Twisted Alexander polynomials of torus links

TL;DR

This paper provides explicit formulas for twisted Alexander polynomials of torus links, demonstrating their local constancy on the $SL(2, C)$-character variety and exploring related invariants.

Contribution

It introduces explicit formulas for twisted Alexander polynomials of torus links and analyzes their behavior on the character variety, extending to higher-dimensional cases.

Findings

Twisted Alexander polynomial formulas are explicitly derived for torus links.

The polynomial is shown to be locally constant on the $SL(2, C)$-character variety.

Connections to higher-dimensional twisted Alexander polynomials and Reidemeister torsion are discussed.

Abstract

In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.

Equations

$$G(L)=⟨x_1,\dots ,x_l|r_1,\dots ,r_{l-1}⟩$$

$${\alpha }_L:G(L)\to H_1(E_L;\mathbb{Z})\cong {\mathbb{Z}}^{\oplus \mu }=⟨t_1⟩\oplus \cdots \oplus ⟨t_{\mu }⟩$$

$$\Phi :\mathbb{Z}[F_l]\to M\left(2,\mathbb{C}[t_1^{\pm 1},\dots ,t_{\mu }^{\pm 1}]\right)$$

$$\Phi \left(\frac{\partial r_i}{\partial x_j}\right)\in M\left(2,\mathbb{C}[t_1^{\pm 1},\dots ,t_{\mu }^{\pm 1}]\right),$$

$${\Delta }_{L,\rho }(t_1,\dots ,t_{\mu })=\frac{\det A_j}{\det \Phi (x_j-1)}$$

$$G(L)=⟨m_1,\cdots ,m_{\mu },x,y,\ell \mid m_{\mu }{m}_{\mu -1}\cdots m_1=x^r{y}^s,[\ell ,m_i]=1(1\le i\le \mu ),\ell =x^p=y^q⟩$$

$${\alpha }_L(m_i)=t_i(1\le i\le \mu ),{\alpha }_L(x)=t^q,{\alpha }_L(y)=t^p$$

$$G(L)=⟨m_1,\cdots ,m_{\mu -1},x,y\mid [x^p,m_i]=1(1\le i\le \mu -1),x^p=y^q⟩.$$

$$\mathcal{X}(\mu p,\mu q)=\mathcal{R}(\mu p,\mu q)/\sim .$$

$$t_x=\mathrm{tr}(X),t_y=\mathrm{tr}(Y),t_i=\mathrm{tr}(M_i),t_{xy}=\mathrm{tr}(XY),t_{xi}=\mathrm{tr}(X{M}_i),t_{xyi}=\mathrm{tr}(XY{M}_i)$$

$$\rho :G(L)=⟨m_1,x,y\mid [x^p,m_1]=1,x^p=y^q⟩\to \mathrm{SL}(2,\mathbb{C})$$

$$\begin{array}{rl}{t}_x& =\mathrm{tr}(X)=\alpha (a)+\alpha (a{)}^{-1}=2\cos (a\pi /p),\\ t_y& =\mathrm{tr}(Y)=\beta (b)+\beta (b{)}^{-1}=2\cos (b\pi /q).\end{array}$$

$$X=\left(\begin{array}{cc}\alpha (a)& 0\\ 0& \alpha (a{)}^{-1}\end{array}\right),t_x\ne \pm 2;Y=\left(\begin{array}{cc}s& 1\\ u& v\end{array}\right),u=sv-1,t_y\ne \pm 2.$$

$$\alpha (a)=\frac{t_x+\sqrt{t_x^2-4}}{2}\text{ or }\alpha (a)=\frac{t_x-\sqrt{t_x^2-4}}{2}.$$

$$\begin{array}{rl}{t}_{xy}& =\alpha (a)s+\alpha (a{)}^{-1}v\\ & =s(\alpha (a)-\alpha (a{)}^{-1})+\alpha (a{)}^{-1}{t}_y.\end{array}$$

$$\begin{array}{rl}s& =\frac{t_{xy}-\alpha (a{)}^{-1}{t}_y}{\alpha (a)-\alpha (a{)}^{-1}},\\ v& =t_y-s\\ & =-\frac{t_{xy}-\alpha (a)t_y}{\alpha (a)-\alpha (a{)}^{-1}},\\ u& =sv-1\\ & =-\frac{t_{xy}^2-t_{xy}{t}_x{t}_y+t_y^2+t_x^2-4}{t_x^2-4}.\end{array}$$

$$\begin{array}{rl}{t}_{x1}& =\mathrm{tr}(X{M}_1)\\ & =\alpha (a)\gamma +\alpha (a{)}^{-1}\zeta \\ & =\alpha (a)\gamma +\alpha (a{)}^{-1}(t_1-\gamma )\\ & =(\alpha (a)-\alpha (a{)}^{-1})\gamma +\alpha (a{)}^{-1}{t}_1,\end{array}$$

$$\gamma =\frac{t_{x1}-\alpha (a{)}^{-1}{t}_1}{\alpha (a)-\alpha (a{)}^{-1}}$$

$$\begin{array}{rl}\zeta & =t_1-\gamma \\ & =-\frac{t_{x1}-\alpha (a)t_1}{\alpha (a)-\alpha (a{)}^{-1}}.\end{array}$$

$$\begin{array}{rl}{t}_{y1}& =\mathrm{tr}(Y{M}_1)\\ & =s\gamma +\epsilon +u\delta +v\zeta \\ & =s\gamma +\epsilon +u\delta +v(t_1-\gamma )\\ & =(s-v)\gamma +u\delta +\epsilon +v{t}_1,\end{array}$$

$$Z^2-(t_{y1}-v{t}_1-(s-v)\gamma )Z+u(\gamma \zeta -1)=0$$

$$\begin{array}{rl}X& =\left(\begin{array}{cc}\alpha (a)& 0\\ 0& \alpha (a{)}^{-1}\end{array}\right),\alpha (a)\ne \pm 1;\\ Y& =\left(\begin{array}{cc}s& 1\\ u& v\end{array}\right),u=sv-1\ne 0;\\ M_1& =\left(\begin{array}{cc}\gamma & {\theta }_1/u\\ {\theta }_2& \zeta \end{array}\right)\mathrm{or}\left(\begin{array}{cc}\gamma & {\theta }_2/u\\ {\theta }_1& \zeta \end{array}\right).\end{array}$$

$$t_{xy1}=\alpha (a)s\gamma +\alpha (a{)}^{\pm 1}{\theta }_1+\alpha (a{)}^{\mp 1}{\theta }_2+\alpha (a{)}^{-1}v\zeta .$$

$$t_{xy1}^2-(t_x{t}_{y1}+t_y{t}_{x1}+t_1{t}_{xy}-t_x{t}_y{t}_1)t_{xy1}$$

$$+t_x^2+t_y^2+t_1^2+t_{xy}^2+t_{x1}^2+t_{y1}^2+t_{xy}{t}_{x1}{t}_{y1}-t_x{t}_y{t}_{xy}-t_x{t}_1{t}_{x1}-t_y{t}_1{t}_{y1}-4=0.$$

$$X=\left(\begin{array}{cc}\alpha (a)& 0\\ 0& \alpha (a{)}^{-1}\end{array}\right),Y=\left(\begin{array}{cc}s& 1\\ u& t_y-s\end{array}\right),$$

$$\alpha (a{)}^{\pm 1}=\frac{t_x\pm \sqrt{t_x^2-4}}{2},s=\frac{t_{xy}-\alpha (a{)}^{-1}{t}_y}{\alpha (a)-\alpha (a{)}^{-1}},u=-\frac{t_{xy}^2-t_{xy}{t}_x{t}_y+t_y^2+t_x^2-4}{t_x^2-4},$$

$$M_1=\left(\begin{array}{cc}\gamma & \delta \\ \epsilon & t_1-\gamma \end{array}\right),\gamma =\frac{t_{x1}-\alpha (a{)}^{-1}{t}_1}{\alpha (a)-\alpha (a{)}^{-1}}$$

$$\epsilon (u\delta )=u(\gamma t_1-{\gamma }^2-1),\epsilon +u\delta =t_{y1}-(t_y-s)t_1-\gamma (2s-t_y).$$

$$\big{(}t_{x}=2\cos(\pi a/p),t_{y}=2\cos(\pi b/q),t_{1},t_{xy},t_{x1},t_{y1},t_{xy1}\big{)}$$

$$X=\left(\begin{array}{cc}\alpha (a)& 0\\ 0& \alpha (a{)}^{-1}\end{array}\right)\mathrm{and}Y=\left(\begin{array}{cc}\beta (b{)}^{\pm 1}& 1\\ 0& \beta (b{)}^{\mp 1}\end{array}\right).$$

Full text

2010 Mathematics Subject Classification. Primary 57M27, Secondary 57M25.

Key words and phrases. character variety, torus link, twisted Alexander polynomial.

1. Introduction

The twisted Alexander polynomial is a generalization of the classical Alexander polynomial of a knot in the $3$-sphere $S^{3}$. It was first introduced by Lin [10] for knot groups and by Wada [14] for finitely presentable groups which include link groups. Recently this polynomial invariant has been widely investigated by lots of authors and recognized as a powerful tool in low dimensional topology. As for recent development of this topic and related references, see the survey papers [2], [7] and [12].

In this paper we consider twisted Alexander polynomials associated to irreducible $\mathrm{SL}(2,\mathbb{C})$-representations of groups of knots or links in $S^{3}$. In particular we investigate the behavior of the twisted Alexander polynomial as a function on the $\mathrm{SL}(2,\mathbb{C})$-character variety, that is, the set of conjugacy classes of irreducible $\mathrm{SL}(2,\mathbb{C})$-representations. As for this kind of property, it is known that there is a hyperbolic knot such that the twisted Alexander polynomial varies continuously on its character variety (see [3]). On the other hand, it is known that such kind of phenomenon never happen for any torus knot. Actually in our previous paper [9] we showed that every coefficient of the twisted Alexander polynomial of a torus knot is a locally constant function of the $\mathrm{SL}(2,\mathbb{C})$-character variety. Furthermore we reproved a result of Johnson that the Reidemeister torsion of a torus knot is locally constant on the $\mathrm{SL}(2,\mathbb{C})$-character variety (see [5]).

The purpose of this paper is to generalize the above result to torus links in $S^{3}$ and moreover to discuss similar things for higher dimensional twisted Alexander polynomials. To this end we give an explicit formula for the (higher dimensional) twisted Alexander polynomial, which depends only on the eigenvalues of the $2\times 2$ matrices corresponding to the cores of the two solid tori of the standard genus one Heegaard splitting of $S^{3}$. The property of the twisted Alexander polynomial and the Reidemeister torsion mentioned above immediately follows from the formula.

This paper is organized as follows. In the next section, we quickly review the definition of the twisted Alexander polynomial of a link in $S^{3}$. In Section 3 we study the $\mathrm{SL}(2,\mathbb{C})$-character variety of a torus link. In Section 4 we describe a formula for the twisted Alexander polynomial of a torus link. In the last section, we investigate the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.

2. Twisted Alexander polynomials

Let $L=L_{1}\sqcup\cdots\sqcup L_{\mu}$ be a $\mu$-component oriented link in $S^{3}$ and $E_{L}=S^{3}\setminus\mathrm{int}(N(L))$ the exterior of $L$ in $S^{3}$. Here $N(L)$ is a closed tubular neighborhood of $L$. We denote $\pi_{1}(E_{L})$ by $G(L)$ and call it the link group. We choose and fix a Wirtinger presentation of $G(L)$:

[TABLE]

where every generator corresponds to an arc in a regular projection $D(L)$ of the link $L$ and every relator comes from a crossing in $D(L)$. The abelianization homomorphism

[TABLE]

is given by assigning to each generator $x_{i}$ the meridian element $t_{k}\in H_{1}(E_{L};\mathbb{Z})$ of the corresponding component $L_{k}$ of $L$. Here we denote the sum in each $\mathbb{Z}$ multiplicatively.

In this paper we consider a representation of $G(L)$ into the two-dimensional special linear group $\mathrm{SL}(2,\mathbb{C})$, say $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$. The maps $\rho$ and $\alpha_{L}$ naturally induce two ring homomorphisms $\tilde{\rho}:{\mathbb{Z}}[G(L)]\rightarrow M(2,{\mathbb{C}})$ and $\tilde{\alpha}_{L}:{\mathbb{Z}}[G(L)]\rightarrow{\mathbb{Z}}[t_{1}^{\pm 1},\ldots,t_{\mu}^{\pm 1}]$, where ${\mathbb{Z}}[G(L)]$ is the group ring of $G(L)$ and $M(2,{\mathbb{C}})$ is the matrix algebra of degree $2$ over ${\mathbb{C}}$. Then $\tilde{\rho}\otimes\tilde{\alpha}_{L}$ defines a ring homomorphism ${\mathbb{Z}}[G(L)]\to M\left(2,{\mathbb{C}}[t_{1}^{\pm 1},\ldots,t_{\mu}^{\pm 1}]\right)$. Let $F_{l}$ denote the free group on generators $x_{1},\ldots,x_{l}$ and

[TABLE]

the composition of the surjection $\tilde{\phi}:{\mathbb{Z}}[F_{l}]\to{\mathbb{Z}}[G(L)]$ induced by the presentation of $G(L)$ and the map $\tilde{\rho}\otimes\tilde{\alpha}_{L}:{\mathbb{Z}}[G(L)]\to M(2,{\mathbb{C}}[t_{1}^{\pm 1},\ldots,t_{\mu}^{\pm 1}])$.

Let us consider the $(l-1)\times l$ matrix $A$ whose $(i,j)$-entry is the $2\times 2$ matrix

[TABLE]

where $\frac{\partial}{\partial x}:\mathbb{Z}[F_{l}]\to\mathbb{Z}[F_{l}]$ denotes the free differential. We call $A$ the Alexander matrix  of the link group $G(L)$ associated to $\rho$. For $1\leq j\leq l$, let us denote by $A_{j}$ the $(l-1)\times(l-1)$ matrix obtained from $A$ by removing the $j$th column. We regard $A_{j}$ as a $2(l-1)\times 2(l-1)$ matrix with coefficients in ${\mathbb{C}}[t_{1}^{\pm 1},\ldots,t_{\mu}^{\pm 1}]$. Then Wada’s twisted Alexander polynomial [14] of a link $L$ associated to a representation $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$ is defined to be the rational function

[TABLE]

and well-defined up to multiplication by $t_{1}^{2k_{1}}\cdots t_{\mu}^{2k_{\mu}}~{}(k_{j}\in{\mathbb{Z}})$. In particular, it does not depend on a choice of a presentation of $G(L)$.

Remark 2.1*.*

By definition, $\Delta_{L,\rho}(t_{1},\ldots,t_{\mu})$ is a rational function in the variables $t_{1},\ldots,t_{\mu}$, but it will be a Laurent polynomial if $L$ is a link with two or more components [14, Proposition 9], or $L$ is a knot $K$ and $\rho$ is non-abelian [8, Theorem 3.1]. We note that if $\rho,\rho^{\prime}:G(L)\to\mathrm{SL}(2,\mathbb{C})$ are conjugate representations, then $\Delta_{L,\rho}(t_{1},\ldots,t_{\mu})=\Delta_{L,\rho^{\prime}}(t_{1},\ldots,t_{\mu})$ holds (see [14, Section 3]).

3. Character varieties of torus links

In this section, we discuss the $\mathrm{SL}(2,\mathbb{C})$-character variety of the group of a torus link. To this end, we first review a presentation of the group of a torus link.

3.1. A presentation of the group of a torus link

Let $L=T(\mu p,\mu q)$ be a $\mu$-component torus link in $S^{3}$ where $p,q$ are coprime integers. The link group $G(L)=\pi_{1}(E_{L})$ has the following presentation (see [13, Lemma 2.2]):

[TABLE]

where $x,y$ represent the cores of the two solid tori of the Heegaard splitting defined by the torus $T\subset S^{3}$ (namely $T$ determines the Heegaard decomposition of genus one in $S^{3}$), $\ell$ represents a parallel of the torus knot $T(p,q)$ on $T$, $m_{i}$ is a meridian of each component of $L$ and $r,s$ satisfy $ps+qr=1$. We note that $G(L)$ contains $G(p,q)=\langle x,y\,|\,x^{p}=y^{q}\rangle$, the group of the torus knot $T(p,q)$, as a subgroup.

Remark 3.1*.*

• (1)

The center of $G(L)$ is an infinite cyclic group $\mathbb{Z}$ generated by $x^{p}=y^{q}$.

• (2)

The abelianization homomorphism $\alpha_{L}:G(L)\rightarrow\langle t_{1}\rangle\oplus\cdots\oplus\langle t_{\mu}\rangle$ is given by

[TABLE]

where we put $t=t_{1}\cdots t_{\mu}$.

• (3)

It is known that the bridge number of the torus link $T(\mu p,\mu q)$ is equal to $\min(\mu p,\mu q)$ (see [13, Corollary 1.5]).

Using the relation $m_{\mu}m_{\mu-1}\cdots m_{1}=x^{r}y^{s}$, the above presentation of $G(L)$ can be reduced to

[TABLE]

The latter presentation will be useful for calculating the (higher dimensional) twisted Alexander polynomial in Sections 4 and 5.

3.2. Character varieties of torus links

In this subsection we describe the character variety of $G(L)$ following the paper [4].

Now let us consider an irreducible  representation $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$. Namely there is no nontrivial proper invariant subspace of $\mathbb{C}^{2}$ under the natural action of $\rho(G(L))$. A representation $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$ is called reducible  if it is not irreducible.

Let $\mathcal{R}(\mu p,\mu q)$ be the set of irreducible $\mathrm{SL}(2,\mathbb{C})$-representations of the link group $G(L)$ of a $\mu$-component torus link $L=T(\mu p,\mu q)$. We denote the quotient space by conjugation action of $\mathrm{SL}(2,\mathbb{C})$ by

[TABLE]

Remark 3.2*.*

We call $\mathcal{R}(\mu p,\mu q)$ the irreducible representation variety of $L=T(\mu p,\mu q)$ and $\mathcal{X}(\mu p,\mu q)$ the irreducible character variety of $L$. Actually, for a character $\chi_{\rho}:G(L)\to\mathbb{C}$ defined by $\chi_{\rho}(\gamma)={\mathrm{tr}\,}\rho(\gamma),\,\gamma\in G(L)$, it is known that $\chi_{\rho}=\chi_{\rho^{\prime}}\,(\rho,\rho^{\prime}\in\mathcal{R}(\mu p,\mu q))$ if and only if $\rho,\rho^{\prime}$ are conjugate (see [1, Proposition 1.5.2]).

We also denote the $2\times 2$ identity matrix by $I$ and the image of each generator of $G(L)$ by its capital letter.

Lemma 3.3**.**

If $\rho$ is irreducible, then $X^{p}=Y^{q}=\pm I$ hold.

Proof.

Assume that $X^{p}=Y^{q}\neq\pm I$. Since $X^{p}=Y^{q}$ is in the center of $\rho(G(L))$, $X^{p}=Y^{q}$ commutes with any one of $X,Y$ and $M_{j}~{}(j=1,\ldots,\mu-1)$. Thus an eigenvector of $X^{p}=Y^{q}$ is also an eigenvector of any of them. It contradicts the irreducibility of $\rho$. ∎

By the above lemma, we can write the eigenvalues of $X$ and $Y$ as $\alpha(a)=\exp(\sqrt{-1}\pi a/p),~{}$ $0\leq a\leq p$ and $\beta(b)=\exp(\sqrt{-1}\pi b/q),~{}0\leq b\leq q$ respectively.

Remark 3.4*.*

Since $X^{p}=(-I)^{a}=Y^{q}=(-I)^{b}$, it holds that $a\equiv b~{}(\mathrm{mod}~{}2)$.

For $\rho\in\mathcal{R}(\mu p,\mu q)$, it will be defined by a set of matrices $(X,Y,M_{1},\ldots,M_{\mu-1})$ which satisfy the relations coming from the presentation of $G(L)$. We note that the relations $[x^{p},m_{i}]=1~{}(1\leq i\leq\mu-1)$ give no restriction to the matrices $(X,Y,M_{1},\ldots,M_{\mu-1})$, because $X^{p}=Y^{q}=\pm I$ hold by Lemma 3.3.

Notation. We will use the following notations:

[TABLE]

for $1\leq i\leq\mu-1$.

3.2.1. $\mu=2$

We observe how a representative of an irreducible representation

[TABLE]

in each conjugacy class can be determined by using traces. Here we fix two non negative integers $a,b~{}(0\leq a\leq p,~{}0\leq b\leq q,~{}a\equiv b\text{ mod }2)$ and then fix

[TABLE]

Case 1. $\rho|_{G(p,q)}$ is non-abelian. By [4, Proposition 4.3], up to conjugation, we can put

[TABLE]

By $\alpha(a)^{2}-t_{x}\alpha(a)+1=0$, one sees that

[TABLE]

Since $\begin{pmatrix}\alpha(a)&0\\ 0&\alpha(a)^{-1}\end{pmatrix}$ is conjugate to $\begin{pmatrix}\alpha(a)^{-1}&0\\ 0&\alpha(a)\end{pmatrix}$ by conjugation action of $\begin{pmatrix}0&1\\ -1&0\end{pmatrix}$, the matrix $X$ is perfectly determined by $t_{x}$ up to conjugation.

Next we determine $Y$ by using $t_{y},t_{xy}$. Since $s+v=t_{y}$, we have $v=t_{y}-s$. To determine $s$, we use

[TABLE]

Thus we have

[TABLE]

Therefore $t_{x},t_{y},t_{xy}$ determine $X$ and $Y$ uniquely, up to conjugation.

Remark 3.5*.*

Note that $\alpha(a)\not=\alpha(a)^{-1}$ since $t_{x}\neq\pm 2$. Moreover, $u\not=0$ if and only if $t_{xy}^{2}-t_{xy}t_{x}t_{y}+t_{y}^{2}+t_{x}^{2}-4\not=0$.

Case 1.1. $u\neq 0$. This is equivalent to $t_{xy}^{2}-t_{xy}t_{x}t_{y}+t_{y}^{2}+t_{x}^{2}-4\not=0$. In this case $\rho|_{G(p,q)}$ is irreducible.

We now determine $M_{1}$ by using $t_{1},t_{x1},t_{y1},t_{xy1}$. Put $M_{1}=\begin{pmatrix}\gamma&\delta\\ \varepsilon&\zeta\end{pmatrix}$ with $\gamma\zeta-\delta\varepsilon=1$ and $\gamma+\zeta=t_{1}$. Also we have

[TABLE]

namely

[TABLE]

and

[TABLE]

Further we have

[TABLE]

then $u\delta+\varepsilon=t_{y1}-vt_{1}-(s-v)\gamma$ holds.

On the other hand, $\gamma\zeta-\delta\varepsilon=1$ implies $(u\delta)\varepsilon=u(\gamma\zeta-1)$. Hence $u\delta$ and $\varepsilon$ are the solutions $\theta_{1},\theta_{2}$ of the quadratic equation

[TABLE]

whose coefficients are functions of $t_{x},t_{y},t_{1},t_{xy},t_{x1}$ and $t_{y1}$. For any values of $t_{x},t_{y},t_{1},t_{xy},t_{x1}$ and $t_{y1}$, there exists at most two solutions $\theta_{1},\theta_{2}$. Therefore these six coordinates determine two triples $(X,Y,M_{1})$ and $(X,Y,M_{1}^{\prime})$, that is,

[TABLE]

These two triples yield the two possible values of the coordinate $t_{xy1}={\mathrm{tr}\,}(XYM_{1})$:

[TABLE]

In other words, if we also fix the value of $t_{xy1}$, the triple $(X,Y,M_{1})$ is uniquely determined. Note that, by writing in terms of traces, $t_{xy1}$ satisfies the following quadratic equation

[TABLE]

See [4, Section 5].

Combining all that we have obtained at this point, we have

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Fixing the value of $t_{xy1}$, the matrix $M_{1}$ and hence the triple $(X,Y,M_{1})$ is perfectly determined.

As a conclusion, in Case 1.1, a representation $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$, i.e. the triple $(X,Y,M_{1})$ of matrices is well determined by the coordinates

[TABLE]

where $t_{xy}^{2}-t_{xy}t_{x}t_{y}+t_{y}^{2}+t_{x}^{2}-4\not=0$ and $(t_{1},t_{xy},t_{x1},t_{y1},t_{xy1})$ satisfies equation (3.1). Here $0<a<p$, $0<b<q$, and $a\equiv b\text{ mod }2$.

So there are $\frac{(p-1)(q-1)}{2}$ components, which are labeled by $t_{x},t_{y}$. Further each of them has the complex dimension $4$ parametrized by $t_{1},t_{xy},t_{x1},t_{y1},t_{xy1}$ which satisfy equation (3.1).

Note the followings:

• •

Any point with $t_{xy}^{2}-t_{xy}t_{x}t_{y}+t_{y}^{2}+t_{x}^{2}-4=0$ (i.e. $t_{xy}=\alpha(a)\beta(b)^{\pm 1}+\alpha(a)^{-1}\beta(b)^{\mp 1}$) belongs to one of the spaces in Case 1.2 below.

• •

$t_{xy1}$ can have only two values after fixing other coordinates $t_{x},t_{y},t_{1},t_{xy},t_{x1}$ and $t_{y1}$.

Case 1.2. $u=0$. This is equivalent to $t_{xy}^{2}-t_{xy}t_{x}t_{y}+t_{y}^{2}+t_{x}^{2}-4=0$. In this case $\rho|_{G(p,q)}$ is reducible. Up to conjugation, we may assume that

[TABLE]

The matrices $X$ and $Y$ are determined by $t_{x},t_{y}$ and $t_{xy}=\alpha(a)\beta(b)^{\pm 1}+\alpha(a)^{-1}\beta(b)^{\mp 1}$. Moreover, if we fix $t_{x}$ and $t_{y}$, there are 2 choices for $t_{xy}$.

For $M_{1}=\begin{pmatrix}\gamma&\delta\\ \varepsilon&\zeta\end{pmatrix}$, we can see

[TABLE]

as in Case 1.1. Since

[TABLE]

we can see

[TABLE]

In this case, the representation $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$ is irreducible if and only if $\varepsilon\neq 0$. This open condition is

[TABLE]

Hence there are $\frac{(p-1)(q-1)}{2}\times 2=(p-1)(q-1)$ components, and each of them has the complex dimension $3$ parametrized by $t_{1},t_{x1},t_{y1}$ which satisfy condition (3.2).

Case 2. $\rho|_{G(p,q)}$ is abelian. In this case, $(X,Y)$ is conjugate in $\mathrm{SL}(2,\mathbb{C})$ to a pair of diagonal matrices, so it is enough to consider the following cases.

Case 2.1. $\rho|_{G(p,q)}\not=\{\pm I\}$. Then $X\not=\pm I$ or $Y\not=\pm I$. Without loss of generality, we assume $X\not=\pm I$. As in Case 1.2, we have

[TABLE]

We then see that the representation $\rho$ is irreducible if and only if $\delta\neq 0$ and $\varepsilon\neq 0$. These conditions are equivalent to $t_{1}\not=\pm 2$ and $\gamma\zeta\neq 1$. Here

[TABLE]

and then the conditions are

[TABLE]

In this case each component has the dimension $2$ parametrized by $t_{1}$ and $t_{x1}$ which satisfy condition (3.3).

Case 2.2. $\rho|_{G(p,q)}=\{\pm I\}$. In this case $\rho$ is an abelian representation. This is a contradiction.

3.2.2. $\mu\geq 3$

In this case we are able to explicitly describe the irreducible representations in the following cases. The other cases remain unknown.

Case 1.1. $\rho|_{G(p,q)}$ is irreducible. In this case, similar to the previous subsection, the representation $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$, i.e. the $(\mu+1)$-tuple $(X,Y,M_{1},\ldots,M_{\mu-1})$ is well determined by the coordinates

[TABLE]

where $t_{xy}^{2}-t_{x}t_{y}t_{xy}+t_{x}^{2}+t_{y}^{2}-4\not=0$ and each quadruple $(t_{i},t_{xi},t_{yi},t_{xyi})$ satisfies the following quadratic equation

[TABLE]

The coordinates $t_{x}$ and $t_{y}$ are determined by $a~{}(0<a<p)$ and $b~{}(0<b<q)$ which satisfy $a\equiv b~{}(\mathrm{mod}~{}2)$. So there are $\frac{(p-1)(q-1)}{2}$ components, which are labeled by $t_{x},t_{y}$. Further each of them has the complex dimension $1+4(\mu-1)-(\mu-1)=3\mu-2$ parametrized by $t_{i},t_{xy},t_{xi},t_{yi},t_{xyi}$, which satisfy equation (3.4).

Case 2.2. $\rho|_{G(p,q)}=\{\pm I\}$. In this case $\mathcal{X}(\mu p,\mu q)$ can be identified with the irreducible character variety of a free group $F_{\mu-1}$. Hence the dimension of each component is given by $3(\mu-1)-3=3\mu-6$. See [4, Section 5] for details.

Problem 3.6**.**

Describe $\mathcal{X}(\mu p,\mu q)$ when $\rho|_{G(p,q)}$ is reducible and nontrivial.

4. Twisted Alexander polynomials of torus links

In this section, we give an explicit formula for the twisted Alexander polynomial of any torus link associated to an irreducible $\mathrm{SL}(2,\mathbb{C})$-representation.

Let $\rho\in\mathcal{R}(\mu p,\mu q)$, namely $\rho:G(L)\rightarrow\mathrm{SL}(2,\mathbb{C})$ is an irreducible representation. By Lemma 3.3, all eigenvalues of $X$ and $Y$ are roots of unity and we may assume that $X$ is conjugate to $\begin{pmatrix}\alpha&0\\ 0&\alpha^{-1}\end{pmatrix}$ and $Y$ is to $\begin{pmatrix}\beta&0\\ 0&\beta^{-1}\end{pmatrix}$ where $\alpha=\alpha(a)=\exp(\sqrt{-1}\pi a/p),\,0\leq a\leq p$ and $\beta=\beta(b)=\exp(\sqrt{-1}\pi b/q),\,0\leq b\leq q$.

Recall that $G(L)$ has the presentation

[TABLE]

We put relators $r_{i}=x^{p}m_{i}x^{-p}m_{i}^{-1}~{}(1\leq i\leq\mu-1)$ and $r_{\mu}=x^{p}y^{-q}$. Applying the free differential to each $r_{i}\ (1\leq i\leq\mu-1)$, we have

[TABLE]

Moreover

[TABLE]

We consider the square matrix $A_{\mu+1}$ which is obtained from the Alexander matrix $A$ by removing the $(\mu+1)$th column. By the definition of the twisted Alexander polynomial, we have

[TABLE]

where $\alpha^{p}=(-1)^{a}$ and we put $t=t_{1}\cdots t_{\mu}$.

It is clear that $\Delta_{L,\rho}(t_{1},\ldots,t_{\mu})$ depends only on the eigenvalues of $X$ and $Y$, which does not vary continuously. Therefore every coefficient of $\Delta_{L,\rho}(t_{1},\ldots,t_{\mu})$ is locally constant.

To sum up we have the following.

Theorem 4.1**.**

The twisted Alexander polynomial of the torus link $L=T(\mu p,\mu q)$ is given by

[TABLE]

where $t=t_{1}\cdots t_{\mu}$. Moreover every coefficient of $\Delta_{L,\rho}(t_{1},\ldots,t_{\mu})$ is locally constant on the irreducible character variety $\mathcal{X}(\mu p,\mu q)$.

Remark 4.2*.*

If $\mu=1$, that is, $L$ is the torus knot $T(p,q)$, then the above formula gives the same one as in [9, Section 4] and [12, Theorem 5.16].

5. Higher dimensional twisted Alexander polynomials

In this section we investigate the higher dimensional twisted Alexander polynomial associated to an irreducible representation of $\mathrm{SL}(2,\mathbb{C})$.

5.1. Irreducible representations of $\mathrm{SL}(2,\mathbb{C})$

The group $\mathrm{SL}(2,\mathbb{C})$ acts naturally on the vector space $\mathbb{C}^{2}$. Then the symmetric product $\text{Sym}^{n-1}(\mathbb{C}^{2})$ and the induced action by $\mathrm{SL}(2,\mathbb{C})$ gives an $n$-dimensional irreducible representation of $\mathrm{SL}(2,\mathbb{C})$. In fact, $\text{Sym}^{n-1}(\mathbb{C}^{2})$ can be identified with the vector space $V_{n}$ of homogeneous polynomials on $\mathbb{C}^{2}$ with degree $n-1$, i.e.

[TABLE]

The action of $P\in\mathrm{SL}(2,\mathbb{C})$ on $V_{n}$ is expressed as

[TABLE]

This action defines a representation $\sigma_{n}:\mathrm{SL}(2,\mathbb{C})\to\mathrm{GL}(V_{n})$.

It is known that the image of $\sigma_{n}$ is actually contained in $\mathrm{SL}(n,\mathbb{C})$, and every $n$-dimensional irreducible representation of $\mathrm{SL}(2,\mathbb{C})$ is equivalent to $(V_{n},\sigma_{n})$. For a representation $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$, we denote the composition $\sigma_{n}\circ\rho:G(L)\to\mathrm{SL}(n,\mathbb{C})$ by $\rho_{n}$. Note that $\rho_{2}=\rho$ holds.

We now study the twisted Alexander polynomial $\Delta_{L,\rho_{n}}(t_{1},\ldots,t_{\mu})$ for the torus link $L=T(\mu p,\mu q)$. Let $\rho:G(L)\to\mathrm{SL}(2,\mathbb{C})$ be an irreducible representation. Assume that $X$ is conjugate to $\begin{pmatrix}\alpha&0\\ 0&\alpha^{-1}\end{pmatrix}$ and $Y$ is to $\begin{pmatrix}\beta&0\\ 0&\beta^{-1}\end{pmatrix}$ where $\alpha=\exp(\sqrt{-1}\pi a/p),\,0\leq a\leq p$ and $\beta=\exp(\sqrt{-1}\pi b/q),\,0\leq b\leq q$ as before. We also put $t=t_{1}\cdots t_{\mu}$.

Theorem 5.1**.**

If $n$ is even, then

[TABLE]

If $n$ is odd, then

[TABLE]

We give a proof of the theorem in the next subsection. As an immediate corollary, we have the following.

Corollary 5.2**.**

Every coefficient of $\Delta_{L,\rho_{n}}(t_{1},\ldots,t_{\mu})$ is locally constant on the irreducible character variety $\mathcal{X}(\mu p,\mu q)$.

For a link $L$ in $S^{3}$, by [6], we have $\mathbb{T}_{L,\rho_{n}}=\Delta_{L,\rho_{n}}(1,\ldots,1)$, where $\mathbb{T}_{L,\rho_{n}}$ is the Reidemeister torsion (see [11] for the definition). Hence, for the torus link $L=T(\mu p,\mu q)$, we obtain the following.

Corollary 5.3**.**

Suppose $n$ is even and $a\equiv b\equiv 1\pmod{2}$. Then

[TABLE]

Remark 5.4*.*

For the torus knot $T(p,q)$ (i.e. $\mu=1$), the above formula give the same one as in [16, Proposition 4.1]. Moreover, by a similar argument to [15, Proposition 3.8], one can show that

[TABLE]

where $p^{\prime}=\frac{p}{(a,p)}$ and $q^{\prime}=\frac{q}{(b,q)}$.

5.2. Proof of Theorem 5.1

Recall $\rho(x)=\begin{pmatrix}\alpha&0\\ 0&\alpha^{-1}\end{pmatrix}$ and $\rho(y)$ is conjugate to $\begin{pmatrix}\beta&0\\ 0&\beta^{-1}\end{pmatrix}$, where $\alpha=\exp(\sqrt{-1}\pi a/p),\,0\leq a\leq p$ and $\beta=\exp(\sqrt{-1}\pi b/q),\,0\leq b\leq q$. Then it is easy to check that $\rho_{n}(x)=\text{diag}(\alpha^{n-1},\alpha^{n-3},\ldots,\alpha^{-(n-1)})$ and $\rho_{n}(y)$ is conjugate to $\text{diag}(\beta^{n-1},\beta^{n-3},\ldots,\beta^{-(n-1)})$, which are diagonal matrices of degree $n$. Hence

[TABLE]

If $n$ is even, then

[TABLE]

Since $\alpha^{p}=(-1)^{a}$, we have

[TABLE]

Hence

[TABLE]

Similarly, if $n$ is odd, then

[TABLE]

This completes the proof of Theorem 5.1.

Acknowledgements

The first and second authors have been partially supported by JSPS KAKENHI Grant Numbers 16K05161 and 17K05261 respectively. The third author has been partially supported by a grant from the Simons Foundation (#354595 to AT).