9
35
(K9a
40
)
A knot diagram
1
Linearized knot diagam
6 8 7 9 2 1 3 5 4
Solving Sequence
2,8 3,5
6 9 1 4 7
c
2
c
5
c
8
c
1
c
4
c
7
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, a − 1, u
3
− u
2
+ 3u − 1i
I
u
2
= hb − u, −u
3
+ a − 2u + 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
I
u
3
= hu
3
− u
2
+ b + 2u − 1, u
3
+ 2a + u + 1, u
4
− 2u
3
+ 3u
2
− 3u + 2i
I
u
4
= hu
3
+ u
2
+ b + 3u + 1, a − 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
I
u
5
= hb + u, a + 1, u
2
+ 1i
* 5 irreducible components of dim
C
= 0, with total 17 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= hb − u, a − 1, u
3
− u
2
+ 3u − 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
5
=
1
u
a
6
=
−u + 1
u
a
9
=
−u
−u
2
+ u
a
1
=
u
2
− u + 1
−u
2
a
4
=
u
2
+ 1
−2u + 1
a
7
=
u
u
2
− 2u + 1
a
7
=
u
u
2
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6u
2
+ 6u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
u
3
+ u
2
+ 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
y
3
+ 5y
2
+ 7y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.361103
a = 1.00000
b = 0.361103
−0.595615 −16.6160
u = 0.31945 + 1.63317I
a = 1.00000
b = 0.31945 + 1.63317I
17.5696 − 7.9406I −0.69212 + 3.53846I
u = 0.31945 − 1.63317I
a = 1.00000
b = 0.31945 − 1.63317I
17.5696 + 7.9406I −0.69212 − 3.53846I
5
II. I
u
2
= hb − u, −u
3
+ a − 2u + 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
5
=
u
3
+ 2u − 1
u
a
6
=
u
3
+ u − 1
u
a
9
=
−2u
3
− 2u
2
− 5u − 3
−1
a
1
=
u
3
+ 2u
2
+ 3u + 2
−u
2
a
4
=
u
2
+ 1
−u
3
− u
2
− 2u − 1
a
7
=
u
u
3
+ u
a
7
=
u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
− 4u
2
− 12u − 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
u
4
− u
3
+ 3u
2
− 2u + 1
c
4
, c
8
, c
9
u
4
+ 2u
3
+ 3u
2
+ 3u + 2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
4
, c
8
, c
9
y
4
+ 2y
3
+ y
2
+ 3y + 4
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = −1.54742 + 1.12087I
b = −0.395123 + 0.506844I
3.07886 + 1.41510I −5.82674 − 4.90874I
u = −0.395123 − 0.506844I
a = −1.54742 − 1.12087I
b = −0.395123 − 0.506844I
3.07886 − 1.41510I −5.82674 + 4.90874I
u = −0.10488 + 1.55249I
a = −0.452576 − 0.585652I
b = −0.10488 + 1.55249I
10.08060 + 3.16396I −2.17326 − 2.56480I
u = −0.10488 − 1.55249I
a = −0.452576 + 0.585652I
b = −0.10488 − 1.55249I
10.08060 − 3.16396I −2.17326 + 2.56480I
9
III. I
u
3
= hu
3
− u
2
+ b + 2u − 1, u
3
+ 2a + u + 1, u
4
− 2u
3
+ 3u
2
− 3u + 2i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
5
=
−
1
2
u
3
−
1
2
u −
1
2
−u
3
+ u
2
− 2u + 1
a
6
=
1
2
u
3
− u
2
+
3
2
u −
3
2
−u
3
+ u
2
− 2u + 1
a
9
=
−
3
2
u
3
+ 2u
2
−
5
2
u +
5
2
−u
3
+ 2u
2
− u + 3
a
1
=
−
1
2
u
3
−
1
2
u +
1
2
−u
3
+ 2u
2
− 2u + 3
a
4
=
u
2
+ 1
2u
3
− u
2
+ 3u − 2
a
7
=
u
u
3
+ u
a
7
=
u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u − 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
8
, c
9
u
4
− u
3
+ 3u
2
− 2u + 1
c
2
, c
3
, c
7
u
4
+ 2u
3
+ 3u
2
+ 3u + 2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
8
, c
9
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
2
, c
3
, c
7
y
4
+ 2y
3
+ y
2
+ 3y + 4
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.956685 + 0.641200I
a = −0.826150 − 1.069070I
b = −0.10488 − 1.55249I
10.08060 − 3.16396I −2.17326 + 2.56480I
u = 0.956685 − 0.641200I
a = −0.826150 + 1.069070I
b = −0.10488 + 1.55249I
10.08060 + 3.16396I −2.17326 − 2.56480I
u = 0.043315 + 1.227190I
a = −0.423850 + 0.307015I
b = −0.395123 − 0.506844I
3.07886 − 1.41510I −5.82674 + 4.90874I
u = 0.043315 − 1.227190I
a = −0.423850 − 0.307015I
b = −0.395123 + 0.506844I
3.07886 + 1.41510I −5.82674 − 4.90874I
13
IV. I
u
4
= hu
3
+ u
2
+ b + 3u + 1, a − 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
5
=
1
−u
3
− u
2
− 3u − 1
a
6
=
u
3
+ u
2
+ 3u + 2
−u
3
− u
2
− 3u − 1
a
9
=
−u
−1
a
1
=
u
3
+ 2u
u
2
+ u + 2
a
4
=
u
2
+ 1
−u
3
− u
2
− 2u − 1
a
7
=
u
u
3
+ u
a
7
=
u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
− 4u
2
− 12u − 10
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u
4
+ 2u
3
+ 3u
2
+ 3u + 2
c
2
, c
3
, c
4
c
7
, c
8
, c
9
u
4
− u
3
+ 3u
2
− 2u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
y
4
+ 2y
3
+ y
2
+ 3y + 4
c
2
, c
3
, c
4
c
7
, c
8
, c
9
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = 1.00000
b = 0.043315 − 1.227190I
3.07886 + 1.41510I −5.82674 − 4.90874I
u = −0.395123 − 0.506844I
a = 1.00000
b = 0.043315 + 1.227190I
3.07886 − 1.41510I −5.82674 + 4.90874I
u = −0.10488 + 1.55249I
a = 1.00000
b = 0.956685 − 0.641200I
10.08060 + 3.16396I −2.17326 − 2.56480I
u = −0.10488 − 1.55249I
a = 1.00000
b = 0.956685 + 0.641200I
10.08060 − 3.16396I −2.17326 + 2.56480I
17
V. I
u
5
= hb + u, a + 1, u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
−1
a
5
=
−1
−u
a
6
=
u − 1
−u
a
9
=
−u
u + 1
a
1
=
−u
1
a
4
=
0
−1
a
7
=
u
0
a
7
=
u
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
u
2
+ 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
(y + 1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −1.00000
b = − 1.000000I
4.93480 0
u = − 1.000000I
a = −1.00000
b = 1.000000I
4.93480 0
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
(u
2
+ 1)(u
3
+ u
2
+ 3u + 1)(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· (u
4
+ 2u
3
+ 3u
2
+ 3u + 2)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
(y + 1)
2
(y
3
+ 5y
2
+ 7y −1)(y
4
+ 2y
3
+ y
2
+ 3y + 4)
· (y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
23
