsions -up to a
sign-, and that this ratio of torsions is equal to the modulus of $\rho$ when
it is a boundary-parabolic representation ([18, Proposition 4.7]). Our point
of view permits to fix and compute the sign ambiguity. Moreover, our results
Proposition 1.2 and Theorem 1.3 are more general, since they do not require
the Reidemeister torsion to be well-defined, for instance they hold for high
dimensional components of the character variety.

Finally, we consider _ideal_ points of the A-polynomial, those are points
added at infinity in a compactification of the curve $\\{A(L,M)=0\\}$ in
$\mathbb{C}^{2}$. In [9], Culler and Shalen constructed incompressible
surfaces in $M_{K}$ associated to such points. Those surfaces have a non-empty
boundary, whose slope is determined by rational number $p/q$. We prove the
following theorem:

######  Theorem 1.5.

Let $y\in\mathbb{C}\mathbb{P}^{2}$ be an ideal point of the curve
$\\{A_{K}(L,M)=0\\}$. The value of $s_{K}$ at $y$ equal _minus_ the slope of
the Culler–Shalen incompressible surface associated to $y$.

This theorem sheds some light on the main theorem of [7], which states that
the boundary slopes of the Culler–Shalen surfaces are boundary slopes of the
Newton polygon of the A-polynomial. Indeed it is well-known that the
logarithmic Gauss map converges at those ideal points to the value of the
slope of the corresponding boundary of the Newton polygon.

To conclude this introduction, we mention that the slope invariant can be
extended to orthogonal (real) representations of link groups. In this more
general setting, the first twisted homology space $H_{1}(\partial M_{K},\rho)$
can have an arbitrary dimension higher than $2$ and the kernel
$\mathcal{Z}(K,\rho)$ might not be a line anymore. However, the space
$H_{1}(\partial M_{K},\rho)$ carries a natural symplectic structure given by
the (twisted) intersection form on $\partial M_{K}$, and $\mathcal{Z}(K,\rho)$
is still a Lagrangian subspace. A construction of Arnold [1] related to the
Maslov index allows to construct a _generalized slope_ for this context, lying
in $S^{1}\subset\mathbb{C}^{*}$. As it turns out, in the case of a
representation $\rho\colon\pi_{1}(M_{K})\to\operatorname{SU}(2)$, both
theories coincide via the natural isomorphism $\mathbb{R}\mathbb{P}^{1}\simeq
S^{1}$. We postpone rigorous definitions and further study of this invariant
to an upcoming article.

### Organisation of the paper

In Section 2 we collect basic definitions on character varieties and
A-polynomials. In Section 3 we define the slope invariant and we prove
Proposition 1.2 and Proposition 1.4. In Section 4 we prove Theorem 1.3.
Finally, in Section 5 we prove Theorem 1.5.

### Aknowledgements

Léo Bénard is a member of the Research Training Group 2491 "Fourier Analysis
and Spectral Theory", University of Göttingen. The authors thank Julien
Marché, Stepan Yu. Orevkov and Joan Porti for related conversations.

##  2\. Representation varieties, character varieties and A-polynomial.

This section is devoted to definitions and properties of representations
spaces and character varieties (Section 2.1). We compute the character variety
of the group $\mathbb{Z}^{2}$ in Section 2.2 and define the A-polynomial of
knots in Section 2.3. References for character varieties are [19, 20], the
A-polynomial was first defined in [7], see also [6].

###  2.1. Representation and character varieties

Let $\Gamma$ be a finitely generated group. The _representation variety_ is
the affine algebraic set

$R(\Gamma)=\text{Hom}(\Gamma,\operatorname{SL}_{2}(\mathbb{C})).$

If $\Gamma$ is generated by $n$ elements, the representation variety is an
algebraic subset of $\operatorname{SL}_{2}(\mathbb{C})^{n}$ given by
polynomial relations corresponding to the relations of the group $\Gamma$. Two
different presentations yield naturally isomorphic algebraic sets.

A representation $\rho:\Gamma\rightarrow\operatorname{SL}_{2}(\mathbb{C})$ is
_abelian_ is $\rho(\Gamma)$ is an abelian subgroup of
$\operatorname{SL}_{2}(\mathbb{C})$. A representation $\rho$ is _reducible_ if
there exists a proper subspace of $\mathbb{C}^{2}$ invariant under the action
of $\rho(\Gamma)$. Equivalently, $\rho(\Gamma)$ is conjugated to a subgroup of
the group of upper-triangular matrices in $\operatorname{SL}_{2}(\mathbb{C})$.
Abelian representations are reducible, but the converse does not hold. Non-
reducible representations are _irreducible_.

Two representations $\rho$ and $\rho^{\prime}$ in $R(\Gamma)$ are equivalent
if they have the same trace:

$\rho\sim\rho^{\prime}\text{ if and only if
}\operatorname{Tr}\rho(\gamma)=\operatorname{Tr}\rho^{\prime}(\gamma)\text{,
for any }\gamma\in\Gamma.$

The set of equivalence classes of representations coincides with the algebro-
geometric quotient of $R(\Gamma)$ by the action of
$\operatorname{SL}_{2}(\mathbb{C})$ by conjugation. This quotient is usually
constructed through invariant theory, and is denoted

$X(\Gamma)=R(\Gamma)/\\!/\operatorname{SL}_{2}(\mathbb{C}).$

Points of the character variety are called _characters_. The equivalence class
of a representation $\rho$ (the character of $\rho)$ is denoted by
$\chi_{\rho}:\Gamma\rightarrow\mathbb{C}$ with
$\chi_{\rho}(\gamma)=\operatorname{Tr}(\rho(\gamma))$ for $\gamma\in\Gamma$.
If $\Gamma$ is the fundamental group of a manifold $W$, we simply write $R(W)$
and $X(W)$ for the representation and character varieties of the manifold $W$.

Despite being abelian is not a well-defined notion on the character variety,
the notion of being reducible makes sense there, since a reducible
representation $\rho\colon\Gamma\to\operatorname{SL}_{2}(\mathbb{C})$ can be
characterized by the fact that for any $\gamma,\delta\in\Gamma$, the following
equality holds (see for instance [9, Lemma 1.2.1]):

(2) $\operatorname{Tr}\rho\left(\gamma\delta\gamma^{-1}\delta^{-1}\right)=2.$

The character variety $X(\Gamma)$ can be decomposed as

$X(\Gamma)=X^{\text{irr}}(\Gamma)\cup X^{\text{red}}(\Gamma),$

where $X^{\text{red}}(\Gamma)$ is the set of reducible characters, and its
complement $X^{\text{irr}}(\Gamma)$ is the set of irreducible characters. Eq.
2 implies that $X^{\text{red}}(\Gamma)$ is a Zariski closed subset of
$X(\Gamma)$.

An algebraic set is _reducible_ if it can be written as a union of two proper
algebraic subset, else it is _irreducible_. An irreducible component of an
algebraic set is a maximal irreducible algebraic subset.

######  Remark 2.1.

Despite $R(\Gamma)$ or $X(\Gamma)$ are called _varieties_ , they are not quite
algebraic varieties in general: they are actually reducible, and might not be
reduced as schemes (some points or subspaces might have multiplicity). On the
other hand, any irreducible component is irreducible, and in particular
reduced, by definition.

Two representations $\rho$ and $\rho^{\prime}$ are _conjugate_ if there exists
a matrix $M\in\operatorname{SL}_{2}(\mathbb{C})$ such that
$\rho(\gamma)=M\rho^{\prime}(\gamma)M^{-1}$ for every $\gamma\in\Gamma$. Two
conjugate representations define the same character; the converse is false in
general, but true for elements of $X^{\text{irr}}(\Gamma)$. More precisely,
the following holds.

######  Proposition 2.2.

[9, Proposition 1.5.2] If $\rho$ and $\rho^{\prime}$ are two representations
$\Gamma\^{A}~\to\operatorname{SL}_{2}(\mathbb{C})$ with $\rho$ irreducible and
$\chi_{\rho}=\chi_{\rho^{\prime}}$, then $\rho$ and $\rho^{\prime}$ are
conjugate (and $\rho^{\prime}$ is irreducible as well).

Two non-conjugate representations having the same character in $X(\Gamma)$
must be reducible. If $\Gamma$ is a knot group, Burde and De Rham [5, 12]
showed that the set of characters containing non-conjugate representations is
finite.

###  2.2. The character variety of $\mathbb{Z}^{2}$

We describe explicitly the character variety of a 2-torus $S^{1}\times S^{1}$.
Pick a basis $m,\ell$ of $\pi_{1}(S^{1}\times S^{1})=\mathbb{Z}^{2}$. Any
representation in $\operatorname{SL}_{2}(\mathbb{C})$ is conjugate to a
representation $\rho$ given by two commuting matrices of the form

$\rho(m)=\begin{pmatrix}M&*\\\
0&M^{-1}\end{pmatrix},\quad\rho(\ell)=\begin{pmatrix}L&*\\\
0&L^{-1}\end{pmatrix}$

for $M,L\in\mathbb{C}^{*}$. Each point of the character variety $X(S^{1}\times
S^{1})$ has a pre-image in $R(S^{1}\times S^{1})$ of the form

(3) $\rho(m)=\begin{pmatrix}M&0\\\
0&M^{-1}\end{pmatrix},\quad\rho(\ell)=\begin{pmatrix}L&0\\\
0&L^{-1}\end{pmatrix},\quad M,L\in\mathbb{C}^{*}.$

This pre-image is unique up to the involution $\sigma$ of
$(\mathbb{C}^{*})^{2}$ sending $(L,M)$ to $(L^{-1},M^{-1})$, and
$X(S^{1}\times S^{1})$ can be identified with the singular affine complex
surface $(\mathbb{C}^{*})^{2}/\sigma$. It embedds in $\mathbb{C}^{3}$ as the
zeros of the polynomial

$\Delta=x^{2}+y^{2}+z^{2}-xyz-4.$

Indeed, the function algebra of $X(S^{1}\times S^{1})$ naturally identifies
with the $\sigma$-invariant sub-algebra $\mathbb{C}[M+M^{-1},L+L^{-1}]$ of
$\mathbb{C}[L^{\pm 1},M^{\pm 1}]$. This algebra of invariant functions is
isomorphic with $\mathbb{C}[x,y,z]/(\Delta)$ through $M+M^{-1}\mapsto
x,L+L^{-1}\mapsto y,ML+(ML)^{-1}\mapsto z$. From this description, one sees
that the singular locus of $X(S^{1}\times S^{1})$ consists on the four points
$\\{(L,M)=(\pm 1,\pm 1)\\}$.

###  2.3. The A-polynomial

Let $K$ be an oriented knot in $S^{3}$ with exterior $M_{K}$. The inclusion
$\partial M_{K}\subset M_{K}$ induces an injective group homomorphism
$\pi_{1}(\partial M_{K})\hookrightarrow\pi_{1}(M_{K})$. Let $r$ be the
restriction map:

$r\colon X(M_{K})\longrightarrow X(\partial M_{K})\simeq X(S^{1}\times
S^{1}).$

For short we denote by $\rho_{\partial}=r(\rho)$ the restriction of $\rho$ to
$\pi_{1}(\partial M_{K})$. By Section 2.2, the choice of the longitude $\ell$
and the meridian $m$ induces an identification of $X(S^{1}\times S^{1})$ with
a quotient of $(\mathbb{C}^{*})^{2}$. The image of $r$ is a union of points
and curves, possibly with multiplicities, see for instance [13, Lemma 2.1].
Discarding the $0$-dimensional components, the _A-polynomial_ of $K$ is the
unique polynomial $A_{K}(L,M)$ in $\mathbb{C}[L,M]$ whose zero locus in
$\mathbb{C}^{2}$ is exactly mapped onto the image of $r$. Note that
$A_{K}(L,M)$ is always divisible by $L-1$. This factor corresponds to the
curve of reducible characters. Boyer, Zhang, Dunfield and Garoufalidis have
shown the following result.

######  Theorem 2.3 ([4, 13]).

Let $K$ be a knot in $S^{3}$. The A-polynomial $A_{K}(L,M)$ is equal to
$(L-1)^{k}$ for some $k$, if and only if $K$ is the trivial knot (and in this
case $k=1$).

##  3\. The Slope invariant

In this section we define the slope of an admissible representation (Section
3.1), and observe that generic
$\operatorname{SL}_{2}(\mathbb{C})$-representations are admissible. In Section
3.2 we show that the slope is invariant by conjugation of the representation.
We prove in Section 3.3 that it yields a rational function on irreducible
components of the character variety and that the slope of a real
representation is a real number. Then we prove in Section 3.4 that the slope
can be written as a quotient of Reidemeister torsions. Finally, in Section 3.5
we describe a procedure to compute the slope with an Alexander matrix.

###  3.1. Admissible representations

Let $V$ be a finite dimensional $\mathbb{C}$-vector space, and
$\rho\colon\pi_{1}(M_{K})\to\operatorname{GL}(V)$ be a representation. The
representation extends to a ring homomorphism and $V$ can be viewed as a right
$\mathbb{Z}[\pi_{1}(M_{K})]$-module $V_{\rho}$. The twisted homology
$H_{*}(M_{K},\rho)$ is the homology of the complex of $\mathbb{C}$-vector
spaces:

$C_{*}(M_{K};\rho)=V_{\rho}\otimes_{\mathbb{Z}[\pi_{1}(M_{K})]}C_{*}(M_{K};\mathbb{Z}[\pi_{1}(M_{K})]).$

######  Definition 3.1.

A representation $\rho\colon\pi_{1}(M_{K})\to\operatorname{GL}(V)$ is
_admissible_ if it satisfies:

  * -

there exists $v_{\rho}\in V$ such that $\\{v_{\rho}\otimes\ell,v_{\rho}\otimes
m\\}$ is a basis of $H_{1}(\partial M_{K},V_{\rho})\simeq\mathbb{C}^{2}$,

  * -

the kernel of the homomorphism induced by inclusion:

$\mathcal{Z}(K,\rho)=\ker\left(H_{1}(\partial
M_{K},V_{\rho})\stackrel{{\scriptstyle
i_{*}}}{{\longrightarrow}}H_{1}(M_{K},V_{\rho})\right)$

has dimension one.

We restrict to representations
$\rho\colon\pi_{1}(M_{K})\rightarrow\operatorname{SL}_{2}(\mathbb{C})$. The
composition of $\rho$ with the adjoint action $\operatorname{Ad}$ of
$\operatorname{SL}_{2}(\mathbb{C})$ on $\mathfrak{sl}_{2}(\mathbb{C})$ induces
the following representation:

$\begin{array}[]{rccc}\operatorname{Ad}\circ\rho:&\pi_{1}(M_{K})&\longrightarrow&\mathrm{Aut}(\mathfrak{sl}_{2}(\mathbb{C}))\\\
&\gamma&\longmapsto&\left(v\mapsto\rho(\gamma)v\rho(\gamma)^{-1}\right).\end{array}$

######  Definition 3.2.

Let $\rho\colon\pi_{1}(M_{K})\to\operatorname{SL}_{2}(\mathbb{C})$ be such
that $\operatorname{Ad}\circ\rho$ is admissible. Let
$a\,(v_{\rho}\otimes\ell)+b\,(v_{\rho}\otimes m)$ be a generator of
$\mathcal{Z}(K,\operatorname{Ad}\circ\rho)$ for some
$[a:b]\in\mathbb{C}\mathbb{P}^{1}$ . The _slope_ of the knot $K$ at the
representation $\rho$ is

$s_{K}(\rho)=-\frac{b}{a}\in\mathbb{C}\cup\infty.$

######  Definition 3.3.

A representation
$\rho\colon\pi_{1}(M_{K})\to\operatorname{SL}_{2}(\mathbb{C})$ is _boundary-
parabolic_ if the restriction $\rho_{\partial}:\pi_{1}(\partial
M_{K})\to\operatorname{SL}_{2}(\mathbb{C})$ is parabolic, that is
$\operatorname{Tr}\rho(\gamma)=\pm 2$ for any $\gamma\in\pi_{1}(\partial
M_{K})$.

A boundary-parabolic character is the character of a boundary-parabolic
representation.

######  Lemma 3.4.

Let $\rho\colon\pi_{1}(M_{K})\to\operatorname{SL}_{2}(\mathbb{C})$ be a non-
parabolic representation. The vector space $H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)$ is isomorphic to $\mathbb{C}^{2}$, and the
kernel $\mathcal{Z}(M,\operatorname{Ad}\circ\rho)$ is one-dimensional.
Moreover, if $\rho$ is not boundary-parabolic, then
$\operatorname{Ad}\circ\rho$ is admissible.

###### Proof.

The group $\pi_{1}(M_{K})$ is generated by pairwise conjugate meridians. If
$\rho$ is non-parabolic, then the image of a meridian must differ to $\pm
I_{2}$, otherwise we would have $\rho(\pi_{1}(M_{K}))\subset\\{\pm I_{2}\\}$.

Consider the complex $C_{*}(\partial M_{K},\operatorname{Ad}\circ\rho)$ with
one $0$-cell, two $1$-cells corresponding to $\ell$ and $m$ and one $2$-cell.
An explicit computation of the homology of shows that the dimension of
$H_{1}(\partial M_{K},\operatorname{Ad}\circ\rho)$ is two. Moreover, when
$\rho$ is not boundary-parabolic, for
$v_{\rho}\in\mathfrak{sl}_{2}(\mathbb{C})$ invariant by
$\operatorname{Ad}\circ\rho_{\partial}$, the pair of vectors
$(v_{\rho}\otimes\ell,v_{\rho}\otimes m)$ forms a basis of $H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)$. Since the Killing form on the local system
$\mathfrak{sl}_{2}(\mathbb{C})$ yields a Poincaré duality isomorphism
$H_{*}(M_{K},\operatorname{Ad}\circ\rho)\simeq H_{3-*}(M_{K},\partial
M_{K},\operatorname{Ad}\circ\rho)$, the following diagram is commutative

${H_{2}(M,\partial M_{K},\operatorname{Ad}\circ\rho)}$${H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)}$${H_{1}(M_{K},\operatorname{Ad}\circ\rho)}$${H_{1}(M_{K},\operatorname{Ad}\circ\rho)^{\ast}}$${H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)^{\ast}}$${H_{2}(M_{K},\partial
M_{K},\operatorname{Ad}\circ\rho)^{\ast}}$$\scriptstyle{`d}$$\scriptstyle{i_{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{`d^{*}}$

with exact rows and where the vertical arrows are isomorphisms. Exactness
implies

$\dim\ker
i_{*}=\operatorname{rank}\delta=\operatorname{rank}i^{*}=\operatorname{rank}i_{*}$

since the diagram commutes and transposition preserves the rank. Hence
$\dim\mathcal{Z}(K,\operatorname{Ad}\circ\rho)=\dim\ker i_{*}=(1/2)\dim
H_{1}(\partial M,\operatorname{Ad}\circ\rho).$ ∎

As an example, we compute the slope for abelian non boundary-parabolic
representations. Let $\varphi\colon\pi_{1}(M_{K})\to H_{1}(M_{K})=\mathbb{Z}$
be the abelianization. For any $\lambda\in\mathbb{C}^{*}$, there is an abelian
representation

(4) $\displaystyle\rho_{\lambda}\colon\pi_{1}(M_{K})$
$\displaystyle\longrightarrow\operatorname{SL}_{2}(\mathbb{C})$
$\displaystyle\gamma$
$\displaystyle\longmapsto\left(\begin{smallmatrix}\lambda^{\varphi(\gamma)}&0\\\
0&\lambda^{-\varphi(\gamma)}\end{smallmatrix}\right)$

and any abelian, non boundary-parabolic representation is conjugate to a
representation of this form.

######  Lemma 3.5.

For any $\lambda\neq\pm 1$, the slope at the abelian representation
$\rho_{\lambda}$ vanishes:

$s_{K}(\rho_{\lambda})=0.$

###### Proof.

Up to conjugation, the representation $\operatorname{Ad}\circ\rho$ has the
form

$\operatorname{Ad}\circ\rho(\gamma)=\left(\begin{smallmatrix}\lambda^{2\varphi(\gamma)}&0&0\\\
0&1&0\\\ 0&0&\lambda^{-2\varphi(\gamma)}\end{smallmatrix}\right)$

and $\mathfrak{sl}_{2}(\mathbb{C})$ splits as
$\mathbb{Z}[\pi_{1}(M_{K})]$-module as

$\mathfrak{sl}_{2}(\mathbb{C})=\mathbb{C}_{\lambda^{2}}\oplus\mathbb{C}\oplus\mathbb{C}_{\lambda^{-2}}.$

This yields a splitting in twisted homology (with abelian coefficients), for
$U=\partial M_{K}$ or $U=M_{K}$:

$H_{1}(U,\operatorname{Ad}\circ\rho)=H_{1}\left(U,\mathbb{C}_{\lambda^{2}}\right)\oplus
H_{1}(U,\mathbb{C})\oplus H_{1}\left(U,\mathbb{C}_{\lambda^{-2}}\right)$

Since $\lambda\neq\pm 1$, for $U=\partial M_{K}$ the only non-trivial summand
is $H_{1}(\partial M_{K},\mathbb{C})$, and the map $H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)\xrightarrow{i_{*}}H_{1}(M_{K},\operatorname{Ad}\circ\rho)$
coincides with the map induced by the inclusion in homology with trivial
coefficients $H_{1}(\partial M_{K},\mathbb{C})\to H_{1}(M_{K},\mathbb{C})$,
whose kernel is generated by $\ell$. ∎

###  3.2. The slope of characters

By the following lemma, the slope does not depend on the conjugacy class of an
irreducible representation. Combined with Proposition 2.2, it follows that the
slope of an irreducible representation depends only on its character.

######  Lemma 3.6.

Let $\rho$ and
$\rho^{\prime}\colon\pi_{1}(M_{K})\to\operatorname{SL}_{2}(\mathbb{C})$ be two
irreducible, non boundary-parabolic representations. If $\rho$ and
$\rho^{\prime}$ are conjugate, then $s_{K}(\rho)=s_{K}(\rho^{\prime})$.

###### Proof.

Let $A$ be a matrix in $\operatorname{GL}_{2}(\mathbb{C})$ such that
$\rho^{\prime}=A\,\rho\,A^{-1}$. Any $\operatorname{Ad}\circ\rho$-invariant
vector $v_{\rho}\^{A}~\in\mathfrak{sl}_{2}(\mathbb{C})$ yields an
$\operatorname{Ad}\circ\rho^{\prime}$-invariant vector
$v^{\prime}_{\rho}=A\,v_{\rho}\,A^{-1}$, and the conjugation by $A$ induces an
isomorphism

$H_{1}(\partial M_{K},\operatorname{Ad}\circ\rho)\to H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho^{\prime})$

sending the basis $\\{v_{\rho}\otimes\ell,v_{\rho}\otimes m\\}$ to
$\\{v^{\prime}_{\rho}`(`*)\ell,v^{\prime}_{\rho}`(`*)m\\}$ and the subspace
$\mathcal{Z}(K,\operatorname{Ad}\circ\rho)$ to
$\mathcal{Z}(K,\operatorname{Ad}\circ\rho^{\prime})$. Hence
$s_{K}(\rho)=s_{K}(\rho^{\prime})$. ∎

######  Remark 3.7.

There exist pairs of reducible, non-conjugate representations with the same
character. Indeed, let $\chi$ be an arbitrary reducible character in
$X(M_{K})$. Consider a representation $\rho$ of the form
$\left(\begin{smallmatrix}\lambda(\gamma)&*\\\
0&\lambda^{-1}(\gamma)\end{smallmatrix}\right)$, where
$\lambda\colon\pi_{1}(M_{K})\to\mathbb{C}^{*}$ is a group homomorphism,
choosen such that $\chi(\rho)=\chi$. Note that $\lambda$ can further be
written $\lambda(\gamma)=\lambda^{\varphi(\gamma)}$ for some
$\lambda\in\mathbb{C}^{*}$ and $\varphi:\pi_{1}(M_{K})\to\mathbb{Z}$. Hence
the abelian representation $\rho_{\lambda}$ defined in Eq. 4 has also
character $\chi$, but is not conjugated in general to $\rho$. It turns out
that they can have different slope values.

For example, consider the right-handed trefoil knot $T$ in $S^{3}$. The
character variety $X(M_{T})$ is the union of a line $X^{\text{red}}$ and a
conic $X^{\text{irr}}$ in the plane. The line contains only reducible
characters, and any character in the conic is irreducible except the two
intersection points $X^{\text{red}}\cap X^{\text{irr}}$. Let $\chi$ be a point
in $X^{\text{red}}\cap X^{\text{irr}}$. Since $\chi$ is reducible, there exist
$\lambda\in\mathbb{C}^{*}$ such that the abelian representation
$\rho_{\lambda}$ has character $\chi$. By Lemma 3.5, one has
$s_{T}(\rho_{\lambda})=0$. However, we show in Example 3.19 that the slope
defines a constant function on $X^{\text{irr}}$, everywhere equal to $-6$.

###  3.3. Regularity and properties of the slope

We extend the slope to a rational function –locally a quotient of polynomials–
on the character variety $X(M_{K})$.

There is a component $X^{\text{red}}\subset X(M_{K})$ of reducible characters
only. By Remark 3.7 any character in $X^{\text{red}}$ is the character of an
abelian representation. Hence the slope is identically zero on
$X^{\text{red}}$, see Lemma 3.5. Suppose now that $X\subset X(M)$ is an
irreducible component containing an irreducible character.

######  Proposition 3.8.

Let $X\subset X(M)$ be an irreducible component which contains an irreducible
character. The slope extends to a rational function on $X$, still denoted
$s_{K}$. Moreover, if $\chi\in X$ is a boundary-parabolic character then

(5) $s_{K}(\chi)=\tau(\chi),$

where the modulus $\tau(\chi)\in\mathbb{C}$ is defined by taking the
representative $\rho$ of $\chi$ satisfying

$\rho(m)=\begin{pmatrix}\pm 1&1\\\ 0&\pm
1\end{pmatrix},\quad\rho(\ell)=\begin{pmatrix}\pm 1&\tau(\chi)\\\ 0&\pm
1\end{pmatrix}.$

######  Remark 3.9.

If $\chi$ is the character of an irreducible representation and lies at the
intersection of several irreducible components, then the value of the slope at
$\chi$ is well-defined.

The rest of the section is devoted to the proof of Proposition 3.8. Lemma 3.10
asserts that the slope is a rational function in the neighborhood of any
irreducible, non boundary-parabolic character. For boundary parabolic
characters $\chi$, we _define_ the slope by the relation in Eq. 5 and we show
that the result is still a rational function on $X$ in Lemma 3.12.

######  Lemma 3.10.

Let $\chi_{0}$ an irreducible, non-boundary-parabolic character in $X$. The
slope is a rational function in a neighborhood of $\chi_{0}$ in $X$.

###### Proof.

Let $\rho_{0}$ in $R(M_{K})$ be a representation with character $\chi_{0}$. By
Lemma 3.4 one has $H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)\simeq\mathbb{C}^{2}$. The set of complex
lines

$\mathbb{P}(\rho)=\mathbb{P}(H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho))$

is a complex algebraic variety isomorphic to $\mathbb{C}P^{1}$. If $\rho$ and
$\rho^{\prime}$ are conjugate, then there is a natural algebraic isomorphism
$\mathbb{P}(\rho)\simeq\mathbb{P}(\rho^{\prime})$. It defines an algebraic
$\mathbb{C}\mathbb{P}^{1}$-fibration on a neighborhood of $\chi_{\rho_{0}}$,
and for any $\chi$, the complex line
$\mathcal{Z}(M_{K},\operatorname{Ad}\circ\rho)$ is an algebraic section of
this fibration, independent of the choice of representation $\rho$ with
character $\chi$.

It remains to show that the identification
$\mathbb{P}(\rho)\simeq\mathbb{C}\mathbb{P}^{1}$ is algebraic, in other words,
that the choice of the basis $(v_{\rho}\otimes\ell,v_{\rho}\otimes m)$ depends
algebraically on $\rho$. Since $\rho_{0}$ is not boundary-parabolic, we can
shrink the chosen neighborhood so that no representation $\rho$ near
$\rho_{0}$ is boundary-parabolic. Then, since $\rho_{\partial}$ is conjugated
to a diagonal representation, there is a unique
$\operatorname{Ad}\circ\rho_{\partial}$-invariant vector $v_{\rho}$ with norm
$1$ in $\mathfrak{sl}_{2}(\mathbb{C})$. This choice depends polynomially on
the entries of the matrix $\operatorname{Ad}\circ\rho(m)$, and then the basis
$(v_{\rho}\otimes\ell,v_{\rho}\otimes m)$ depends algebraically on $\rho$. ∎

We now consider the case of boundary-parabolic characters.

######  Lemma 3.11.

Let $\rho_{0}$ be a boundary-parabolic representation whose character
$\chi_{\rho_{0}}$ lies in $X$. Then $\rho_{0}$ is irreducible, in particular
$\rho_{0}(m)\neq\pm I_{2}$.

###### Proof.

For $\rho$ reducible in $X$, [5, 12] implies that $\rho(m)$ has eigenvalues
$\lambda,\lambda^{-1}$ in $\mathbb{C}$, whose square is a root of the
Alexander polynomial $\Delta_{M_{K}}(t)$, in particular $\lambda\neq\pm 1$,
and $\rho$ is not boundary-parabolic. Now for irreducible $\rho$, the image of
any meridian must be different of $I_{2}$, since meridians generate the group
$\pi_{1}(M_{K})$. ∎

######  Lemma 3.12.

Let $X\subset X(M)$ be an irreducible component containing an irreducible
character, and $\chi_{0}\in X$ a boundary-parabolic character. Then the slope
function $s_{K}$ is rational in a neighborhood of $\chi_{0}$.

###### Proof.

Suppose first that $X$ contains only boundary-parabolic characters. Any
$\chi\in X$ is the character of a representation $\rho$ such that
$\rho(m)=\left(\begin{smallmatrix}\pm 1&1\\\ 0&\pm 1\end{smallmatrix}\right)$.
Hence $\chi\mapsto\tau(\chi)$ is rational.

Now we assume that $X$ contains a non boundary-parabolic character. By
definition, boundary-parabolic characters form a Zariski closed subset of $X$.
By Lemma 3.10, the slope function is rational on the open, non-empty subset of
$X$ consisting of non boundary-parabolic characters. By analytic continuation,
it is enough to show that

$\lim_{\chi\to\chi_{0}}\^{A}~s_{K}(\chi)=\tau(\chi_{0}).$

By Lemma 3.11 any boundary-parabolic representation $\rho_{0}$ with character
$\chi_{0}$ is irreducible. Moreover, since $\rho_{0}(m)$ can not be trivial,
we can chose such a $\rho_{0}$ satifying

$\rho_{0}(m)=\left(\begin{smallmatrix}\pm 1&1\\\ 0&\pm
1\end{smallmatrix}\right).$

For any $\chi$ close to $\chi_{0}$, we chose similarly a representation $\rho$
with character $\chi$ such that

$\rho(m)=\left(\begin{smallmatrix}M&1\\\ 0&M^{-1}\end{smallmatrix}\right),$

with $M$ close to $\pm 1$ in $\mathbb{C}^{*}$. For such $\rho$, let
$v_{\rho}=\left(\begin{smallmatrix}M-M^{-1}&2\\\
0&M^{-1}-M\end{smallmatrix}\right)$ be an
$\operatorname{Ad}\circ\rho_{\partial}$-invariant vector. The limit at
$\rho_{0}$ of $v_{\rho}$ is the
$(\operatorname{Ad}\circ\rho_{0})_{\partial}$-invariant vector
$v_{\rho_{0}}=\left(\begin{smallmatrix}0&2\\\ 0&0\end{smallmatrix}\right)$.
However, a direct computation shows that $v_{\rho_{0}}\otimes\ell$ and
$v_{\rho_{0}}\otimes m$ are linearly dependent in $H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho_{0})$, and we cannot compute the slope of the
boundary parabolic representation $\rho_{0}$ by means of Definition 3.2.
Nevertheless, the subspace $\mathcal{Z}(K,\operatorname{Ad}\circ\rho_{0})$ is
one-dimensional by Lemma 3.4.

It implies that the map $i_{*}\colon H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)\to H_{1}(M_{K},\operatorname{Ad}\circ\rho)$
has rank one at any representation $\rho$ near $\rho_{0}$, and at $\rho_{0}$
as well. In particular, for any $\rho$ near $\rho_{0}$, the slope can be
computed as the ratio of $i_{*}(v_{\rho}\otimes\ell)$ and
$i_{*}(v_{\rho}\otimes m)$ in $i_{*}(H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho))$. This actually makes sense for
$\rho=\rho_{0}$ as well. An explicit computation of the boundary operator
$\partial_{1}\colon C_{2}(\partial M_{K},\operatorname{Ad}\circ\rho_{0})\to
C_{1}(\partial M_{K},\operatorname{Ad}\circ\rho_{0})$ shows that the vector
$v_{\rho_{0}}\otimes\ell-\tau(\chi_{0})\,v_{\rho_{0}}\otimes m$ belongs to
$\operatorname{im}\partial_{2}$, and the equality

$v_{\rho_{0}}\otimes\ell=\tau(\chi_{0})\,v_{\rho_{0}}\otimes m$

holds in $H_{1}(\partial M_{K},\operatorname{Ad}\circ\rho_{0})$. This implies
that the ratio of $i_{*}(v_{\rho_{0}}\otimes\ell)$ and
$i_{*}(v_{\rho_{0}}\otimes m)$ coincides with the modulus $\tau(\chi_{0})$.
This proves the lemma, and achieves the proof of Proposition 3.8. ∎

We end up this section with the following observation.

######  Proposition 3.13.

Let $X\subset X(M)$ be an irreducible component which contains a non boundary-
parabolic representation. If $\rho\in X$ is a _real_ representation
$\rho\colon\pi_{1}(M_{K})\to\operatorname{SL}_{2}(\mathbb{R})$ or
$\rho\colon\pi_{1}(M_{K})\to\operatorname{SU}(2)$, then the slope is a real
number in $\mathbb{R}\mathbb{P}^{1}$.

###### Proof.

First assume that $\rho$ is non-boundary-parabolic. If $\rho$ is real,
denoting by $\operatorname{Ad}\circ\rho_{\mathbb{R}}$ the action of $\rho$ on
the Lie algebra $\mathfrak{sl}_{2}(\mathbb{R})$ (resp. $\mathfrak{su}(2)$) of
$\operatorname{SL}_{2}(\mathbb{R})$ (resp. $\operatorname{SU}(2)$), then
obviously the Lagrangian $\mathcal{Z}(M_{K},\operatorname{Ad}\circ\rho)\subset
H_{1}(\partial M_{K},\operatorname{Ad}\circ\rho)$ is the complexification of
the real Lagrangian
$\mathcal{Z}(M_{K},\operatorname{Ad}\circ\rho_{\mathbb{R}})$ in the real
symplectic vector space $H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho_{\mathbb{R}})$ and the slope of this real
Lagrangian is the slope of its complexification, a real number. If $\rho$ is
boundary-parabolic and re al, then it takes value into
$\operatorname{SL}_{2}(\mathbb{R})$ and the proposition follows from the
definition of the modulus $\tau$. ∎

###  3.4. Slope and Reidemeister torsion

In this section we show that the slope coincides with the "change of curve
term" for the Reidemeister torsion as stated in Proposition 1.4.

If $\rho$ is an irreducible representation in $X(M_{K})$, we consider the
torsion of the complex $C_{*}(M_{K},\operatorname{Ad}\circ\rho)$ defined in
Section 3.1. This complex is naturally based from a cell decomposition of
$M_{K}$ and a choice of a basis of $\mathfrak{sl}_{2}(\mathbb{C})$, but not
acyclic. The Reidemeister torsion is usually defined for acyclic complexes. In
the case we are considering, one needs to make some additional choices to
define it, namely a basis of each homology group
$H_{*}(M_{K},\operatorname{Ad}\circ\rho)$.

According to [18], one can still define the Reidemeister torsion of the
cellular complex $C_{*}(M_{K},\operatorname{Ad}\circ\rho)$ for representations
$\rho$ in $R(M_{K})$ such that $H_{1}(M_{K},\operatorname{Ad}\circ\rho)$ has
dimension $1$. For a given curve $\gamma\in\pi_{1}(\partial M_{K})$, the
representation $\rho$ is _$\gamma$ -regular_ if there exists a vector
$v_{\rho}\in\mathfrak{sl}_{2}(\mathbb{C})$ such that $v_{\rho}\otimes\gamma$
spans $H_{1}(M_{K},\operatorname{Ad}\circ\rho)$. In this case, since there is
a natural choice of a basis of $H_{2}(M_{K},\operatorname{Ad}\circ\rho)$, the
curve $\gamma$ determines a homology basis of the complex
$C_{*}(M_{K},\operatorname{Ad}\circ\rho)$ and the torsion
$\mathbb{T}_{M_{K},\gamma}(\operatorname{Ad}\circ\rho)\in\mathbb{C}^{*}$ is
defined. Note that this torsion depends only on the conjugacy class of $\rho$,
as well as the property of being $\gamma$-regular.

Let $X\subset X(M_{K})$ the component containing $\chi$, the torsion function
is the rational function

$\mathbb{T}_{M_{K},\gamma}\colon X\to\mathbb{C}$

defined as the Reidemeister torsion of the complex
$C_{*}(M_{K},\operatorname{Ad})$ if $\chi$ is $\gamma$-regular, and by
$T_{M_{K},\gamma}(\chi)=0$ otherwise.

We start with the following lemma, which provides the genuine setting to
define the Reidemeister torsion.

######  Lemma 3.14.

If $X$ has dimension one and contains the character of a scheme-smooth
representation $\rho$, then $\dim H_{1}(M_{K},\operatorname{Ad}\circ\rho)=1$.

###### Proof.

The proof of Lemma 3.14 follows from the isomorphism between
$H^{1}(M_{K},\operatorname{Ad}\circ\rho)$ and the Zariski tangent space of
$X(M_{K})$ at $\rho$, see [20, Theorem 1] . Scheme-smoothness implies that the
Zariski tangent space is the actual tangent space, which is one-dimensional
because $X$ is. ∎

Note that scheme-smoothness is a Zariski open condition.

It turns out that the character variety $X(M_{K})$ of a knot exterior is often
one-dimensional. This is the case if the knot is _small_ (if it does not
contains a closed incompressible surface [7, Proposition 2.4]). This is also
the case for any component $X\subset X(M_{K})$ containing the character of a
lift of the holonomy representation
$\overline{\rho}\colon\pi_{1}(M_{K})\to\operatorname{PSL}_{2}(\mathbb{C})$,
provided that the interior of $M_{K}$ admits a hyperbolic structure.

The following proposition is the main result of this section.

######  Proposition 3.15.

Let $X\subset X(M)$ be an irreducible one-dimensional component which contains
a scheme-smooth, non-boundary parabolic character. For all $\chi\in X$ the
following holds

$s_{K}(\chi)=\frac{\mathbb{T}_{M_{K},\ell}(\chi)}{\mathbb{T}_{M_{K},m}(\chi)}.$

We provide two different proofs of this result: one uses the natural
definition of the torsion while the other relies directly on some results on
the _torsion form_ proved by the first author in [2].

####  3.4.1. Torsion and chain complexes

This section is devoted to the proof of Proposition 3.15 by using the chain
complex of $M_{K}$. The proof is very similar to [10, Theorem 3.21] or [10,
Theorem 6.7]. We use the following technical lemma.

######  Lemma 3.16.

Let $\gamma$ be a curve in $\pi_{1}(\partial M_{K})$, and $\chi$ be an
irreducible $\gamma$-regular character in $X(M_{K})$. There exists a Zariski
open neighborhood of $\chi$ such that any character in this neighborhood is
irreducible and $\gamma$-regular.

###### Proof.

Being irreducible is a Zariski open condition, see Eq. 2. The
$\gamma$-regularity follows from lower semi-continuity of the rank of a linear
map. Indeed the dimension of $H_{1}(M_{K},\operatorname{Ad}\circ\rho)$ is
upper semi-continuous. It is at least one (the dimension of $X$) again because
it is isomorphic to the Zariski tangent space hence it is locally constant
equal to one. On the other hand, the rank of the linear map
$H_{1}(\gamma,\operatorname{Ad}\circ\rho)\to
H_{1}(M_{K},\operatorname{Ad}\circ\rho)$ sending $v_{\rho}\otimes\gamma$ to
itself is lower semi-continuous. It is at most one (the dimension of
$H_{1}(\gamma,\operatorname{Ad}\circ\rho)$ and it cannot decrease on a
neighborhood of $\chi$. We deduce that
$H_{1}(\gamma,\operatorname{Ad}\circ\rho)\to
H_{1}(M_{K},\operatorname{Ad}\circ\rho)$ is an isomorphism on a Zariski open
subset. ∎

###### Proof of Proposition 3.15.

Let $\chi$ be an irreducible, scheme-smooth, non boundary-parabolic character
and let $\rho$ be a representation in $R(M_{K})$ with character $\chi$. We
first assume that $\rho$ is $\ell$ and $m$-regular, that is for
$v\in\mathfrak{sl}_{2}(\mathbb{C})$ an
$\operatorname{Ad}\circ\rho_{\partial}$-invariant vector, both $v\otimes\ell$
and $v\otimes m$ provide a basis of $H_{1}(M_{K},\operatorname{Ad}\circ\rho)$.

The calculation of the torsions $\mathbb{T}_{M_{K},\ell}(\chi)$ and
$\mathbb{T}_{M_{K},m}(\chi)$ involves different choices of homology basis of
$C_{*}(M_{K},\operatorname{Ad}\circ\rho)$. By [18, Proposition 3.18], the
bases of $H_{2}(M_{K},\operatorname{Ad}\circ\rho)$ are determined by the
fundamental class of $H_{2}(\partial M_{K};\mathbb{C})$ and can be choosen to
be the same. Hence, if $b_{1}$ is a basis of $\mathrm{im}(\partial_{1})$, the
ratio of torsions corresponding to the choice of $m$ or of $\ell$ is reduced
to

$\frac{\mathbb{T}_{M_{K},\ell}(\chi)}{\mathbb{T}_{M_{K},m}(\chi)}=\frac{\det(b_{1}\oplus(v\otimes\ell),c_{1})}{\det(b_{1}\oplus(v\otimes
m),c_{1})}.$

In parallel, consider the affine equation in
$C_{1}(M_{K},\operatorname{Ad}\circ\rho)$:

$y\,b_{1}+x\,v\otimes m=v\otimes\ell,$

with at least a solution $y=0$ and $x=s_{K}(\rho)$. The Cramer determinants
expressed in the common basis $c_{1}$ show that $s_{K}(\rho)$ coincides with
the ratio of torsions.

If there exists a character in $X$ which is $\ell$-regular and a character in
$X$ which is $m$-regular, then Proposition 3.15 holds on the whole component
$X$ by Lemma 3.16.

Assume that $X$ contains only characters that are not (say) $\ell$-regular.
Since the map $H_{1}(\partial M_{K},\operatorname{Ad}\circ\rho)\to
H_{1}(M_{K},\operatorname{Ad}\circ\rho)$ is not trivial (by Lemma 3.4), it is
onto on a Zariski open subset $U\in X$, again because
$H_{1}(M_{K},\operatorname{Ad}\circ\rho)$ has dimension one generically. Thus
all characters in $U$ must be $m$-regular, and it follows from the definition
that the slope and the quotient of torsions are identically zero on $X$. A
similar argument works replacing $\ell$ by $m$ and zero by infinity. ∎

####  3.4.2. The torsion form

In this paragraph, we present an alternative proof of Proposition 3.15. We
follow a slightly different point of view on the torsion, as a volume form on
the character variety. The following lemma asserts that the cotangent space of
the character variety [20, Section 8] is isomorphic to the first
$\operatorname{Ad}\circ\rho$-twisted homology group.

######  Lemma 3.17.

Let $\chi$ be an irreducible character in $X(M_{K})$, and a representation
$\rho$ with character $\chi$. Let $T^{*}_{\chi}X(M_{K})$ be the Zariski
tangent space of $X(M_{K})$ at $\chi$. There is a natural isomorphism

$H_{1}(M_{K},\operatorname{Ad}\circ\rho)\simeq T^{*}_{\chi}X(M_{K}).$

Moreover, if $\chi$ is not boundary-parabolic, then

$H_{1}(\partial M_{K},\operatorname{Ad}\circ\rho)\simeq
T^{*}_{r(\chi)}X(\partial M_{K}).$

The proof of Lemma 3.17 follows from [20, Theorem 1]. Note that through the
isomorphim, the space $\mathcal{Z}(K,\operatorname{Ad}\circ\rho)$ is the
Zariski conormal bundle of $r(X(M_{K}))$ in $X(\partial M_{K})$.

If $X\subset X(M_{K})$ is a one-dimensional component of the character variety
which contains a scheme-smooth character, the first author proved in [2,
Proposition 5.1] that the _torsion form_ can be written as

(6)
$\operatorname{tor}(M_{K})=\frac{1}{{\mathbb{T}}_{M_{K},\ell}}r^{*}\left(\frac{dL}{L}\right)=\frac{1}{{\mathbb{T}}_{M_{K},m}}r^{*}\left(\frac{dM}{M}\right)$

where $r^{*}$ is the cotangent map

$r^{*}\colon T^{*}X(\partial M_{K})\to T^{*}X(M_{K}).$

###### Proof of Proposition 3.15.

By Eq. 6, the ratio of torsions can be written as

$\frac{{\mathbb{T}}_{M_{K},\ell}}{{\mathbb{T}}_{M_{K},m}}=\frac{r^{*}(dL/L)}{r^{*}(dM/M)}.$

If $\chi$ is a non boundary-parabolic character, the character variety
$X(\partial M_{K})$ is diffeomorphic to $(\mathbb{C}^{*})^{2}$ in a
neighborhood of $r(\chi)$. A local chart of $X(\partial M_{K})$ is given by
taking $\mathfrak{l},\mathfrak{m}\in\mathbb{C}$ satisfying
$\exp\mathfrak{l}=L$ and $\exp\mathfrak{m}=M$. The latter ratio of torsions
can be written

$\frac{{\mathbb{T}}_{M_{K},\ell}}{{\mathbb{T}}_{M_{K},m}}=\frac{r^{*}(d\,\mathfrak{l})}{r^{*}(d\,\mathfrak{m})}.$

Lemma 3.17 implies that the cotangent map $r^{*}\colon
T^{*}_{r(\chi)}X(\partial M_{K})\to T^{*}_{\chi}X(M_{K})$ coincides with the
homomorphism in homology $H_{1}(\partial M_{K},\operatorname{Ad}\circ\rho)\to
H_{1}(M_{K},\operatorname{Ad}\circ\rho)$, thus by Lemma 3.4 the range of the
map $r^{*}$ is one-dimensional, and the images of the elements
$d\,\mathfrak{l},d\,\mathfrak{m}$ are colinear. It turns out that the ratio
$\frac{r^{*}(d\,\mathfrak{l})}{r^{*}(d\,\mathfrak{m})}$ coincides with the
slope by its very definition.

Finally, the formula extends to the whole $X$ since irreducible, non boundary-
parabolic character are Zariski dense in $X$. ∎

###  3.5. Compute the slope

In this section we compute the slope $s_{K}(\rho)$ when $\rho$ is an
irreducible non-boundary parabolic representation, with Fox calculus,
similarly to [10]. Note that for the boundary-parabolic case, the slope can be
computed directly from the representation using Proposition 3.8.

Consider a presentation of the knot group

$`p_{1}(M_{K})=\left<x_{1},\ldots,x_{p}\mid r_{1},\ldots,r_{q}\right>$

obtained from a Wirtinger presentation. We also assume that $m=x_{1}$ is the
preferred meridian and add the preferred longitude $\ell=x_{2}$, with the
relation $[m,\ell]=1$. Consider the complex of
$\mathbb{Z}[`p_{1}(M_{K})]$-modules

$S_{*}:=S_{2}"-{{}^{\partial_{1}}}>"S_{1}"-{{}^{\partial_{0}}}>"S_{0}$

where

$S_{2}=\bigoplus_{j=1}^{q}\mathbb{Z}[`p_{1}(M_{K})]`(`*)r_{j},\
S_{1}=\bigoplus_{j=1}^{p}\mathbb{Z}[`p_{1}(M_{K})]`(`*)dx_{i},\
S_{0}=\mathbb{Z}[`p_{1}(M_{K})]$

and $dx_{i}$ is a formal generator corresponding to $x_{i}$. Let
$\partial/\partial
x_{i}\colon\mathbb{Z}[\pi_{1}(M_{K})]\to\mathbb{Z}[\pi_{1}(M_{K})]$ denote the
Fox derivatives. For every $i`:\\{1,\dots,p\\}$ and $j`:\\{1,\dots,q\\}$ the
boundary operators are defined by

$\partial_{1}:r_{j}"|->"dr_{j},\quad\partial_{0}:dx_{i}"|->"x_{i},$

where $dw$ is the Fox differential of the word $w`:`p_{1}(M_{K})$:

$dw:=\sum_{i=1}^{p}\frac{\partial w}{\partial x_{i}}dx_{i}`:S_{1}.$

Now consider the $\operatorname{Ad}\circ\rho$-twisted chain complex
$S_{*}(`r):=\mathfrak{sl}_{2}(\mathbb{C})\>{`(`*)}_{\mathbb{Z}[`p_{1}(M_{K})]}\>S_{*}$,
where elements of $\mathfrak{sl}_{2}$ are identified with line vectors in
$\mathbb{C}^{3}$. The ($\operatorname{Ad}\circ\rho$-twisted) _Alexander
matrix_ of $M_{K}$ associated with the presentation is the matrix of
$\partial_{1}(`r)$, with coefficients in $\mathbb{C}$ and given by the
blockwise definition:

${\left((\operatorname{Ad}\circ\rho)\left(\dfrac{\partial r_{i}}{\partial
x_{j}}\right)\right)}_{1\leq i\leq q,\>1\leq j\leq p}$

The computation of the slope using $S_{*}(`r)$ is achieved with the following
result:

######  Proposition 3.18.

If $`r$ is irreducible and non-boundary parabolic, then there exists
$a,b`:\mathbb{C}$ and an $\operatorname{Ad}\circ\rho_{\partial}$-invariant
vector $v_{\rho}\in\mathfrak{sl}_{2}(\mathbb{C})$ such that

$\mathrm{im}\left(\partial_{1}(`r)\right)\cap\left<v_{`r}`(`*)d\ell,v_{`r}`(`*)dm\right>=\left<a\,(v_{`r}`(`*)d\ell)+b\,(v_{`r}`(`*)dm)\right>$

and the slope is $s_{K}(\rho)=-\frac{b}{a}$.

###### Proof.

Set a base point $p$ on $\partial M_{K}$. Following Crowell [8], the homology
space $H_{1}(S_{*}(`r))$ is isomorphic to
$H_{1}(M_{K},p,\operatorname{Ad}\circ\rho)$. The subcomplex
${S_{*}(`r_{\partial})}$ defined by considering only the generators $x_{1}=m$,
$x_{2}=\ell$ and the relation $[m,\ell]=1$ computes the space $H_{1}(\partial
M_{K},p;\operatorname{Ad}\circ\rho)$ as well. There are natural
identifications

$H_{1}\left(\partial M_{K},p;\operatorname{Ad}\circ\rho\right)=H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)\text{ and }H_{1}(\partial
M_{K},p;\operatorname{Ad}\circ\rho)\hookrightarrow H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)$

and the following diagram commutes:

${{S_{1}(`r_{\partial})}}$${S_{1}(`r)}$${H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)}$${H_{1}(M_{K},\operatorname{Ad}\circ\rho)}$${H_{1}(M_{K},{p},\operatorname{Ad}\circ\rho)}$$\scriptstyle{h_{\partial
M_{K}}}$$\scriptstyle{h}$$\scriptstyle{i_{*}}$

where $h$ and $h_{\partial M_{K}}$ are the quotient maps.

Let $u\in\mathfrak{sl}_{2}(\mathbb{C})$ be an
$\operatorname{Ad}\circ\rho$-invariant vector and $\gamma\in\pi_{1}(M_{K})$.
Any element $u`(`*)\gamma$ of $H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)$ can be lifted to $u`(`*)dw$ in
$S_{1}(`r_{\partial})$. Since $\rho$ is admissible, there exists
$a,b`:\mathbb{C}$ such that $\mathcal{Z}(K,\operatorname{Ad}\circ\rho)=\ker
i_{*}=\left<a\,(v_{`r}`(`*)\ell)+b\,(v_{`r}`(`*)m)\right>$. Then
$a\,(v_{`r}`(`*){d\ell})+b\,(v_{`r}`(`*){dm})`:\ker(h)=\mathrm{im}\left(\partial_{1}(`r)\right)$.

Reciprocally, suppose that there exists complex numbers $a,b`:\mathbb{C}$ such
that $dz:=a\,(v_{`r}`(`*){d\ell})+b\,(v_{`r}`(`*){dm})$ is a non-zero vector
belonging to $\mathrm{im}\left(\partial_{1}(`r)\right)$. Then $h_{\partial
M_{K}}(dz)=a\,(v_{`r}`(`*)\ell)+b\,(v_{`r}`(`*){m})$ must be non-zero since
$(v_{`r}`(`*)\ell,v_{`r}`(`*){m})$ is a free basis of $H_{1}(\partial
M_{K},\operatorname{Ad}\circ\rho)$. However, $h(dz)=i_{*}\left(h_{\partial
M_{K}}(dz)\right)=0$; hence $h_{\partial M_{K}}(dz)`:\ker i_{*}$. Since $\ker
i_{*}$ is one-dimensional, then $\ker i_{*}=\left<h_{\partial
M_{K}}(dz)\right>$, and the slope is $-\frac{b}{a}$. ∎

######  Example 3.19.

_The trefoil knot_ Let $T$ be the exterior of the right-handed trefoil knot,
with group $\pi_{1}(M_{K})=\langle u,v\mid uvu=vuv\rangle.$ Any irreducible
representation is conjugate to $\rho$ with

$\rho(u)=\begin{pmatrix}M&1\\\
0&M^{-1}\end{pmatrix},\quad\rho(v)=\begin{pmatrix}M^{-1}&0\\\
-1&M\end{pmatrix}$

where $M`:\mathbb{C}$. If $\ell=vuv^{-1}uvu^{-3}$ is the prefered longitude
with corresponding meridian $m=u$, we obtain

$\rho(\ell)=\left(\begin{smallmatrix}-M^{-6}&M^{5}+M^{3}+M+M^{-1}+M^{-3}+M^{-5}\\\
0&-M^{6}\end{smallmatrix}\right).$

Whenever $M\neq\pm 1$, the vector
$v_{\rho}=\begin{pmatrix}0,1,\frac{1}{M-M^{-1}}\end{pmatrix}$ is right
$\operatorname{Ad}\circ\rho_{\partial}$-invariant. By Section 3.5, the
Alexander matrix (acting on the right on the coefficients) whose rowspace is
generating $\mathrm{im}\left(\partial_{1}\right)$ is given by

$\begin{array}[]{rr*{3}{c}:*{3}{c}:*{3}{c}l}&&\lx@intercol\hfil\mathfrak{sl}_{2}(\mathbb{C})`(`*)d\ell\hfil\lx@intercol&\lx@intercol\hfil\mathfrak{sl}_{2}(\mathbb{C})`(`*)dm\hfil\lx@intercol&\lx@intercol\hfil\mathfrak{sl}_{2}(\mathbb{C})`(`*)dv\hfil\lx@intercol&\\\
\hbox{\multirowsetup$dr_{1}$}&\ldelim({6}{*}&0&0&0&1&0&0&-{M^{-2}}&0&0&\rdelim){6}{*}\\\
&&0&0&0&0&1&0&-M^{-1}&-1&0&\\\ &&0&0&0&0&0&1&1&2\,M&-M^{2}&\\\
\cdashline{3-11}\hbox{\multirowsetup$dr_{2}$}&&-1&0&0&-2{\left(2-M^{-2}\right)}&0&0&1+M^{-2}&0&0&\\\
&&0&-1&0&2{M}^{-1}&-2&0&{M}^{-1}&2&0\\\
&&0&0&-1&-2&-4\,M&2\,M^{2}-4&-1&-2\,M&M^{2}+1&\\\ \end{array}$

where $r_{1}$ is $uvu=vuv$ and $r_{2}$ is the longitude definition. By
Proposition 3.18, the space $\mathcal{Z}(M_{K},\operatorname{Ad}\circ\rho)$
has generator

$\begin{pmatrix}0,1,\frac{1}{M-M^{-1}},0,6,\frac{6}{M-M^{-1}},0,0,0\end{pmatrix}$

in the 2-dimensional subspace spanned by

$\left\\{\begin{pmatrix}0,1,\frac{1}{M-M^{-1}},0,0,0,0,0,0\end{pmatrix},\begin{pmatrix}0,0,0,0,1,\frac{1}{M-M^{-1}},0,0,0\end{pmatrix}\right\\}$

and the slope is $s_{T}(\operatorname{Ad}\circ\rho)=-6.$ In particular it does
not depend on $\rho$.

######  Example 3.20.

_The figure-eight knot_ Let $K$ be the figure-eight knot. There is a unique
component $X\subset X(M_{K})$ containing irreducible characters (see for
instance [2, Examples 1.6.2 and 5.5]). This component is a plane curve given
by the equation

$\\{2x^{2}+y^{2}-x^{2}y-y-1=0\\}\subset\mathbb{C}^{2},$

where $x$ it the coordinate function given by $\chi\mapsto\chi(m)$. Note that
the coordinate function of the longitude is
$\chi\mapsto\chi(\ell)=x^{4}-5x^{2}+2$. Using [18, Théorème 4.1 (ii)] and
Proposition 3.15 we compute

$\displaystyle
s_{K}(x,y)^{2}=\frac{x^{2}-4}{(x^{4}-5x^{2}+2)^{2}-4}(4x^{3}-10x)^{2}=\frac{4(2x^{2}-5)^{2}}{(x^{2}-5)(x^{2}-1)}$

Expanding the denominator with the relation $x^{2}=\frac{y^{2}-y-1}{y-2}$, we
obtain, up to sign

$s_{K}(x,y)=\pm\frac{2(2x^{2}-5)(y-2)}{(y-1)(y-3)}.$

##  4\. Slope and A-polynomial

In this section, we express the slope function in terms of the A-polynomial of
the knot. As mentioned in Section 2.3, $r(X)$ might have $0$-dimensional
components but they are omitted in the definition of the A-polynomial.

######  Theorem 4.1.

Let $X\subset X(M_{K})$ be an irreducible component such that $r(X)$ has
dimension $1$. For all $\chi\in X$ with $r(\chi)=(L,M)$, the following holds

$s_{K}(\chi)=-\frac{M}{L}\cdot\frac{\partial_{M}A(L,M)}{\partial_{L}A(L,M)},$

where $A(L,M)=A_{K}(L,M)$ and $\partial_{L}$ and $\partial_{M}$ are the
partial derivatives.

######  Remark 4.2.

Combining Proposition 3.15 with [18, Corollaire 4.9], the result of Theorem
4.1 follows directly, up to sign, in the case where $X$ has itself dimension
$1$. We resolve those two issues. Moreover Theorem 4.1 does not require the
characters in $X$ to be scheme-reduced, and the factors of the A-polynomial
might have multiplicities greater than $1$.

###### Proof.

From Lemma 3.17 it follows that the Lagrangian
$\mathcal{Z}(M_{K},\operatorname{Ad}\circ\rho)$ generically identifies with
the Zariski conormal bundle of $r(X(M_{K}))$ in $X(\partial M_{K})$. Picking
local coordinates $\mathfrak{l}=\log L,\mathfrak{m}=\log M$ around $r(\chi)$,
the kernel of the cotangent map is generated by

$dA(e^{\mathfrak{l}},e^{\mathfrak{m}})=\partial_{\mathfrak{l}}A(e^{\mathfrak{l}},e^{\mathfrak{m}})d\mathfrak{l}+\partial_{\mathfrak{m}}A(e^{\mathfrak{l}},e^{\mathfrak{m}})d\mathfrak{m}$

in $\mathbb{C}^{2}=\langle d\mathfrak{l},d\mathfrak{m}\rangle$. Using the
chain rule, we obtain that it is generated by the vector

$\left(L\frac{\partial A(M,L)}{\partial L},M\frac{\partial A(M,L)}{\partial
M}\right)$

and the proposition follows. ∎

######  Remark 4.3.

Let $T$ be the right-handed trefoil knot, with $A_{T}(L,M)=1+LM^{6}$. Theorem
4.1 gives

$s_{T}=-\frac{M}{L}\cdot\frac{6M^{5}L}{M^{6}}=-6.$

Compare to Example 3.19.

##  5\. The slope at an ideal point

In this section we prove Theorem 1.5. The context of this result is the work
of Culler–Shalen (see for instance [19]) which associates incompressible
surfaces in $M_{K}$ to ideal points of curves of $X(M_{K})$.

Let $X\subset X(M_{K})$ be an irreducible component whose image $r(X)=Y$ is a
curve in $X(\partial M_{K})$, defined as the zero locus of an irreducible
factor $P$ of $A_{K}(L,M)$. Its function ring is usually denoted by
$\mathbb{C}[Y]=\mathbb{C}[L,M]/(P)$, and its function field is
$\mathbb{C}(Y)=\operatorname{Frac}(\mathbb{C}[Y])$.

To any point $y$ in $Y$ one can associate a _discrete valuation_ $v$ on the
multiplicative group $\mathbb{C}(Y)^{*}$ in the field $\mathbb{C}(Y)$ of
rational functions on $Y$. A discrete valuation
$v\colon\mathbb{C}(Y)^{*}\to\mathbb{Z}$ is a group epimorphism satisfying
$v(f+g)\geq\min(v(f),v(g))$. The valuation associated to a smooth point $y$ is
simply the map

$f\mapsto v_{y}(f)=\operatorname{ord}_{y}f$

given the vanishing order of $f$ at the point $y$. More generally, the smooth
projective model $\overline{Y}$ of $Y$ is smooth compact curve birational to
$Y$, canonically defined up to isomorphism, and the points of $\overline{Y}$
are bijectively associated to discrete valuations on the function field
$\mathbb{C}(Y)\simeq\mathbb{C}(\overline{Y})$.

An _ideal_ point $y$ of $Y$ is a point added "at infinity" in the smooth
projective model $\overline{Y}$, it corresponds to a valuation $v_{y}$ on
$\mathbb{C}(Y)$ such that not every regular function $f\in\mathbb{C}[Y]$ has
non-negative valuation $v_{y}(f)$. In other words, some regular functions (at
least one) should have poles at $y$.

In [9], Marc Culler and Peter Shalen gave a procedure to construct an
_incompressible_ surface $\Sigma$ in $M_{K}$ from the data of an ideal point
$x$ in a subcurve $C$ of $X(M_{K})$ together with the valuation
$v_{x}\^{A}~\colon\mathbb{C}(C)^{*}\to\mathbb{Z}$. Not any ideal point $x\in
X(M_{K})$ yields an ideal point $y=r(x)\in X(\partial M_{K})$.

In this special case, the ideal point $y$ in $Y$ gives an incompressible
surface in $M_{K}$ of a particular kind: as observed in [7, Proposition 3.1],
the incompressible surface $\Sigma$ must have non-empty boundary
$\partial\Sigma\subset\partial M_{K}$. The curve $\partial\Sigma$ is a finite
union of parallel circles in $\partial M_{K}$ and uniquely determines a
_boundary slope_ in $\mathbb{Q}\cup\\{\infty\\}$: the slope of $a\ell+bm$ in
$H_{1}(\partial M_{K};\mathbb{Z})$ is the rational number $\frac{b}{a}$.

On the other hand, the Newton polygon of
$A_{K}(L,M)=\sum_{i,j}a_{i,j}L^{i}M^{j}$ is the convex hull in
$\mathbb{C}^{2}$ of the points $\\{(i,j)\in\mathbb{Z}^{2}\mid a_{i,j}\neq
0\\}$. It is a convex polygon of $\mathbb{C}^{2}$ with integral vertices,
whose sides have a slope in $\mathbb{Q}\cup\\{\infty\\}$. In [7], Culler,
Cooper, Gillet, Long and Shalen proved the following result:

######  Theorem 5.1.

[7, Theorem 3.4] The slopes of the sides of the Newton polygon of the
$A$-polynomial $A_{K}(L,M)$ are boundary slopes of incompressible surfaces in
$M_{K}$ which correspond to ideal points of one-dimensional components of
$r^{*}(X(M_{K}))$ in $X(\partial M_{K})$.

Our next statement (Theorem 1.5 in the introduction) states that the slope
invariant studied in this paper coincides with the slopes of [7] at ideal
points.

######  Theorem 5.2.

Let $y$ be an ideal point in a one-dimensional component $Y$ of the
$A$-polynomial. Then the value of the slope function at the ideal point $y$
equals _minus_ the boundary slope of an incompressible surface corresponding
to $y$ or _minus_ the slope of the corresponding side of the Newton polygon of
the $A$-polynomial.

###### Proof.

The coordinate functions $L,M$ define rational functions on $Y$, in particular
their valuations $v_{y}(L)$ and $v_{y}(M)$ are well-defined. Since $y$ is an
ideal point and $L,M$ generate the coordinate ring $\mathbb{C}[Y]$ of the
curve $Y$, at least one of this valuation must be negative, and at least one
of these coordinate functions must have a pole at $y$.

######  Claim.

The value of $s_{K}$ at the ideal point $y$ is $\frac{v_{y}(L)}{v_{y}(M)}$.

###### Proof of the claim.

From the proof of Proposition 3.15, we deduce that the value of the slope at
$y$ is given by

$s_{K}(y)=\lim_{(L,M)\to y}\frac{r^{*}(dL/L)}{r^{*}(dM/M)}.$

The following argument is an algebraic analogue of taking Taylor expansion of
the functions $L$ and $M$ around the ideal point $y$. We pick $t$ a local
coordinate around $y$. It is characterized by $v_{y}(t)=1$, and we can write

$L=u_{1}t^{v_{y}(L)}$

for $u_{1}\in\mathbb{C}(Y),v_{y}(u_{1})=0$, and similarly

$M=u_{2}t^{v_{y}(M)}$

for $u_{2}\in\mathbb{C}(Y),v_{y}(u_{2})=0$. Moreover, near $y$ it follows that

$\frac{r^{*}(dL/L)}{r^{*}(dM/M)}=\frac{v_{y}(L)/t}{v_{y}(M)/t}=\frac{v_{y}(L)}{v_{y}(M)}$

and the claim follows. ∎

Now Theorem 1.5 follows directly from the claim, because it is proven in [7,
Proposition 3.1] that the quantity $-\frac{v_{y}(L)}{v_{y}(M)}$ is the
boundary slope of an incompressible surface corresponding to $y$. ∎

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