Cluster Algebra and Complex Volume (Intelligence of Low-dimensional Topology)

(1)

Cluster

Algebra

and

Complex

Volume

Kazuhiro HIKAMI

#,

Rei INOUE

# Faculty

of

Mathematics,

Kyushu

University,

Fukuoka

819-0395

Department

of

Mathematics

and

Informatics,

Faculty of Science,

Chiba

University,

Chiba

263-8522

1 Introduction

The volume conjecture [13, 15] indicates anintimate relationshipbetween quantum

in-variants of knots andcomplex volume of knotcomplements. Generallyquantum

invari-ants of knots

are

constructed combinatoricallybasedon$R$-matrix, andit is expected that

complex volume

can

be formulated based on knot diagram. Indeed in [24, 3, 2, 1]

con-structedwasthe Neumann-Zagierpotentialfunction [20]_based_on_the$R$-matrices of the

Kashaevinvariant and thecolored Jones polynomialatroot ofunity.

Our purpose in this article is to study complex volume of knot complements from

viewpoint of clusteralgebra. Seeourworks [9, 10]_{for detail.}

2 Cluster

Algebra

and

3-Dimensional

Hyperbolic

Geometry

2.1 ClusterAlgebra

Wefollow adefinition of cluster algebras in [6, 7]. _Let$(\mathbb{P}, \oplus, \cdot)$beasemifield endowed an

auxiliaryaddition $\oplus$, which iscommutative, associative, and distributive with respectto

the group multiplication. in$\mathbb{P}$

.

Let

$\mathbb{Q}\mathbb{P}$ denote the quotient field of the group ring$\mathbb{Z}\mathbb{P}$of $\mathbb{P}$

.

Fix_{$N\in \mathbb{Z}_{>0}.$}

Definition2.1. $A$seedis atriple$(x,\epsilon,B)$,where

.

acluster$x=(x_{1,\ldots,N}x)$isan$N$-tupleof$N$algebraically independentvariables with

coefficients in$\mathbb{Q}\mathbb{P},$

acoefficient tuple $\epsilon=(\epsilon_{1}, \ldots,\epsilon_{N})$is an$N$-tuple of elements in$\mathbb{P},$

.

an

exchange matrix$B=(b_{ij})$is

an

$NxN$ skew symmetric integer matrix.

(2)

Definition 2.2. Let $(x,\epsilon,B)$ be

a

seed. For each _$k=1,$$\ldots,N$, we define the mutation of

$(x,\epsilon,B)$ by$\mu_{k}$ as

$\mu_{k}(x,\epsilon,B)=(\tilde{x},\tilde{\epsilon},\tilde{B})$,

where

the cluster$\tilde{x}=(\tilde{x}_{1}, \ldots,x_{N})$is

$\tilde{x}_{i}=1^{x_{i}}\frac{\epsilon_{k}}{1\oplus\epsilon_{k}}\cdot\frac{1}{x_{k}}\prod_{j:b_{jk}>0}x_{j}^{b_{jk}}+\frac{1}{1\oplus\epsilon_{k}}\cdot\frac{1}{xh}\prod_{j:b_{jk}<0}x;^{b_{jk}}, forifori=h\neq h,$

’

(1)

the coefficienttuple$\tilde{\epsilon}=(\tilde{\epsilon}_{1}, \ldots,\tilde{\epsilon}_{N})$is

$\tilde{\epsilon}_{i}=1_{\epsilon_{i}(}^{\epsilon_{h}^{-1}}\epsilon_{i}(\frac{\epsilon_{k}}{1\oplus\epsilon_{k})1\oplus\epsilon_{k}})^{b_{ki}}-b_{ki}, fori\neq h,b_{ki}\leq 0fori\neq k,b_{hi}\geq 0fori=h,$

,

(2)

the exchange matrix $\tilde{B}=(\tilde{b}_{ij})$is

$\tilde{b}_{ij}=\{\begin{array}{ll}-b_{ij}, for i=k or j=k,b_{ij}+\frac{1}{2}(|b_{ik}|b_{kj}+b_{ik}|b_{kj}|) , otherwise.\end{array}$ (3)

Note that the resultedtriplet$(\tilde{x},\tilde{\epsilon},\tilde{B})$ is againaseed.

Bystartingfroman initial seed$(x,\epsilon,B)$,

we

iterate mutations and collect all obtained

seeds. The cluster algebra $\mathscr{A}(x,\epsilon,B)$ is the $\mathbb{Z}\mathbb{P}$-subalgebra oftherational function field

$\mathbb{Q}\mathbb{P}(x)$ generated by all thecluster variables. Weusethefollowing.

Proposition2.3([7]). Let$y$bean$N$-tuple $y=(y_{1}, \ldots,y_{N})$,

defined

by use

of

cluster$x$and

coefficient

$\epsilon$as

$y_{j}= \epsilon_{j}\prod_{k}x_{k}^{b_{kj}}$

.

(4)

Thenwehavea mutation,

$\mu_{k}ty,B)=(\tilde{y},\tilde{B})$, (5)

where

$\tilde{y}=(\tilde{y}_{1}, \ldots,\tilde{y}_{N})$isanalogous to(2),

(3)

$.\tilde{B}=(\tilde{b}_{ij})$is (3).

For the later use,

we

introducethepermutation acting

on

seeds.

Definition 2.4. For $i,j\in\{1, \ldots,N\}$and $i\neq j$, let$s_{i,j}$be

a

permutation of subscripts$i$ and

$j$inseeds. For example permutated cluster $s_{i,j}(x)$is definedby

$s_{i,j}(\cdots,x_{i},\cdots,x_{j},\cdots)=(\cdots,x_{j},\cdots,x_{i},\cdots)$

.

Actions

on

$\epsilon$ and $B$

are

defined in the

same

manner.

They induce

an

action on

$y$, and

$s_{i,j}(y)$has

a

same

form.

2.2 Hyperbolic Geometry

A fundamental object in three-dimensional hyperbolic geometry is

an

ideal hyperbolic

tetrahedron $\triangle$ in Fig. 1 [23]. The tetrahedron is parameterized byamodulus$z\in \mathbb{C}$, and

eachdihedralangle isgiven

as

inthe figure. We

mean

$z’$and$z”$for given modulus $z$ by

$z’=1- \frac{1}{z}, z"=\frac{1}{1-z}$

.

(7)

The

cross

section by the horosphere at each vertex is similar to the triangle in $\mathbb{C}$ with

vertices $0,1$, and$z$. We have assigned avertex ordering following [27], which is crucial

incomputingthe complexvolume oftetrahedra modulo$\pi^{2}$

.

See

Fig. 1.

Figure 1: An ideal hyperbolic tetrahedron$\triangle$withmodulus$z$

.

Dihedralanglesaregivenby$z,$$z’=1$-llz,

and$z”=1/(1-z)$.Each$v_{a}$denotesavertexordering. Wegiveanorientation toanedgefrom$v_{a}$to$v_{b}(a<b)$.

The hyperbolic volume of

an

ideal tetrahedron$\triangle$withmodulus_$z$isgiven by the Bloch–

Wigner function

$D(z)=\Im Li_{2}(z)+\arg(1-z)\log|z|$_, (8)

where$Li_{2}(z)$is the dilogarithm function,

$Li_{2}(z)=-\int_{0}^{z}\log(1-s)\frac{ds}{s}.$

See,e.g., [26].

Aset of ideal tetrahedra$t\triangle_{v}$} is glued togethertoconstruct acusped hyperbolic

man-ifold$M.$ $A$modulus$z_{v}$ of eachideal tetrahedron$\triangle_{v}$is determined from bothgluing

(4)

volume of$M$is givenby

$Vol(M)=\sum_{v}D(z_{v})$. (9)

The complex volume,$Vol(M)+iCS(M)$, is given interms ofanextended Rogers

dilog-arithm function

$L([z;p,q])=Li_{2}(z)+\frac{1}{2}\log z\log(1-z)+\frac{\pi i}{2}(q\log z+p\log(1-z))-\frac{\pi^{2}}{6}$_, ₍₁₀₎

where$p,q\in \mathbb{Z}$

.

To compute the complex volume, we need an additional structure to the

moduliofideal tetrahedra:

Definition2.5 ([19]). $A$flattening ofanideal tetrahedron $\triangle$is

$(w0,w_{1},w_{2})=(\log z+p\pi i,-\log(1-z)+q\pi i,\log(1-z)-\log z-(p+q)\pi i)$, (11)

where $z$ isthe modulus of$\triangle$ and_{$p,q\in \mathbb{Z}$}

.

Weuse $[z;p,q]$to denote theflattening

of$\triangle.$

In [19], _{the extended}pre-Blochgroup is defined as the free abelian group

on

flatten-ings subject to

a

five-term relation, and shown is that the flattening gives the complex

volume.

Proposition2.6([19]). _{The complex volume}$ofM$is

$i(Vol(M)+iCS(M))=\sum_{v}$sign(v)$L([z_{v};p_{v},q_{v}])$, (12)

where $[z_{v};p_{v},q_{v}]$ and _{$sign(v)=\pm 1$} respectively denote aflattening and a vertex ordering

of

a tetrahedron $\triangle_{v}.$

Fora tetrahedron $\triangle$ in Fig. 1, let

$c_{ab}$ be a complexnumber assigned to an edge

con-necting vertices $v_{a}$ and $vb$

.

Zickert clarified that the flattening$(z;p,q)$ of$\triangle$ is given by

$c_{ab}$ asfollows.

Proposition2.7([27]). _When_we_have

$\frac{c_{03}c_{12}}{c_{02}c_{13}}=\pm z, \frac{c01^{\mathcal{C}}23}{c_{03}c_{12}}=\pm(1-\frac{1}{z}) , \frac{c_{02}c_{13}}{c_{01}c_{23}}o=\pm\frac{1}{1-z}$, (13)

the flattening$(z;p,q)$_isgiven by

$lz+\pi i=lc_{03+lc_{12}-l-\log_{C}13},$

(14)

$-\log(1-z)+q\pi i=\log c_{02}+\log c_{13}-\log c_{01}-\log_{\mathcal{C}}23.$

Remark 2.8. Ingluingtetrahedra toconstruct$M$, identical edgeshavethesamecomplex

numbers.

2.3 Interrelationship

Correspondence between the cluster algebra and the hyperbolic geometry

can

be

seen

in

(5)

Fig. 2. 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 depictedin thefigure. Note that the exchangematrix$B=(b_{ij})$ofquiveris

$b_{ij}=\#\{$

arrows

from$i$ to_$j\}-\#\{$

arrows

from$j$to $i\}.$

Bydefinition(4), themutation$\mu_{3}(y,B)=(\tilde{y},\tilde{B})$, isexplicitlywritten

as

$\tilde{y}_{1}=y_{1}(1+y_{3})$, $\tilde{y}_{2}=y_{2}(1+y_{3^{-1}})^{-1},$ $\tilde{y}_{3}=y_{3^{-1}}$, (15) $\tilde{y}_{4}=y_{4}(1+y_{3^{-1}})^{-1},$ $\tilde{y}_{5}=y_{5}(1+y_{3})$

.

$arrow$

Figure2: [Left]_{Triangulation ofa punctured}surface.Associated quiver is depicted in red.[Right]Flipand attachment ofpleatedtetrahedron.

On the otherhand,wemayregardaflipinFig.2as anattachment ofideal tetrahedron

$\triangle$ with modulus

$z$ whose faces are pleated. When

we

denote $zk$

as a

dihedral angle

on

edge $k$,dihedral angle$\tilde{z}k$ afterattaching$\triangle$is given by

$\tilde{z}_{1}=z_{1}z’,$

$\tilde{z}=z,$

$\tilde{z}3=z$, (16)

$\tilde{z}_{4}=z_{4^{Z"}},$

$\tilde{z}_{5}=z_{5^{Z’}},$

with

a

hyperbolic gluing condition

(6)

Comparing(15)_with(16),

we

observethat the cluster$y$-variable is relatedto dihedral

angle by

$y=-z,$

andespecially amodulus ofideal tetrahedron$\triangle$ isgiven by

$z=-\underline{1}$

, (17)

$\mathcal{Y}3$

where asubscript“3”is adirection ofmutation.

3 Braid Relation

3.1 $R\cdot operator$

We set the exchange matrix$B$ as

$B=\ovalbox{\tt\small REJECT} 001000 -1000001 -1000001 -1-100101 -1000001 -1001000 -1001000]$

.

(18)

By regarding thematrix elementas

$b_{ij}=\#t$

arrows

from $i$ to$jI-\#${arrows from

$j$to $i$}, (19)

exchange matrix $B$ corresponds to quiver, which is dual to _triangulated

surface (see,

e.g., [5]$)$

.

In our case (18), we have the

quiver and the triangulated surface depicted

in Fig. 3.

7

Figure3: Quiver and triangulated surface

Definition3.1 ([10]). _{We define the} $R$-operatorby

(7)

Note that the inverse of the $R$-operator is givenby

$R^{-1}=s_{3,6}s_{2,5}s_{3,5}\mu_{4}\mu_{5}\mu_{3}\mu_{4}$

.

(21)

The permutations are included in the $R$-operator so that the exchange matrix$B(18)$

is invariant under R. We

use

atrivial semi-field [7], andwe set all cluster coefficients to

be 1. Exphcitly

we

have

$R^{\pm 1}(x,B)=(R^{\pm 1}(x),B)$_, ₍₂₂₎

where

$R(x)=[\frac{X_{1^{X_{3}x_{4}x_{5}+x_{3^{X}5^{X}7+x_{1}x_{2}x_{6}x_{7}}}}\frac{xxx+xxx+xxxx_{5}x_{1}}{4^{2_{x_{5}+x_{1}x_{3}x_{6}x_{7}+x_{3}x_{4}x}}x_{2}x_{4}}}{\frac{xxx_{\’{o}}+x_{3}xx_{7}+xxxxx_{5}x_{6}}{x_{4}x_{6}},xx_{7}^{3}}1^{T}$

$R^{-1}(x)=[x_{7}\frac{x_{1}x_{2}x_{4}x_{6+x_{2x_{4}^{2}x_{6}+x_{1}x_{3}x_{5}x_{7}+x_{1}x_{2}xx_{7}+xx_{4}x67}^{\frac{x_{135126246}xx+xxx+xxxx_{1}}{x_{3}x_{4}}}}x_{6}62x}{}\frac{xxx_{6}+x3x_{5}x7+x2xxx_{3}x_{4}x_{5}x_{2}}{x4x_{5}}]^{T}$ (23)

Correspondingly,actions ofthe$R$-operator, (20)and(21),onthe_$y$-variable are

respec-tively givenasfollows:

$R(y)=[\frac{\frac{}{}\frac{y1(1_{\mathcal{Y}2y_{4\mathcal{Y}5\mathcal{Y}6}}+\mathcal{Y}2+\mathcal{Y}2\mathcal{Y}4)}{1^{+y_{2}+\mathcal{Y}6+y_{2}y_{6}+y_{2}y_{4\mathcal{Y}6}}1_{+2++y_{6}+y^{2}y_{4\mathcal{Y}6}}y_{y_{4}}2\mathcal{Y}4}}{t_{\frac{}{}}),\frac{1+y2+y_{2}y_{4})(1+\mathcal{Y}6+y_{4\mathcal{Y}6}1+y_{2}+y_{6}+y_{2}y6+y_{2\mathcal{Y}4\mathcal{Y}6}y2\mathcal{Y}3\mathcal{Y}4y_{6}y_{4\mathcal{Y}6}}{1+y2+y6+y_{2}y6+\mathcal{Y}2\mathcal{Y}4y_{6}(1+y_{6}+y_{4}y_{6})y_{7}}}T$ $R^{-1}(y)=y(2(1+3\mathcal{Y}4\mathcal{Y}5)]^{T}$ (24)

Duringtheworkshop, R. Kashaev kindlyinformedusthat anessentially sameaction

(8)

It should be noted that the$R$-operator(20)canbe writtenas

$R=s_{2,5}s_{3,6}\mu_{2}\mu_{6}\mu_{4}\mu_{2}\mu_{6}$

.

(25)

3.2 Braid Relation

WegeneralizethequiverinFig. 3 to that inFig.4. Therein alsogivenis the triangulated

surface, and

an

exchangematrix$B$is given bythe rule(19)

as

ageneralizationof(18).

$2 5 3i-1 3n-1$

Figure 4:Quiverand triangulated surface.

Definition 3.2. By use of(20), we define the $R$-operator $Ri$ for $i=1,$_$\ldots,n-1$ associated

with thequiverinFig. 4by

$R\mu_{3i+1}\mu_{3i-1}\mu_{3i+3}\mu_{3i+1}i_{=s_{3i,3i+2^{S}3i-1,3i+2^{S}3i,3i+3}}$

.

₍₂₆₎

Note that

$R^{-1}=s_{3i,3i+3}s_{3i-1,3i+2}s_{3i,3i+2}\mu_{3i+1}\mu_{3i+2}\mu_{3i}\mu_{3i+1}i$

.

₍₂₇₎

Theexchangematrix associatedtoFig.4 is invariant under the action ofthe $R$-operators

$R^{i_{\pm 1}}.$

The explicit forms ofthe actions onthe clustervariable $x=(x_{1},x_{2,\ldots,3n+1}x)$ and the

y-variable$y=(y_{1},y_{2}, \ldots,y_{3n+1})$ are as follows.

$R^{\pm 1}(x)=(x_{1},\ldots,x_{3i-3}, R^{\pm 1}(x_{3i-2},\ldots,x_{3i+4}),x_{3i+5},\ldots,x_{3n+1})i$_, ₍₂₈₎

$R^{\pm 1}(y)=(y_{1}, \ldots,y_{3i-3}, R^{\pm 1}(y_{3i-2}, \ldots,y_{3i+4}),y_{3i+5,\ldots,\mathcal{Y}3n+1)}i$_, ₍₂₉₎

(9)

Theorem3.3([10]). The $R$-operator

satisfies

thebraid relation, namely

we

have

$RRRii+1i=RRRi+1ii+1$_,

for

$i=1,2,$$\ldots,n-2_{J}$ (30)

$RRi=RRji$

,

_for

$|i-j|>1$

.

(31)

3.3 Octahedron

Based

on

3-dimensional interpretation of the cluster mutation given in

Section

2.3,

we

can

see

thatthe $R$-operator(20)isrealized

as

an

octahedron in Fig. 5,which is composed

of four tetrahedra $t\triangle_{N},\triangle s,\triangle_{W},\triangle_{E}$}. The four tetrahedra originate from fourmutations

in the $R$-operator, (20)and(21). In octahedra,the cluster variables_$Xk$ and$\tilde{x}k$ definedby

$\tilde{x}=R^{\pm 1}(x)$,

areassignedto edges,andwe have used

$x_{c}= \frac{x_{2}x_{6}+x_{3}x_{5}}{x_{4}}$

.

(32)

Note thatwehave fixedvertexorderingforourconvention, and thatedgeswith the

same

complexparameters$(e.g., two$pairs$of$edges $v_{0}-v_{2}, v_{1}-v_{3})$

are

identical.

Figure5:Octahedron for$R$(left)and$R^{-1}$_(right)

As the $R$-operator satisfies the braid relation (Theorem 3.3), we can interpret that

each octahedron is assigned to every crossing of knot diagram

as

in Fig. 6. This

re-minds a fact [22] _{that octahedron} was assigned to the Kashaev $R$-matrix [12] (see also,

[8, 1, 3, 24]$)$

.

Note thatanother expression(25)ofthe

same

$R$-operator corresponds to a

decomposition ofoctahedron intofive tetrahedra,whichwasusedinstudies of the colored

(10)

$\triangle_{N}$ $\triangle_{N}$

$\triangle s \triangle s$

Figure6: Dihedral angleatcrossings, $R$(left)and$R^{-1}$(right).

Taking into account ofthe vertex ordering oftetrahedra, we can determine moduli

ofeach tetrahedron from (13) as in Table 1. From these results, we define dilogarithm

functions foreverycrossing by

$L([R^{\pm 1}])=\sum_{t\in\{N,S,W,E\}}sign(\triangle_{t})L([z_{\triangle_{t}};p\triangle_{t},q_{\triangle_{t}}])$

.

(33)

Here integers$p\triangle_{t}$ and $q\triangle_{t}$

are

given from (14)by useof Table 1. Forinstance, $p\triangle_{E}$ and $q\triangle_{E}$ intheoperator

$R$aregiven as

$p_{\triangle_{E}} \pi i=-\log(\frac{\tilde{x}_{5}x_{6}}{x_{3}x_{5}})+\log(\tilde{x}_{5})+l(X6)(X3,$

$q_{\triangle_{E}} \pi i=-\log(-\frac{xx}{x_{c}x_{7}})+\log(x3)+\log(x_{5})-\log(x_{c})-\log(x_{7})$

.

$R R^{-1}$

$\triangle$ Volume sign(A)

$z_{\triangle}$ $\frac{1}{1-z_{\triangle}}$ sign(A) $z_{\triangle}$ $\frac{1}{1-z_{\triangle}}$

$\Delta_{N} D(-\frac{1}{y_{4}}) - -\frac{x_{2}x_{6}}{x_{3^{X}6}} \frac{x_{3^{X}6}}{X4^{\chi_{C}}} + -\frac{x_{3}x_{5}}{x_{2}x_{6}} \frac{x_{2}x_{6}}{x4^{X}c}$

$\triangle s D(-\tilde{y}_{4}) - -\frac{\tilde{x}\tilde{x}}{x3^{X}5} \frac{xx}{\tilde{x}_{4}x_{c}} + -\frac{\tilde{x}_{2}\tilde{x}_{6}}{x_{2}x_{6}} \frac{xx}{x_{c}\tilde{x}_{4}}$ $\triangle w D(\frac{\tilde{y}_{1}}{y_{1}}) + \frac{x_{2^{\tilde{\chi}}3}}{xsx_{5}} -\frac{x_{3^{X}5}}{x_{1}x_{c}}=_{--} -- \frac{\tilde{x}_{2}\alpha_{3}}{x_{2}x_{6}} -\frac{X2^{X}6}{x_{1}x_{c}}$ $\triangle_{E} D(\frac{\tilde{y}_{7}}{y_{7}}1 + \frac{\tilde{x}_{5}x_{6}}{xx} -\frac{x_{3}x_{5}}{x_{c}x_{7}} - \frac{x_{5}\tilde{x}_{6}}{x_{2}x_{6}} -\frac{xx}{x_{c}x_{7}}$

Table1: Moduli of four tetrahedra assigned to operators$R$and$R^{-1}.$ _$Sgn+$”(resp. “-,,)meansthat vertex

orderingoftetrahedron issame(resp. inverse)withFig. 1.

It should be remarkedthat,toidentify the$R$-operator with

a

hyperbolicoctahedron,

we

needaconsistencycondition aroundacentraledgelabeledby$x_{c}$ inFig. 5. This condition

isautomaticallysatisfiedby

(11)

where$\tilde{y}=R^{\pm 1}(y)(24)$

.

In Fig. 6denoted

are

dihedral angles around centralaxisassigned

to eachcrossing.

3.4 GluingOctahedra

Theorem3.4. Let knot$K$have abraidgrouppresentation$\sigma_{k_{1}}^{\epsilon_{1}}\sigma_{k_{2}}^{\epsilon_{2}}\cdots\sigma_{k_{m}}^{\epsilon_{m}}$, where$\epsilon_{j}=\pm 1$

and

$\mathscr{R}_{n}=\langle\sigma_{1},\sigma_{2},\ldots,\sigma_{n-1}|\sigma_{i}\sigma_{i+1}\sigma_{i^{\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}fo}}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}7_{ori=1,2,\ldots,n-2}^{|i-j|>1}\rangle.$

We

_define

a clusterpattern

_for

$x[.i]=(x[i]_{1}, \ldots,x[j]_{3n+1})$by

$x[1]arrow x[2]R^{\epsilon_{1}}k_{1}arrow\cdots x[m+1]R^{E}2\underline{R^{\epsilon_{m}}}k_{2k_{m}}$

, (34)

with the exchange matrix associated to Fig. 4. Weassume thatthe initial cluster variable

$x[1]$

satisfies

$x[1]=x[m+1]$

_.

(35)

Then there exist

an

algebraic solution of(35)such that the complex volume $ofK$isgiven

$by$

$i(Vol(S^{3}\backslash K)+iCS(S^{3}\backslash K))=\sum_{j=1}^{m}L([R^{\epsilon_{j}}])k_{j}$

.

(36)

We study a

case

of trefoil $3_{1}$

.

The braid group presentation is $\sigma_{1}^{3}$, and its cluster

pattern is

$x[1]arrow R1x[2]arrow R1x[3]arrow R1x[4].$

We solve$x[1]=x[4]$by choosinganinitial cluster variableas

$x[1]=(1,x, -x, 1, -x^{2},1,1)$,

and get $x= \frac{1\pm i\sqrt{15}}{4}$

.

We check numerically that (36)gives -$8.22467 \cdots\simeq-\frac{5}{6}\pi^{2}$

.

It

agrees

with the Chern-Simons invariant of$3_{1}$, which is also given from asymptotic limit of the

Kashaev invariant [14,25, 11].

4 2-Bridge

Knots

Computation from the cluster pattem basedonthe$R$-operator(20)is muchinvolved, and

itcanbe much simplifiedwhenweknow asimple triangulation ofhyperbolic manifolds,

suchas once-puncturedtorus bundle and 2-bridgeknotcomplements. In thissection,we

employacanonical triangulationof2-bridgeknot complementsstudied in[21].

Let$K_{q/p}$ be a hyperbolic 2-bridgeknot or link (see e.g. [16]). Herewe assume that$p$

(12)

continuedfractionexpressionof$q/p,$

$q/p= \frac{1}{\alpha 1+\frac{1}{a2+\frac{1}{1}}}$

, (37)

$+-a_{n}$

where$n\geq 1,$ $aJ\in \mathbb{Z}_{>0}$, and $a_{n}\geq 2$

.

We set

$c= \sum_{i=1}^{n}a_{i}$

.

(38)

We then setasequence ofsymbols to denoteflipsas

$F_{1}F_{2}\cdots F_{c-3}=\{\begin{array}{l}R^{a_{1}-1}L^{a}R^{a}\cdots R^{a_{n-1}}L^{a_{n}-2}, when n is even,R^{a_{1}-1}L^{a_{2}}R^{a_{3}}\cdots L^{a_{n-1}}R^{a_{n}-2}, when n is odd.\end{array}$ (39)

where$F_{k}$ denotesasymbol,$F_{h}=R$ or$L.$

Figure7:$A$quiverassociated to triangulation of four-puncturedsphere$\Sigma_{0,4}.$

A setup forclusteralgebrais asfollows. Weuseanexchange matrix

$B= (_{-}010_{1}1 -1-10011 -1-10011 -1-10011 -1-10011 -1-00111)$ , (40)

whose quiver is in Fig. 7. We introduce the flips$R$ and$L$ in (39) as cluster mutations

definedby

(13)

Permutations $St,j$

are

used

so

that the exchange matrix$B(40)$ is invariant under these

flips. The actionson$y$-variablesareexplicitlygivenby

$R(y)=[1^{-1}2^{-1}y_{2}^{-1}T$ $L(y)=(^{y_{1}(1+y)^{-1}(1+y)^{-1}}\mathcal{Y}2(1yy_{5}+6((y_{3^{-1}}11+y_{3})(1+y_{4}^{4})$

Theorem4.1 ([9]). We set$y[k]$recursively by

$y[k]arrow y[h+1]F_{k}$ ₍₄₃₎

where$F_{k}$ is$R$or$L$ in(39), andan initial_$y$-variable is given by

$y[1]=(y,y,- \frac{1}{y},-\frac{1}{y},-1,-1)$

.

(44)

Here$y$ isageometric solution

of

$\{\begin{array}{l}y[c-2]_{3}=y[c-2]_{4}=-1, ifn is even,y[c-2]_{1}=y[c-2]_{2}=-1, ifn is odd,\end{array}$ (45)

such that each modulus$z\iota[h]$

for

$i=1,2$and$k=1,2,\ldots,c-3$

defined

by

$z_{i}[k]=\{$ $-\perp$ $ifF_{k}=R,$ $y[k]_{j}$’ $- \frac{1}{y[h]_{2+i}},$ $ifF_{h}=L,$ (46)

is in the upper halfplane $\mathbb{H}.$

Then $z_{i}[k]$denotesa modulus

of

tetrahedron $\triangle_{i}(F_{k})_{J}$and the hyperbolic volume

of

the knotcomplement$S^{3}\backslash K_{q/p}$ is given by

$v_{o1(S^{3}\backslash K_{q/p})=\sum_{k=1}^{c-3}\sum_{i=1,2}D(z\iota[h])}$

.

(47)

We

can

compute thecomplexvolumeof 2-bridgeknotcomplement by

use

ofthecluster

variables. In this case, we need aspecific semi-field to fulfill a“folding condition” at the

end, and it is tediousto fixan orientationof tetrahedra. See [9].

Acknowledgments

(14)

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FacultyofMathematics

KyushuUniversity

Fukuoka606-8502

JAPAN

$E$-mail address: [email protected]

$7\iota^{J})\backslash b|\star\not\cong\cdot\ovalbox{\tt\small REJECT}^{\backslash }\downarrow$

fflffi

Departmentof Mathematics andInformatics, FacultyofScience,

Chiba University

Chiba263-8522

JAPAN

$E$-mailaddress: reiiy@math.$s$.chiba-u.ac.jp