1.1 Quiver mutation

In this paper, we always assume that a quiver has

• ∙ the vertex set $I=\{1,\ldots ,n\}$ , and

• ∙ no loops and oriented $2$ -cycles (see Figure 1).

Figure 1.

A loop (left) and an oriented $2$ -cycle (right) of a quiver.

For vertices $i$ and $j\in I$ , we define

$$\begin{eqnarray}Q(i,j)=\sharp \{\text{arrows from }i\text{ to }j\},\qquad \overline{Q}(i,j)=Q(i,j)-Q(j,i).\end{eqnarray}$$

Note that the quiver $Q$ is uniquely determined by the skew-symmetric matrix $\overline{Q}(i,j)$ (or equivalently $Q(i,j)$ ) under the assumption aboveFootnote ¹ .

For the vertex $k$ , we define a new quiver $\unicode[STIX]{x1D707}_{k}Q$ (mutation of $Q$ at vertex $k$ ) by an antisymmetric matrix

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D707}_{k}Q}(i,j)=\left\{\begin{array}{@{}ll@{}}-\overline{Q}(i,j),\quad & i=k\text{ or }j=k,\\ \overline{Q}(i,j)+Q(i,k)Q(k,j)-Q(j,k)Q(k,i),\quad & i,j\neq k.\end{array}\right.\end{eqnarray}$$

1.2 Cluster variables

Given a sequence $\mathbf{k}=(k_{1},\ldots ,k_{l})$ of vertices and “time” parameters $t=0,\ldots ,l$ , we define

$$\begin{eqnarray}Q_{0}:=Q,\qquad Q_{t}:=\unicode[STIX]{x1D707}_{k_{t-1}}\cdots \unicode[STIX]{x1D707}_{k_{1}}Q\quad (t>0).\end{eqnarray}$$

For initial values $x_{i}(0)=x_{i}$ and $y_{i}(0)=y_{i}$ , we define the cluster $x$ -variables $x_{i}(t)$ and the cluster $y$ -variables (coefficients) $y_{i}(t)$ ( $i\in I$ ) by

(1) $$\begin{eqnarray}x_{i}(t+1)=\frac{\mathop{\prod }_{j}x_{j}(t)^{Q_{t}(i,j)}+\mathop{\prod }_{j}x_{j}(t)^{Q_{t}(j,i)}}{x_{i}(t)},\end{eqnarray}$$

and

(2) $$\begin{eqnarray}y_{i}(t+1)=\left\{\begin{array}{@{}ll@{}}y_{k}(t)^{-1},\quad & i=k,\\ y_{i}(t)y_{k}(t)^{Q_{t}(k,i)}(1+y_{k}(t))^{\overline{Q_{t}}(i,k)},\quad & i\neq k.\end{array}\right.\end{eqnarray}$$

2.1 Quiver associated to a triangulation

For a triangulation $\unicode[STIX]{x1D70F}$ without self-folded arcs we will define a quiver $Q_{\unicode[STIX]{x1D70F}}$ whose vertex set $I$ is the set of arcs in $\unicode[STIX]{x1D70F}$ .

For a triangle $\unicode[STIX]{x1D6E5}$ and arcs $i$ and $j$ , we define a skew-symmetric integer matrix $\overline{Q}^{\unicode[STIX]{x1D6E5}}$ by

$$\begin{eqnarray}\overline{Q}^{\unicode[STIX]{x1D6E5}}(i,j):=\left\{\begin{array}{@{}ll@{}}1\quad & \unicode[STIX]{x1D6E5}\text{ has sides }i\text{ and }j,\text{with }i\text{ following }j\\ \quad & \text{in the clockwise order},\\ -1\quad & \text{the same holds, but in the counter-clockwise order,}\\ 0\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

We define

$$\begin{eqnarray}\overline{Q}_{\unicode[STIX]{x1D70F}}:=\mathop{\sum }_{\unicode[STIX]{x1D6E5}\in \unicode[STIX]{x1D70F}}\overline{Q}^{\unicode[STIX]{x1D6E5}},\end{eqnarray}$$

where the sum is taken over all triangles in $\unicode[STIX]{x1D70F}$ . Let $Q_{\unicode[STIX]{x1D70F}}$ denote the quiver associated to the matrix $\overline{Q}_{\unicode[STIX]{x1D70F}}$ .

For an arc $i$ in the triangulation $\unicode[STIX]{x1D70F}$ , we can flip the edge $i$ to get a new triangulation $f_{i}(\unicode[STIX]{x1D70F})$ . This operation is compatible with a mutation at vertex  $i$ :

$$\begin{eqnarray}Q_{f_{i}(\unicode[STIX]{x1D70F})}=\unicode[STIX]{x1D707}_{i}(Q_{\unicode[STIX]{x1D70F}}).\end{eqnarray}$$

2.2 Mapping class group action

For a triangulation $\unicode[STIX]{x1D70F}$ , we define

$$\begin{eqnarray}T=T(\unicode[STIX]{x1D70F}):=\mathbb{C}(y_{e})_{e\in \unicode[STIX]{x1D70F}_{1}},\qquad T^{\vee }=T^{\vee }(\unicode[STIX]{x1D70F}):=\mathbb{C}(x_{e})_{e\in \unicode[STIX]{x1D70F}_{1}}.\end{eqnarray}$$

For a puncture $m\in M$ , take a sufficiently small circle around $m$ and let $e_{1},\ldots ,e_{n}$ be the sequence of arcs which intersect with the circle, where $e_{1},\ldots ,e_{n}$ may have multiplicity. We define

(3) $$\begin{eqnarray}\displaystyle y_{m}:=\mathop{\prod }_{i=1}^{n}y_{e_{i}} & & \displaystyle\end{eqnarray}$$

and

$$\begin{eqnarray}\text{}\underline{T}=\text{}\underline{T}(\unicode[STIX]{x1D70F}):=\mathbb{C}[y_{e},y_{e}^{-1}]_{e\in \unicode[STIX]{x1D70F}_{1}}\big/(y_{m})_{m\in M}.\end{eqnarray}$$

Let us fix a mapping class $\unicode[STIX]{x1D711}$ . Then the two triangulations $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})$ are related by a sequence of flips, together with appropriate changes of labels. More formally, there exists a sequence

$$\begin{eqnarray}\mathbf{k}=(k_{1},\ldots ,k_{l})\in (\unicode[STIX]{x1D70F}_{1})^{l}\end{eqnarray}$$

such that the two triangulations $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})$ are related by the sequence of flips associated to $\mathbf{k}$ (see [Reference Fomin, Shapiro and ThurstonFST08, Proposition 3.8]). Note that a flip provides a canonical bijection of the edges of the triangulations. We can represent the composition of the bijections by a permutation $\unicode[STIX]{x1D70E}\in \mathfrak{S}_{I}$ . We define the automorphisms

$$\begin{eqnarray}\text{CT}_{\unicode[STIX]{x1D711}}:T(\unicode[STIX]{x1D70F})=T(\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F}))\overset{{\sim}}{\longrightarrow }T(\unicode[STIX]{x1D70F}),\qquad \text{CT}_{\unicode[STIX]{x1D711}}^{\vee }:T^{\vee }(\unicode[STIX]{x1D70F})=T^{\vee }(\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F}))\overset{{\sim}}{\longrightarrow }T^{\vee }(\unicode[STIX]{x1D70F})\end{eqnarray}$$

by

$$\begin{eqnarray}\text{CT}_{\unicode[STIX]{x1D711}}(y_{e})=y_{\unicode[STIX]{x1D70E}(e)}(l),\qquad \text{CT}_{\unicode[STIX]{x1D711}}^{\vee }(x_{e})=x_{\unicode[STIX]{x1D70E}(e)}(l).\end{eqnarray}$$

Thanks to the result [Reference Fomin, Shapiro and ThurstonFST08, Theorem 3.10] and the pentagon relation of cluster transformations, $\text{CT}_{\unicode[STIX]{x1D711}}$ and $\text{CT}_{\unicode[STIX]{x1D711}}^{\vee }$ are independent of the choices of the sequences of flips and provides a well-defined action of the mapping class group on $T(\unicode[STIX]{x1D70F})$ .

4.1 Shape parameters

For an ideal tetrahedron in $\mathbb{H}^{3}$ with vertices $0$ , $1$ , $z$ and $\infty$ (Figure 4), we associate the shape parameter $z$ with the edge connecting $0$ and $\infty$ . For an ideal tetrahedron, a pair of mutually nonintersecting edges has a common shape parameter, and the shape parameters for the three pairs of mutually nonintersecting edges are given by (Figure 5)

(4) $$\begin{eqnarray}z,\qquad 1-z^{-1},\qquad \frac{1}{1-z}.\end{eqnarray}$$

Figure 5.

The three shape parameters of an ideal tetrahedron.

We take a sequence of flips and associated topological decomposition of the mapping torus as in §3. For $t\in \mathbb{Z}$ , let $\unicode[STIX]{x1D6E5}(t)$ denote the $t$ th tetrahedron, where $\unicode[STIX]{x1D6E5}(t)$ and $\unicode[STIX]{x1D6E5}(t+l)$ are identified for any $t$ . Let $Z(t)$ denote the shape parameter of $\unicode[STIX]{x1D6E5}(t)$ at the edge $h(t)$ , the edge flipped at time $t$ . Note that the sequence $(Z(t))$ satisfies shape parameter periodicity

(5) $$\begin{eqnarray}Z(t+l)=Z(t).\end{eqnarray}$$

For a tetrahedron $\unicode[STIX]{x1D6E5}$ , let $\unicode[STIX]{x1D6E5}_{1}$ be the set of edges of $\unicode[STIX]{x1D6E5}$ . We define

$$\begin{eqnarray}\overline{E}:=\coprod _{t\in \mathbb{Z}}\unicode[STIX]{x1D6E5}(t)_{1}.\end{eqnarray}$$

Let $\widetilde{E}$ denote the set of all edges in the tetrahedron decomposition of $\unicode[STIX]{x1D6F4}\times \mathbb{R}$ and $\unicode[STIX]{x1D70B}:\overline{E}\rightarrow \widetilde{E}$ be the canonical surjection.

Given parameters $(Z(t))_{t\in \mathbb{Z}}$ , we can define associated parameter $Z_{e}=Z_{e}(t)$ for any $t\in \mathbb{Z}$ and $e\in \unicode[STIX]{x1D6E5}(t)_{1}$ as the shape parameter of $\unicode[STIX]{x1D6E5}(t)$ on the edge $e$ , which is determined as in (4).

4.2 Edge-gluing conditions

Suppose that the shape parameters $(Z(t))_{t\in \mathbb{Z}}$ give an ideal tetrahedron decompositionFootnote ² . This holds if and only if the following three conditions are satisfied.

First, we need the shape parameter periodicity condition as already discussed in (5). Second, we need

$$\begin{eqnarray}\text{Im}\,Z(t)>0\quad \text{for any}~t\quad (\text{positivity condition}),\end{eqnarray}$$

so that the tetrahedron is positively oriented. Third, for each edge $g\in \widetilde{E}$ , the product of all the shape parameters associated to the elements in $\unicode[STIX]{x1D70B}^{-1}(g)$ must be $1$ ([Reference ThurstonThu79], see Figure 6):

$$\begin{eqnarray}\mathop{\prod }_{\bar{g}\in \unicode[STIX]{x1D70B}^{-1}(g)}Z_{\bar{g}}=1\quad (\text{edge}\text{-}\text{gluing equation}).\end{eqnarray}$$

Figure 6.

The edge-gluing equation around an edge.

4.3 $y$ -variables and gluing conditions

Proposition 4.1. Let $e(t)\in \unicode[STIX]{x1D70F}(t)_{1}$ be the edge which we flip at $t$ and $e^{\prime }(t+1)\in \unicode[STIX]{x1D70F}(t+1)_{1}$ be the edge which appears after the flip. The edge-gluing equation is satisfied for the shape parameters

(6) $$\begin{eqnarray}Z(t):=-y_{e(t)}(t)\left(=-y_{e^{\prime }(t+1)}(t+1)^{-1}\right).\end{eqnarray}$$

Figure 7.

An edge $g$ in a tetrahedron decomposition appears at time $t_{1}$ and disappears at time $t_{2}$ .

Proof. Let $g\in \widetilde{E}$ be an edge which appears at the $t_{1}$ th flip at $\bar{g}^{\prime }$ and disappears at the $t_{2}$ th flip $\bar{g}^{\prime \prime }$ (Figure 7). Let $\bar{g}_{1}$ (resp. $\bar{g}_{2}$ ) be the unique element in $\unicode[STIX]{x1D6E5}(t_{1})_{1}\cap \unicode[STIX]{x1D70B}^{-1}(g)$ (resp. in $\unicode[STIX]{x1D6E5}(t_{2})_{1}\cap \unicode[STIX]{x1D70B}^{-1}(g)$ ). The gluing equation associated with $g$ is

$$\begin{eqnarray}\displaystyle 1 & = & \displaystyle \mathop{\prod }_{t=t_{1}}^{t_{2}}\mathop{\prod }_{\bar{g}\in \unicode[STIX]{x1D6E5}(t)_{1}\cap \unicode[STIX]{x1D70B}^{-1}(g)}Z_{\bar{g}}\nonumber\\ \displaystyle & = & \displaystyle Z(t_{1})\times \left(\mathop{\prod }_{t=t_{1}+1}^{t_{2}-1}\mathop{\prod }_{\bar{g}\in \unicode[STIX]{x1D6E5}(t)_{1}\cap \unicode[STIX]{x1D70B}^{-1}(g)}Z_{\bar{g}}\right)\times Z(t_{2}).\nonumber\end{eqnarray}$$

For this equation, we will show

(7) $$\begin{eqnarray}Z(t_{1})\times \left(\mathop{\prod }_{t=t_{1}+1}^{T}\mathop{\prod }_{\bar{g}\in \unicode[STIX]{x1D6E5}(t)_{1}\cap \unicode[STIX]{x1D70B}^{-1}(g)}Z_{\bar{g}}\right)=-y_{g_{0}}(T+1)^{-1},\end{eqnarray}$$

where $g_{0}$ is the edge corresponding to $g$ which $\unicode[STIX]{x1D70F}(t)$ ( $t=t_{1}+1,\ldots ,t_{2}$ ) have in common. We show the equation above by induction with respect to $T$ . The claim for $T=t_{1}$ trivially follows from the definition (6). Let us assume the above statement for $T\rightarrow T-1$ . To show the statement for $T$ , we need to show

$$\begin{eqnarray}\mathop{\prod }_{\bar{g}\in \unicode[STIX]{x1D6E5}(t)_{1}\cap \unicode[STIX]{x1D70B}^{-1}(g)}Z_{\bar{g}}=y_{g_{0}}(t)/y_{g_{0}}(t+1).\end{eqnarray}$$

We will show this by classifying the positional relation of $g_{0}$ and $e(t)$ .

• ∙ $g_{0}$ and $e(t)$ have no triangle in common: both side of the equation above is $1$ .

• ∙ $g_{0}$ and $e(t)$ have a single triangle in common:

  1. – $\overline{Q_{t}}(e(t),g_{0})=1$ (see Figure 8):

     $$\begin{eqnarray}(\text{LHS})\overset{\text{ equation (4) }}{=}1-Z(t)^{-1}\overset{\text{ equation (2) }}{=}(\text{RHS}),\end{eqnarray}$$

  2. – $\overline{Q_{t}}(e(t),g_{0})=-1$ :

     $$\begin{eqnarray}(\text{LHS})\overset{\text{ equation (4) }}{=}(1-Z(t))^{-1}\overset{\text{ equation (2) }}{=}(\text{RHS}),\end{eqnarray}$$

• ∙ $g_{0}$ and $e(t)$ have two triangles in common:

  1. – $\overline{Q_{t}}(e(t),g_{0})=\pm 2$ (see Figure 9):

     $$\begin{eqnarray}(\text{LHS})\overset{\text{ equation (4) }}{=}(1-Z(t)^{\mp })^{\pm 2}\overset{\text{ equation (2) }}{=}(\text{RHS}),\end{eqnarray}$$

  2. – $\overline{Q_{t}}(e(t),g_{0})=0$ : this cannot happen because we prohibit self-folded edges in this paper (see Figure 10).

Figure 8.

The case with $\overline{Q_{t}}(e(t),g_{0})=1$ .

Figure 9.

The case with $\overline{Q_{t}}(e(t),g_{0})=2$ .

Figure 10.

The case with $\overline{Q_{t}}(e(t),g_{0})=0$ .

4.4 Complete hyperbolic structures

Patching ideal tetrahedra with corners removed, we get a hyperbolic $3$ -manifold with boundaries, each of which is isomorphic to a torus. Note that such a boundary torus has two directions: the direction of “time” parameter $t$ (time direction) and the direction of the original surface (surface direction)Footnote ³ .

The intersection of a removed corner and a boundary torus gives a triangle on the torus with a shape parameter for each angle.

Figure 11.

A Dehn half-twist $\unicode[STIX]{x1D70E}_{1}$ along a circle containing $A$ and $B$ .

Figure 12.

The link corresponding to $\unicode[STIX]{x1D70E}_{1}\unicode[STIX]{x1D70E}_{2}\unicode[STIX]{x1D70E}_{3}^{-1}$ .

Figure 13.

A triangulation of the five-punctured sphere.

Figure 14.

The triangulation of the boundary torus. A triangle with number $t$ represents the $t$ th tetrahedron $\unicode[STIX]{x1D6E5}(t)$ , whose modulus $Z(t)$ corresponds to a dihedral angle represented by a black dot.

Fix a puncture $m\in M$ of the surface and a time parameter $t_{0}$ . Let $F_{i}$ ( $i\in \mathbb{Z}/n\mathbb{Z}$ ) be the triangle in $\unicode[STIX]{x1D70F}(t_{0})$ which is adjacent to $e_{i-1}$ , $m$ and $e_{i}$ , where $(e_{1},\ldots ,e_{n})$ is the sequence of arcs around $m$ as before.

On the boundary torus, $e_{i}$ represents a vertex and $F_{i}$ represents an edge connecting $e_{i-1}$ and $e_{i}$ . The union of $F_{i}$ ’s provides a (piecewise linear) closed curve on the boundary torusFootnote ⁵ . We call this a vertical line (see Figure 15).

Figure 15.

Vertical lines, drawn in the triangulation of Figure 14.

The holonomy of along a cycle in the surface direction is given as follows. A vertical line divides the boundary torus into two parts. We fix one of them. For a vertex $e_{i}$ on the vertical line, we take all angles in the universal cover which have $e_{i}$ as the vertex and which are on the given side of the vertical line. We denote by $H_{i}$ the product of the shape parameters associated to these angles. Then we have

(8) $$\begin{eqnarray}\displaystyle (\text{the holonomy in the surface direction})=\mathop{\prod }_{i}(-H_{i}). & & \displaystyle\end{eqnarray}$$

A hyperbolic structure given by a sequence of shape parameters is complete if and only if the following condition holds:

In Figure 7, we study the set of all tetrahedra which are adjacent to an edge. In this setting, the vertical line divides the set of these tetrahedra into two groups: tetrahedra which appear before/after $t=t_{0}$ . Hence we have

$$\begin{eqnarray}H_{i}=Z(t_{1})\times \left(\mathop{\prod }_{t=t_{1}+1}^{t_{0}-1}\mathop{\prod }_{\bar{e}_{i}\in \unicode[STIX]{x1D6E5}(t)_{1}\cap \unicode[STIX]{x1D70B}^{-1}(e_{i})}Z_{\bar{e_{i}}}\right).\end{eqnarray}$$

By (7), the right hand side equals $-y_{e_{i}}(t_{0})$ . Therefore the holonomy (8) is equivalent with $y_{m}(t_{0})=\prod _{i=1}^{n}y_{e_{i}}(t_{0})$ (recall (3)). We can show this product is independent on the choice of $t_{0}$ , either by induction or by using the edge-gluing conditions (vertical lines at different choices of $t_{0}$ are homologous in the triangulation of the boundary torus).

In summary, we get the following description of the holonomy:

4.5 Main theorem

Let us summarize our results in the form of a theorem:

Theorem 4.4. Let $(y_{e})_{e\in \unicode[STIX]{x1D70F}_{1}}$ be nonzero complex numbers such that $y_{m}=1$ for any puncture $m\in M$ . Assume that $y_{h}(t)\Big|_{y_{e}(0)=y_{e}}$ is well defined for any $h$ and $t$ and that the periodicity equation is satisfied

$$\begin{eqnarray}y_{\unicode[STIX]{x1D70E}(h)}=y_{h}(l)\Big|_{y_{e}(0)=y_{e}}.\end{eqnarray}$$

Let us define the shape parameters $Z(t)$ by

$$\begin{eqnarray}Z(t):=-y_{e(t)}(t)\Big|_{y_{e}(0)=y_{e}},\end{eqnarray}$$

where $e(t)$ is the edge flipped at time $t$ , and suppose that $Z(t)\neq 0,1$ for any $t$ . Then $(Z(t))$ satisfies the edge-gluing equations in §4.2 and the cusp condition in §4.4.

This theorem gives a systematic method to identify for hyperbolic structures on mapping tori, formulated in the language of cluster algebras. For a genuine hyperbolic structure we also need to verify $\text{Im}(Z(t))>0$ ; see the examples in the next section.

5.1 Once-punctured torus and $LR$

Let us start with a once-punctured torus. We take a sequence of two flips as in Figure 16. This is the mapping class studied in [Reference Terashima and YamazakiTY13, §3.1]. Then the shape parameter periodicity conditions are

$$\begin{eqnarray}\displaystyle y_{1} & = & \displaystyle y_{2}^{-1}\left(1+y_{1}^{-1}(1+y_{2}^{-1})^{2}\right)^{-2},\nonumber\\ \displaystyle y_{2} & = & \displaystyle y_{3}(1+y_{2})^{2}\left(1+y_{1}(1+y_{2}^{-1})^{-2}\right)^{2},\nonumber\\ \displaystyle y_{3} & = & \displaystyle y_{1}^{-1}(1+y_{2}^{-1})^{2},\nonumber\end{eqnarray}$$

and the cusp condition is

$$\begin{eqnarray}y_{1}y_{2}y_{3}=1.\end{eqnarray}$$

Solving these equations, we get a solution

$$\begin{eqnarray}y_{1}=1,\qquad y_{2}=\frac{-1-\sqrt{-3}}{2},\qquad y_{3}=\frac{-1+\sqrt{-3}}{2}.\end{eqnarray}$$

By Theorem 4.4, shape parameters

$$\begin{eqnarray}\displaystyle Z(0) & = & \displaystyle -y_{2}(0)=-y_{2}=\frac{1+\sqrt{-3}}{2},\nonumber\\ \displaystyle Z(1) & = & \displaystyle -y_{1}(1)=-y_{1}(1+y_{2}^{-1})^{-2}=\frac{1+\sqrt{-3}}{2},\nonumber\end{eqnarray}$$

satisfy edge-gluing conditions. Moreover, the imaginary parts of $Z(0)$ and $Z(1)$ are positive, and we obtain a complete hyperbolic structure on the mapping torus. The parameters coincide with the ones in [Reference Terashima and YamazakiTY13, §3.1].

5.2 Five-punctured sphere and $\unicode[STIX]{x1D70E}_{1}\unicode[STIX]{x1D70E}_{2}\unicode[STIX]{x1D70E}_{3}^{-1}$

Let us take the example of a five-punctured sphere in Example 4.2.

The cusp conditions are

$$\begin{eqnarray}y_{1}y_{2}y_{3}y_{4}y_{8}y_{9}=y_{1}y_{5}=y_{2}y_{5}y_{8}y_{6}=y_{3}y_{6}y_{9}y_{7}=y_{4}y_{7}=1,\end{eqnarray}$$

and the shape parameter periodicity conditions are

$$\begin{eqnarray}\displaystyle y_{6} & = & \displaystyle \frac{1}{y_{1}y_{5}y_{8}^{2}y_{9}}\Bigl((1+y_{5}+y_{5}y_{8})((1+y_{9}+y_{7}y_{9})(1+y_{9}+y_{8}y_{9})\nonumber\\ \displaystyle & & \displaystyle +\,y_{5}(1+(1+y_{7})(1+y_{8})y_{9})(1+(1+y_{8}+y_{1}y_{8})y_{9}))\Bigr),\nonumber\\ \displaystyle y_{1} & = & \displaystyle y_{2}(1+y_{5}+y_{5}y_{8}),\nonumber\\ \displaystyle y_{2} & = & \displaystyle \frac{y_{3}y_{7}y_{9}(1+y_{9}+y_{8}y_{9}+y_{5}(1+y_{8})(1+(1+y_{8}+y_{1}y_{8})y_{9}))}{(1+y_{9}+y_{7}y_{9})(1+y_{9}+y_{8}y_{9})+y_{5}(1+(1+y_{7})(1+y_{8})y_{9})(1+(1+y_{8}+y_{1}y_{8})y_{9})},\nonumber\\ \displaystyle y_{9} & = & \displaystyle \frac{y_{4}(1+(1+y_{7})(1+y_{8})y_{9})(1+y_{9}+y_{8}y_{9}+y_{5}(1+y_{8})(1+(1+y_{8}+y_{1}y_{8})y_{9}))}{(1+y_{5}+y_{5}y_{8})(1+y_{9}+y_{8}y_{9})},\nonumber\\ \displaystyle y_{8} & = & \displaystyle \frac{y_{1}y_{8}y_{9}}{1+y_{9}+y_{8}y_{9}+y_{5}(1+y_{8})(1+(1+y_{8}+y_{1}y_{8})y_{9})},\nonumber\\ \displaystyle y_{5} & = & \displaystyle \frac{y_{5}y_{6}y_{8}}{1+y_{5}+y_{5}y_{8}},\nonumber\\ \displaystyle y_{4} & = & \displaystyle \frac{(1+y_{9}+y_{7}y_{9})(1+y_{9}+y_{8}y_{9})+y_{5}(1+(1+y_{7})(1+y_{8})y_{9})(1+(1+y_{8}+y_{1}y_{8})y_{9})}{y_{7}y_{8}y_{9}},\nonumber\\ \displaystyle y_{3} & = & \displaystyle \frac{y_{8}(1+y_{9}+y_{8}y_{9})}{(1+(1+y_{7})(1+y_{8})y_{9})(1+y_{9}+y_{8}y_{9}+y_{5}(1+y_{8})(1+(1+y_{8}+y_{1}y_{8})y_{9}))},\nonumber\\ \displaystyle y_{7} & = & \displaystyle \frac{y_{1}y_{5}y_{7}y_{8}y_{9}}{(1+y_{9}+y_{7}y_{9})(1+y_{9}+y_{8}y_{9})+y_{5}(1+(1+y_{7})(1+y_{8})y_{9})(1+(1+y_{8}+y_{1}y_{8})y_{9})}.\nonumber\end{eqnarray}$$

Solving the shape parameter periodicity conditions with cusp conditions, we get $14$ solutions. We take one of the solutions

$$\begin{eqnarray}\displaystyle y_{1} & = & \displaystyle 1.781241-0.294452\times \sqrt{-1},\nonumber\\ \displaystyle y_{2} & = & \displaystyle 1,\nonumber\\ \displaystyle y_{3} & = & \displaystyle -0.304877+0.754529\times \sqrt{-1},\nonumber\\ \displaystyle y_{4} & = & \displaystyle 0.460355+1.139318\times \sqrt{-1},\nonumber\\ \displaystyle y_{5} & = & \displaystyle 0.546473+0.0903361\times \sqrt{-1},\nonumber\\ \displaystyle y_{6} & = & \displaystyle 1.155478+1.893847\times \sqrt{-1},\nonumber\\ \displaystyle y_{7} & = & \displaystyle 0.304877-0.754529\times \sqrt{-1},\nonumber\\ \displaystyle y_{8} & = & \displaystyle 0.304877-0.754529\times \sqrt{-1},\nonumber\\ \displaystyle y_{9} & = & \displaystyle -0.14865-0.664193\times \sqrt{-1}.\nonumber\end{eqnarray}$$

Following the algorithm in Theorem 4.4, we get the following six parameters

$$\begin{eqnarray}\displaystyle & & \displaystyle 0.754529\times \sqrt{-1}-0.304877,\nonumber\\ \displaystyle & & \displaystyle 0.754529\times \sqrt{-1}+0.695123,\nonumber\\ \displaystyle & & \displaystyle 0.294452\times \sqrt{-1}-0.781241,\nonumber\\ \displaystyle & & \displaystyle 0.754529\times \sqrt{-1}+0.695122,\nonumber\\ \displaystyle & & \displaystyle 0.475124\times \sqrt{-1}+0.311704,\nonumber\\ \displaystyle & & \displaystyle 1.139320\times \sqrt{-1}+0.460354,\nonumber\end{eqnarray}$$

whose imaginary parts are positive, which provide a complete hyperbolic structure. The volume of the mapping torus computed from the parameters above isFootnote ⁶

$$\begin{eqnarray}4.851170.\end{eqnarray}$$

This coincides with the value computed by SnapPea/SnapPy [Reference Culler, Dunfield and WeeksCDW].

5.3 Nonhyperbolic example

Our formalism discussed in this paper applies to in general nonhyperbolic 3-manifolds which are themselves not covered in Theorem 4.4. To illustrate this point, let us consider a disk with six points, and we consider the $1/6$ rotation as a mapping class. The mapping class is realized as a sequence of three flips as in Figure 17.

Figure 17.

A nonhyperbolic example, associated with the $1/6$ rotation of the six-punctured disc.

The periodicity conditions are

$$\begin{eqnarray}\displaystyle y_{1} & = & \displaystyle y_{1}(3),\qquad y_{2}=y_{2}(3),\qquad y_{3}=y_{3}(3),\qquad y_{4}=y_{5}(3),\qquad y_{5}=y_{6}(3),\nonumber\\ \displaystyle y_{6} & = & \displaystyle y_{7}(3),\qquad y_{7}=y_{8}(3),\qquad y_{8}=y_{9}(3),\qquad y_{9}=y_{4}(3).\nonumber\end{eqnarray}$$

Note that indices of edges in the boundary are rotated. The $y$ -variables are given by

$$\begin{eqnarray}\displaystyle y_{1}(3) & = & \displaystyle y_{1}^{-1}(1+y_{2})^{-1},\nonumber\\ \displaystyle y_{2}(3) & = & \displaystyle y_{2}^{-1}(1+y_{1}(1+y_{2}))(1+y_{3}(1+y_{2})),\nonumber\\ \displaystyle y_{3}(3) & = & \displaystyle y_{3}^{-1}(1+y_{2})^{-1},\nonumber\\ \displaystyle y_{4}(3) & = & \displaystyle y_{4}(1+y_{1}^{-1}(1+y_{2})^{-1})^{-1},\nonumber\\ \displaystyle y_{5}(3) & = & \displaystyle y_{5}(1+y_{1}(1+y_{2})),\nonumber\\ \displaystyle y_{6}(3) & = & \displaystyle y_{6}(1+y_{2}^{-1})^{-1}(1+y_{1}^{-1}(1+y_{2})^{-1})^{-1},\nonumber\\ \displaystyle y_{7}(3) & = & \displaystyle y_{7}(1+y_{3}^{-1}(1+y_{2})^{-1})^{-1},\nonumber\\ \displaystyle y_{8}(3) & = & \displaystyle y_{8}(1+y_{3}(1+y_{2})),\nonumber\\ \displaystyle y_{9}(3) & = & \displaystyle y_{9}(1+y_{2}^{-1})^{-1}(1+y_{3}^{-1}(1+y_{2})^{-1})^{-1}.\nonumber\end{eqnarray}$$

A solution of the periodicity conditions is

$$\begin{eqnarray}\displaystyle y_{1} & = & \displaystyle {\textstyle \frac{1}{2}},\qquad y_{2}=3,\qquad y_{3}={\textstyle \frac{1}{2}},\qquad y_{4}=\sqrt{3},\qquad y_{5}={\textstyle \frac{1}{\sqrt{3}}},\nonumber\\ \displaystyle y_{6} & = & \displaystyle {\textstyle \frac{2}{\sqrt{3}}},\qquad y_{7}=\sqrt{3},\qquad y_{8}={\textstyle \frac{1}{\sqrt{3}}},\qquad y_{9}={\textstyle \frac{2}{\sqrt{3}}}.\nonumber\end{eqnarray}$$

Shape parameters of three tetrahedra evaluated at the solution above are

$$\begin{eqnarray}\displaystyle Z(0) & = & \displaystyle -y_{2}(0)=-y_{2}=-3,\nonumber\\ \displaystyle Z(1) & = & \displaystyle -y_{1}(1)=-y_{1}(1+y_{2})=-2,\nonumber\\ \displaystyle Z(2) & = & \displaystyle -y_{3}(2)=-y_{3}(1+y_{2})=-2.\nonumber\end{eqnarray}$$

Substituting shape parameters to the Rogers dilogarithm $L(x)$ , we have (with $Z(i)^{\prime \prime }=(1-Z(i))^{-1}$ )

$$\begin{eqnarray}\displaystyle L(Z(0)^{\prime \prime })+L(Z(1)^{\prime \prime })+L(Z(2)^{\prime \prime })=\frac{\unicode[STIX]{x1D70B}^{2}}{6}, & & \displaystyle \nonumber\end{eqnarray}$$

which is the complexified volume of the $3$ -manifold. This is identified with the central charge of $\hat{\text{sl}}(2)$ WZW model at the level $4$ (see Remark 0.3).