Cluster Algebra and Complex Volume of Once-Punctured Torus Bundles and Two-Bridge Knots

0.29 MB·English

by  Kazuhiro Hikami

CLUSTER ALGEBRA AND COMPLEX VOLUME OF
ONCE-PUNCTURED TORUS BUNDLES AND TWO-BRIDGE KNOTS
KAZUHIRO HIKAMI AND REI INOUE
Abstract. We propose a method to compute complex volume of 2-bridge knot com-
plements. Clarified is a relationship between cluster variables with coefficients and
canonical decompositions of knot complements.
3
1
0
1. Introduction
2

n
a The cluster algebra was introduced by Fomin & Zelevinsky in [6], and it has been
J
2 studied extensively since then. The characteristic operation in cluster algebra called
1 “mutation” is related to various notions, and there exists many application of cluster

] algebra to representation theory of Lie algebras and quantum groups, triangulated sur-
T
face [4, 5], Teichmu¨ller theory [3], integrable systems, and so on.
G
.
h Asoneofapplicationstogeometry, theclusteralgebraictechniqueisusedtohyperbolic
t
a structure of fibered bundles [13], where cluster y-variables are identified with moduli of
m
ideal hyperbolic tetrahedra. Our purpose in this paper is to study complex volume of
[
2-bridge knot complements via cluster variables with coefficients. The complex volume

2
v is a complexification of hyperbolic volume,
2
4 Vol(M)+i CS(M),
0
6 where Vol(M) is the hyperbolic volume and CS(M) is the Chern–Simons invariant of
.
2 M. Based on canonical decompositions of 2-bridge knot complements in [17], we clarify
1
a relationship between ideal tetrahedra and cluster mutations. Main observation is that
2
1 the cluster variable with coefficients is closely related to Zickert’s formulation of com-
:
v
plex volume [21], and that the complex volume is given only from the cluster variable
i
X
(Theorem 4.8, also Remark 4.9). We shall also give a formula of complex volume for
r
a once-punctured torus bundle over the circle (Theorem 3.10).
There may be natural extensions of our results. One of them is a quantization of
cluster algebra, which will be helpful in studies of Volume Conjecture [10, 11], a rela-
tionship between a hyperbolic geometry and quantum invariants. Indeed in the case of
once-punctured torus bundle, a classical limit of adjoint action of mutations and its re-
lationship with [1, 9] are studied in [18]. Also a generalization to higher rank [3] remains
for future works.
This paper is organized as follows. In section 2, we briefly review the definition of
cluster algebra and three-dimensional hyperbolic geometry, and explain their interre-
lationship by taking a simple example. Section 3 is devoted to once-punctured torus
Date: December 25, 2012. Revised on January 11, 2013.
1
2 K. HIKAMIANDR. INOUE
bundles over the circle. We formulate hyperbolic volume via y-variables in §3.1, and
complex volume via cluster variables in §3.3. Section 4 is for 2-bridge knots. First we
review a canonical decomposition of 2-bridge knot complements, and we describe it in
terms of cluster algebra. The key is to introduce a tropical semifield (Definition 2.4)
to which the cluster coefficients belong. The hyperbolic volume via y-variables and the
complex volume via cluster variables are respectively formulated in §4.2 and §4.4.
2. Cluster Algebra and 3-Dimensional Hyperbolic Geometry
2.1 Cluster Algebra
We follow a definition of cluster algebras in [6, 7]. Let (P,⊕,·) be a semifield endowed
an auxiliary addition ⊕, which is commutative, associative, and distributive with respect
to the group multiplication · in P. Let QP denote the quotient field of the group ring
ZP of P. Fix N ∈ Z .
>0
Definition 2.1. A seed is a triple (x,ε,B), where
• a cluster x = (x ,...,x ) is an N-tuple of N algebraically independent variables
1 N
with coefficients in QP,
• a coefficient tuple ε = (ε ,...,ε ) is an N-tuple of elements in P,
1 N
• an exchange matrix B = (b ) is an N ×N skew symmetric integer matrix.
ij
We call x a cluster variable, and ε a coefficient.
i i
Definition 2.2. Let (x,ε,B) be a seed. For each k = 1,...,N, we define the mutation
of (x,ε,B) by µ as
k
µ (x,ε,B) = (x,ε,B),
k
where
e e e
• the cluster x = (x ,...,x ) is
1 N
x , for i 6= k,
i
e e e
xi =  εk · 1 x bjk + 1 · 1 x −bjk, for i = k, (2.1)
1⊕ε x j 1⊕ε x j
k k k k
e j:bYjk>0 j:bYjk<0
• the coefficient tuple ε = (ε ,...,ε ) is
 1 N
ε −1, for i = k,
k
e e ε ebki
ε = ε k , for i 6= k, b ≥ 0, (2.2)
i  i 1⊕ε ki
 (cid:18) k(cid:19)
ε (1⊕ε )−bki, for i 6= k, b ≤ 0,
e i k ki


• the exchange matrixB = (b ) is
ij
−b , for i = k or j = k,
ij e e
b = (2.3)
ij  1
b + (|b |b +b |b |), otherwise.
 ij ik kj ik kj
2
e


CLUSTER ALGEBRA & COMPLEX VOLUME 3
Note that the resulted triplet (x,ε,B) is again a seed.
e e e
By starting from an initial seed (x,ε,B), we iterate mutations and collect all obtained
seeds. The cluster algebra A(x,ε,B) is the ZP-subalgebra of the rational function field
QP(x) 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 ([7]). Let y be an N-tuple y = (y ,...,y ), defined by use of cluster x
1 N
and coefficient ε as
y = ε x bkj. (2.4)
j j k
k
Y
Then we have a mutation,
µ (y,B) = (y,B), (2.5)
k
where
e e
• y = (y ,...,y ) is analogous to (2.2),
1 N
y −1, for i = k,
e e e k
y bki
y = y k , for i 6= k, b ≥ 0, (2.6)
i  i 1+y ki
 (cid:18) k(cid:19)
y (1+y )−bki , for i 6= k, b ≤ 0,
e i k ki




• B = (b ) is (2.3).
ij
e e
This propositionholdsfor arbitrarysemifield (P,⊕,·). In thispaper we call y acluster
i
y-variable, or a y-variable. Hereafter we use a tropical semifield [7].
Definition 2.4. Set P = {δk | k ∈ Z}. Let (P,⊕,·) be a semifield generated by a
variable δ with a multiplication · and an addition ⊕,
δk1 ⊕δk2 = δmin(k1,k2). (2.7)
Definition 2.5. We define a map ψ, ψ : P−→{−1,1}, given by substituting δ = −1 in
elements of P.
For the later use, we introduce the permutation acting on seeds.
Definition 2.6. For i,j ∈ {1,...,N} and i 6= j, let s be a permutation of subscripts
i,j
i and j in seeds. For example permutated cluster s (x) is defined by
i,j
s (··· ,x ,··· ,x ,···) = (··· ,x ,··· ,x ,···).
i,j i j j i
Actions on ε and B are defined in the same manner. They induce an action on y, and
s (y) has a same form.
i,j
4 K. HIKAMIANDR. INOUE
2.2 Hyperbolic Geometry
A fundamental object in three-dimensional hyperbolic geometry is an ideal hyperbolic
tetrahedron △ in Fig. 1 [19]. The tetrahedron is parameterized by a modulus z ∈ C,
′ ′′
and each dihedral angle is given as in the figure. We mean z and z for given modulus
z by
1 1
′ ′′
z = 1− , z = . (2.8)
z 1−z
The cross section by thehorosphere ateach vertex issimilar tothe triangleinCwith ver-
tices 0, 1, and z as in Fig. 2. In Fig. 1, we have assigned a vertex ordering following [21],
which is crucial in computing the complex volume of tetrahedra modulo π2.
Figure 1. An ideal hyperbolic tetrahedron △ with modulus z. Dihedral
′ ′′
angles are given by z, z = 1−1/z, and z = 1/(1−z). Each v denotes
a
a vertex ordering. We give an orientation to an edge from v to v (a < b).
a b
z
′
z
′′
z z
0 1
Figure 2. A triangle in C with vertices 0, 1, and z.
The hyperbolic volume of an ideal tetrahedron △ with modulus z is given by the
Bloch–Wigner function
D(z) = ℑLi (z)+arg(1−z) log|z|, (2.9)
2
where Li (z) is the dilogarithm function,
2
z ds
Li (z) = − log(1−s) .
2
s
Z0
Note that
′ ′′
D(z) = D(z ) = D(z )
′ ′′
= −D(1/z) = −D(1/z ) = −D(1/z ). (2.10)
CLUSTER ALGEBRA & COMPLEX VOLUME 5
See, e.g., [20].
A set of ideal tetrahedra {△ } is glued together to construct a cusped hyperbolic
ν
manifold M. A modulus z of each ideal tetrahedron △ is determined from both
ν ν
gluing conditions around each edge and a completeness condition [14, 16, 19]. Then the
hyperbolic volume of M is given by
Vol(M) = D(z ). (2.11)
ν
ν
X
The complex volume, Vol(M)+i CS(M), is related to an extended Rogers dilogarithm
function
1 πi π2
L(z;p,q) = Li (z)+ logz log(1−z)+ (qlogz +p log(1−z))− , (2.12)
2
2 2 6
where p,q ∈ Z. To compute the complex volume, we need an additional structure to the
moduli of ideal tetrahedra:
Definition 2.7 ([15]). A flattening of an ideal tetrahedron △ is
(w ,w ,w ) = (logz +pπi,−log(1−z)+qπi,log(1−z)−logz −(p+q)πi), (2.13)
0 1 2
where z is the modulus of △ and p,q ∈ Z. We use (z;p,q) to denote the flattening of △.
In[15], theextended pre-Blochgroupisdefined asthefreeabeliangrouponflattenings
subject toafive-termrelation, andshownisthattheflatteninggivesthecomplexvolume.
Proposition 2.8 ([15]). The complex volume of M is
i (Vol(M)+i CS(M)) = sgn(ν)L(z ;p ,q ), (2.14)
ν ν ν
ν
X
where (z ;p ,q ) and sgn(ν) = ±1 representing a flattening and a vertex ordering of a
ν ν ν
tetrahedron △ .
ν
For a tetrahedron △ in Fig. 1, let c be a complex number assigned to an edge
ab
connecting vertices v and v . Zickert clarified that the flattening (z;p,q) of △ is given
a b
by c as follows.
ab
Proposition 2.9 ([21]). When we have
c c c c 1 c c 1
03 12 01 23 02 13
= ±z, = ± 1− , = ± , (2.15)
c c c c z c c 1−z
02 13 03 12 (cid:18) (cid:19) 01 23
the flattening (z;p,q) is given by
logz +pπi = logc +logc −logc −logc ,
03 12 02 13
(2.16)
−log(1−z)+qπi = logc +logc −logc −logc .
02 13 01 23
Remark 2.10. IngluingtetrahedratoconstructM, identicaledgeshavethesamecomplex
numbers.
6 K. HIKAMIANDR. INOUE
2.3 Interrelationship
Correspondence between the cluster algebra and the hyperbolic geometry can be seen
in a simple example.1 We study a triangulation of surface and its flip as in Fig. 3.
Triangulation is related to quiver where the number of edges in a triangulation is the
same as the fixed number N in the cluster algebra, and flip can be regarded as mutation,
as depicted in the figure. Note that the exchange matrix B = (b ) of quiver is
ij
b = #{arrows from i to j}−#{arrows from j to i}.
ij
By definition (2.4), the mutation µ (y,B) = (y,B), is explicitly written as
3
y = y (1+y ),
1 1 3
e e
y = y (1+y −1)−1,
2 2 3
ye = y −1, (2.17)
3 3
ye = y (1+y −1)−1,
4 4 3
e
y = y (1+y ).
5 5 3
e
e
2 2
3 3
1 5 → 1 5
4 4
Figure 3. Triangulation of a punctured surface. Associated quiver is
depicted in red.
Figure 4. Flip and attachment of pleated tetrahedron.
Ontheotherhand, wemayregarda flipinFig.3asanattachment ofideal tetrahedron
△ with modulus z whose faces are pleated. See Fig. 4. When we denote z as a dihedral
k
1We thank T. Dimofte.
CLUSTER ALGEBRA & COMPLEX VOLUME 7
angle on edge k, dihedral angle z after attaching △ is given by
k
′
z = z z ,
1 1
e
′′
z = z z ,
2 2
e
z = z, (2.18)
3
e ′′
z = z z ,
4 4
e ′
z = z z ,
5 5
e
with a hyperbolic gluing condition
e
z z = 1.
3
Comparing (2.17) with (2.18), we observe that the cluster y-variable is related to
dihedral angle by
y = −z ,
k k
and especially a modulus of ideal tetrahedron △ is given by
1
z = − , (2.19)
y
3
where a subscript “3” is a direction of mutation.
3. Once-Punctured Torus Bundle over S1
3.1 y-pattern and Hyperbolic Volume
Let Σ be a once-punctured torus, (R2\Z2)/Z2. We set M to be the once-punctured
1,1 ϕ
torus bundle over the circle, whose monodromy is determined by a mapping class ϕ ∈
SL(2;Z). More precisely, via ϕ : Σ → Σ we define identification (x,0) ∼ (ϕ(x),1)
1,1 1,1
for ∀x ∈ Σ , and set M = Σ ×[0,1]/ ∼. It is known that M is hyperbolic when ϕ
1,1 ϕ 1,1 ϕ
has distinct real eigenvalues, and that up to conjugation we have
ϕ = Rs1Lt1 ···RsnLtn, (3.1)
where
1 1 1 0
R = , L = .
0 1 1 1
(cid:18) (cid:19) (cid:18) (cid:19)
To denote a mapping class (3.1) we use a sequence of symbols F F ···F =
1 2 c
R···RL···L···R···RL···L where F = R or L, and
k
s1 t1 sn tn
| {z }| {z } | {z }| {z } n
c = (s +t ). (3.2)
j j
j=1
X
We set a triangulation of Σ as depicted in Fig. 5. It is related to the Farey trian-
1,1
gles [2], and we will explain these Farey triangles later in the subsequent section. The
actions of R and L are interpreted as “flips” of triangulation as shown in the figure. The
8 K. HIKAMIANDR. INOUE
triangulation is translated into the cluster algebra of N = 3 with the exchange matrix
as
0 −2 2
B = 2 0 −2 . (3.3)
 
−2 2 0
This denotes the quiver in Fig. 6, where each vertex has a labeling corresponding to that
of an edge in the triangulation. Then the flips R and L can be identified with mutations
in the cluster algebra (cf. [18]), and we have
R = s µ , L = s µ , (3.4)
1,3 1 2,3 2
where s is a permutation defined at Definition 2.6. See Fig. 5. We have used permu-
i,j
tations so that the exchange matrix (3.3) is invariant under these actions. In this way
the flips R and L act on y-variable respectively as
R L
(y,B) −→ (R(y),B), (y,B) −→ (L(y),B), (3.5)
where
y (1+y −1)−2 ⊤ y (1+y −1)−2 ⊤
3 1 1 2
R(y) = y (1+y )2 , L(y) = y (1+y )2 . (3.6)
2 1 3 2
 y −1   y −1 
1 2
   
2 2
3 3 3
1 1
2 2
2 2 3 3 3
1 1
R
3 3 2 2
1 1 1 ր
2 2
3 3
1 1 1 ց 3 3
L
2 2 2 2
1 1 1
3 3
2 2
1 1 1
3 3
Figure 5. Triangulation of once-punctured torus (left). The vertex de-
notes a puncture. A fundamental region is colored gray. A labeling of each
edge corresponds to that of each vertex in the quiver. Actions of flips, R
and L, are given in the right hand side.
CLUSTER ALGEBRA & COMPLEX VOLUME 9
1
2 3
Figure 6. A quiver associated to a triangulation of once-puncture torus Σ .
1,1
Definition 3.1. A y-pattern of a mapping class ϕ = F ···F (3.1) is y[k] for k =
1 c
1,2,...,c+1 defined recursively by
y[k +1] = F (y[k]). (3.7)
k
△(R)
△(L)
Figure 7. Tetrahedra △(R) and △(L) assigned to flip R and L on once-
punctured torus. Once we fix an orientation of triangulation of Σ , an
1,1
orientation of tetrahedra is induced as illustrated in the figure.
To each flip, R or L, assigned is a single ideal hyperbolic tetrahedron △(F ) as il-
k
lustrated in Fig. 7, where triangulations of Σ in Fig. 5 can be regarded as the top
1,1
and bottom pleated faces of ideal tetrahedron [2]. 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 modulus of each ideal tetrahedron is given from a y-pattern.
See also [13].
Proposition 3.2. Let y[k] be a y-pattern of ϕ with an initial condition,
1
y[1] = y ,y , . (3.8)
1 2
y y
(cid:18) 1 2(cid:19)
Here y and y are geometric solutions of
1 2
y[1] = y[c+1], (3.9)
10 K. HIKAMIANDR. INOUE
such that each modulus z[k] defined by
1
− , when F = R,
k
y[k]
z[k] =  1 (3.10)
 1
− , when F = L,
k
y[k]
2


is in the upper half plane H for k =1,··· ,c.
Then z[k] is a modulus of tetrahedron △(F ).
k
Remark 3.3. We have discarded vertex orderings of tetrahedra in setting of moduli (3.10)
here.
Corollary 3.4. The hyperbolic volume of M is given by
ϕ
c
Vol(M ) = D(z[k]), (3.11)
ϕ
k=1
X
where z[k] is the modulus of △(F ) obtained as (3.10).
k
3.2 Proof of Proposition 3.2
We shall check that both gluing conditions and a completeness condition are fulfilled
for a set of moduli (3.10). For this purpose, we recall a developing map of torus at
infinity given in Fig. 8 (see, e.g., [8]).
First of all, we have
2
−1 1 −1
′′ 2
z[3] (z [2]) z[1] =
y[3] 1+y[2] −1 y[1]
1 (cid:18) 1 (cid:19) 1
−1 −1
= = 1.
y[2] y[1]
3 1
Here we have used y[3] = R(y[2]) in the second equality, and the last equality follows
from y[2] = R(y[1]). See (3.6) for the action of flip R. We see that this equality is
nothing but a gluing condition for the second green circle from the bottom in Fig. 8. In
the same manner, we can check
′′ 2
z[k +1] (z [k]) z[k −1] = 1,
for 2 ≤ k ≤ c−1, which is also corresponds to gluing conditions in the figure. With a
help of periodic condition (3.9), we have
2
−1 1 −1
′′ 2
z[2] (z [1]) z[c] =
y[2] 1+y[1] −1 y[c]
1 (cid:18) 1 (cid:19) 2
= 1,
where we have used y[1] = y[c + 1] = y[c] −1. This coincides with a consistency
3 3 2
condition for the right green semi-circle (top and bottom) in the figure.