Cluster Algebra and Complex Volume of Once-Punctured Torus Bundles and Two-Bridge Links
We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.
The cluster algebra was introduced by Fomin and Zelevinsky in , and it has been studied extensively since then. The characteristic operation in the cluster algebra called “mutation” is related to various notions, and there exist many applications of cluster algebra to the representation theory of Lie algebras and quantum groups, triangulated surface [6, 7], Teichmüller theory , integrable systems, and so on.
In geometry, the cluster algebraic techniques are used to understand hyperbolic structure of fibered bundles , where cluster -variables are identified with moduli of ideal hyperbolic tetrahedra. Our purpose in this paper is to study complex volume of 2-bridge link complements via cluster variables with coefficients. The complex volume is a complexification of hyperbolic volume,
where is the hyperbolic volume and is the Chern–Simons invariant of . Based on canonical decompositions of 2-bridge link complements in , we clarify a relationship between ideal tetrahedra and cluster mutations. Main observation is that the cluster variable with coefficients is closely related to Zickert’s formulation of complex volume , and that the complex volume is given from the cluster variable (Theorem 4.9, also Remark 4.10). We shall also give a formula of complex volume for once-punctured torus bundle over the circle (Theorem 3.9).
There may be natural extensions of our results. One of them is a quantization of the cluster algebra, which will be helpful in studies of Volume Conjecture [12, 13], a relationship between hyperbolic geometry and quantum invariants. Indeed in the case of once-punctured torus bundle, a classical limit of adjoint action of mutations and its relationship with [11, 3] are studied in . Also a generalization to higher rank  remains for future works.
This paper is organized as follows. In Section 2, we briefly review the definition of the cluster algebra and the three-dimensional hyperbolic geometry, and explain their interrelationship by taking a simple example. Section 3 is devoted to the once-punctured torus bundles over the circle. We formulate the hyperbolic volume via -variables, and the complex volume via cluster variables. Section 4 is for 2-bridge links. First we review a canonical decomposition of the 2-bridge link complements, and we reformulate it in terms of the cluster algebra. The key is to introduce the cluster coefficient as an element of the tropical semifield. We give an explicit formula for the complex volume in terms of the cluster variables.
2. Cluster Algebra and 3-Dimensional Hyperbolic Geometry
2.1. Cluster Algebra
We briefly give a definition of the cluster algebras following [8, 9]. See these papers for details. We let be a semifield with a multiplication and an addition . This means that is an abelian multiplicative group endowed with a binary operation which is commutative, associative, and distributive with respect to the group multiplication . Let denote the quotient field of the group ring of . (Note that is an integral domain .) Fix . Let be the rational functional field of algebraically independent variables .
A seed is a triple , where
a cluster is an -tuple of elements in such that is a free generating set of ,
a coefficient tuple is an -tuple of elements in ,
an exchange matrix is an skew symmetric integer matrix.
We call a cluster variable, and a coefficient.
An important tool in the cluster algebra is a mutation, which relates cluster seeds.
Let be a seed. For each , we define the mutation of by as
an -tuple of elements in is
a coefficient tuple is
a skew symmetric integral matrix is
Note that the resulted triple is again a seed. We remark that is involutive, and that and are commutative if and only if .
By starting from an initial seed , we iterate mutations and collect all obtained seeds. The cluster algebra is the -subalgebra of the rational function field generated by all the cluster variables. In fact, in this paper we do not need the cluster algebra itself, but the seeds and the mutations. Further, we use the following:
Proposition 2.3 ().
For a seed , let be an -tuple in defined as
Then the mutation of in Def. 2.2 induces a mutation of a pair ,
This proposition holds for an arbitrary semifield . In this paper we call a cluster -variable, or a -variable. Hereafter we use the following tropical semifield .
Set . Let be a semifield generated by a variable with a multiplication and an addition ,
We define a map , given by substituting in elements of .
For the later use, we introduce the permutation acting on seeds.
For and , let be a permutation of subscripts and in seeds. For example a permuted cluster is defined by
Actions on and are defined in the same manner. They induce an action on , and has a same form.
2.2. Hyperbolic Geometry
A fundamental object in the three-dimensional hyperbolic geometry is an ideal hyperbolic tetrahedron in Fig. 1 . The tetrahedron is parameterized by a modulus , and each dihedral angle is given as in the figure. We mean and for given modulus by
The cross section by the horosphere at each vertex is similar to the triangle in with vertices , , and as in Fig. 2. In Fig. 1, we give an orientation to tetrahedron by assigning a vertex ordering , which is crucial in computing the complex volume of tetrahedra modulo .
The hyperbolic volume of an ideal tetrahedron with modulus is given by the Bloch–Wigner function
Here is the dilogarithm function,
for , and the analytic continuation for is given by
See, e.g.,  for details of the dilogarithm function.
We study the case that a cusped hyperbolic -manifold is triangulated into a set of ideal tetrahedra . It is known that the modulus of each ideal tetrahedron is determined from two conditions. One is a set of gluing equations, which means that dihedral angles around each edge sum up to . Another is a cusp condition so that has a complete hyperbolic structure. Then the hyperbolic volume of is given by
The complex volume, , of is a complexification of (2.11), and in view of the Bloch–Wigner function (2.9), it is natural to study the dilogarithm function . Although, in contrast to , the dilogarithm function is a multi-valued function, and we need a “flattening”, i.e., the moduli of ideal tetrahedra with additional parameters and .
Definition 2.7 ().
A flattening of an ideal tetrahedron is
where is the modulus of and . We use to denote the flattening of .
By use of the flattening , we introduce  an extended Rogers dilogarithm function by
where . Here and hereafter, we mean the principal branch in the logarithm. In , the extended pre-Bloch group is defined as the free abelian group on the flattenings subject to a lifted five-term relation, and it is shown that the complex volume can be given as follows.
Proposition 2.8 ().
The complex volume of is
where is a flattening of , and is (resp. ) when the orientation of is same with (resp. opposite to) that of in Fig. 1.
Zickert clarified that the flattening can be given by complex parameters assigned to the edges of an ideal tetrahedron in the following way.
Proposition 2.9 ().
For the ideal tetrahedron in Fig. 1, let be a complex parameter on the edge connecting vertices and for . When these complex parameters satisfy
the flattening is given by
In gluing tetrahedra to construct a three-manifold , edges identified in are required to have the same complex numbers.
It was demonstrated in  that these complex parameters can be read from a developing map such as Figs. 8 and 13. To summarize, when we have a triangulation with modulus of a hyperbolic 3-manifold , the flattering can be given from the above with edge parameters , and we get the complex volume of from (2.14).
Correspondence between the cluster algebra and the hyperbolic geometry can be seen in a simple example.111We thank T. Dimofte. We study a triangulated surface and its flip as in Fig. 3. A triangulation is related to a quiver, i.e., a directed graph, where the number of edges in the triangulation is equal to the fixed number in the cluster algebra . A flip can be regarded as a cluster mutation, as depicted in the figure. Note that the exchange matrix can be read from the quiver by
By definition (2.4), the mutation , is explicitly written as
On the other hand, we may regard the flip in Fig. 3 as an attachment of an ideal tetrahedron with modulus whose faces are pleated. See Fig. 4. When we denote by the dihedral angle on edge labeled as in Fig. 3, a dihedral angle after attaching is given by
with a hyperbolic gluing condition
and especially a modulus of the ideal tetrahedron is given by
See that a subscript “3” denotes a label of vertex in the quiver where the mutation () was applied.
Our idea on a geometrical role of the cluster variable is as follows. We know from (2.4) that the -variable is written as
We also see from (2.1) that the mutation sends a cluster variable to
When we apply the map in Def. 2.5 to the above, the coefficient parts including are . By comparing these formulae with (2.15), we notice that the cluster variables play a role of Zickert’s parameters .
3. Once-Punctured Torus Bundle over
3.1. -pattern and Hyperbolic Volume
Let be a once-punctured torus, . We set to be the once-punctured torus bundle over the circle, whose monodromy is determined by a mapping class . More precisely, via we define an identification for , and set . It is known that is hyperbolic when has distinct real eigenvalues. Up to conjugation we have
To denote a mapping class (3.1) we use a sequence of symbols where or , and
We use a triangulation of as depicted in Fig. 5. It is related to the Farey triangles , which we will explain in Section 4.1. The actions of and are interpreted as “flips” of triangulation as shown in the figure. The triangulation is translated into the cluster algebra of with the exchange matrix as
This denotes the quiver in Fig. 6, where each vertex has a labeling corresponding to that of an edge in the triangulation. Then the flips and can be identified with the mutations in the cluster algebra (cf. ), and we have
where is the permutation defined in Def. 2.6. See Fig. 5. We have used the permutations so that the exchange matrix (3.3) is invariant under these actions. In this way the flips and act on the -variable respectively as
A -pattern of a mapping class (3.1) is for defined recursively by
We have seen in Section 2.3 that the cluster mutation is interpreted as an attachment of an ideal tetrahedron. Thus to each flip , or defined in (3.4), we can assign a single ideal hyperbolic tetrahedron as illustrated in Fig. 7. The triangulations of in Fig. 5 can be regarded as the top and bottom pleated faces of ideal tetrahedron in Fig. 7 . In a cross section by horosphere at a vertex we have four triangles, and each of them has one vertex not shared with any of the other three as in Fig. 7.
Our first claim is that the modulus of each ideal tetrahedron is given from a -pattern. See also .
Let be a -pattern of with an initial condition,
Here and are solutions of
such that each modulus defined by
is in the upper half plane for . Then is the modulus of the tetrahedron .
The hyperbolic volume of is given by
where is the modulus of given in (3.10).
3.2. Proof of Proposition 3.2
We have discussed in Section 2 that the cluster -variable is related to the modulus of ideal tetrahedron. To prove that the -pattern given by (3.7) describes a complete hyperbolic structure of , we need to check [21, 18] that both gluing conditions and a completeness condition are fulfilled. The gluing conditions may be trivial once we know the fact in Section 2.3. Though, we need to check the gluing conditions at edges on top/bottom triangulated surfaces, and the completeness condition is far from trivial. In the following, we shall check all of them by use of a developing map of torus at infinity of (3.1), which is depicted in Fig. 8 (see, e.g., ).
First of all, when , we have
Here we have used in the second equality, and the last equality follows from . See (3.6) for the action of flip . We see that this equality is nothing but a gluing condition for the second circle from the bottom in Fig. 8. A case of can be checked in the similar manner.
In the same manner, we can check
for , which also corresponds to a gluing condition in the figure. Using the periodic condition (3.9), we have
where we have used . This coincides with a consistency condition for the right semi-circle (top and bottom) in the figure.
We can further check that
due to (3.9). This identity corresponds to a gluing condition for the left semi-circle (top and bottom) in the figure. As another example of this type of equality, we have
which denotes a gluing condition for the right middle large circle. In this way, it is straightforward to check consistency conditions in the figure.
A completeness condition can be checked similarly. We have
where the last equality follows from the initial condition (3.8). We can see that this equality denotes a completeness condition from the curve in the figure.
This completes the proof.
3.3. Cluster Pattern and Complex Volume
In the previous Section 3.2, we have proved that the -variables give the hyperbolic volume of the manifold . We shall discuss that the cluster variables give the complex volume of .
A cluster pattern of (3.1) is for defined recursively by
We set an initial seed by
and (3.3). Note that for all we have
When we set a periodic condition
and (3.17) supports the periodicity of -variable, . We choose such that the -variable (3.18) is a geometric solution of (3.9). As was shown in Section 2.3, the cluster variable can be regarded as Zickert’s edge parameters of . See Remark 2.10.
To get the complex volume modulo , we need to take into account of the orientation of . When we assign an orientation to the triangulations of , it induces an orientations of as illustrated in Fig. 7 . The tetrahedron has the same orientation with in Fig. 1, while the tetrahedron has the opposite orientation. Using a relationship between the vertex ordering and dihedral angles in Fig. 1, we obtain the following.
Let be a cluster pattern satisfying the condition (3.17). Then the modulus of is given by
Due to orientation of the tetrahedron, a solution is geometric if and only if (resp. ) for (resp. ).
We obtain the flattening of as follows.
We follow the setting at Lemma 3.6. The flattening for the tetrahedron is given by
When we discard the vertex ordering of the ideal hyperbolic tetrahedra , the resulting complex volume is defined modulo . Ignoring orientations of , we may simply set the moduli of tetrahedra from (3.10) and (3.18) as
for . Then we obtain the flattening of from
With these flattening, we have the complex volume modulo  by
We take an example . A cluster pattern is
An initial cluster variable is solved up to constant from the periodic condition (3.17), :
By setting , geometric solutions are . From (3.20) the solution with plus sign gives the flattening parameters for as , , , while the minus sign solution gives , , . Both solutions give