10
63
(K10a
51
)
A knot diagram
1
Linearized knot diagam
5 10 9 6 2 8 1 4 3 7
Solving Sequence
4,8
9 3 10
1,2
7 6 5
c
8
c
3
c
9
c
2
c
7
c
6
c
4
c
1
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
12
+ 2u
11
+ 9u
10
+ 14u
9
+ 29u
8
+ 34u
7
+ 40u
6
+ 32u
5
+ 20u
4
+ 7u
3
− u
2
+ b − 2u − 1,
− u
12
− 3u
11
− 10u
10
− 21u
9
− 35u
8
− 51u
7
− 52u
6
− 48u
5
− 29u
4
− 11u
3
− 2u
2
+ 2a + 2u,
u
13
+ 3u
12
+ 12u
11
+ 25u
10
+ 51u
9
+ 75u
8
+ 96u
7
+ 96u
6
+ 77u
5
+ 45u
4
+ 16u
3
− 4u − 2i
I
u
2
= h−2u
8
a + 2u
8
+ ··· − 4a + 3,
− u
7
+ u
5
a + u
6
− 2u
4
a − 5u
5
+ 4u
3
a + 5u
4
− 6u
2
a − 8u
3
+ a
2
+ 3au + 7u
2
− 2a − 4u + 2,
u
9
− u
8
+ 6u
7
− 5u
6
+ 11u
5
− 7u
4
+ 6u
3
− 2u
2
+ u − 1i
I
u
3
= hb + 1, 2a − u, u
2
+ 2i
I
v
1
= ha, b − 1, v + 1i
* 4 irreducible components of dim
C
= 0, with total 34 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
= hu
12
+2u
11
+· · ·+b−1, −u
12
−3u
11
+· · ·+2a+2u, u
13
+3u
12
+· · ·−4u−2i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
3
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
1
2
u
12
+
3
2
u
11
+ ··· + u
2
− u
−u
12
− 2u
11
+ ··· + 2u + 1
a
2
=
u
3
+ 2u
u
5
+ 3u
3
+ u
a
7
=
1
2
u
12
+
3
2
u
11
+ ··· − 2u − 1
u
8
+ u
7
+ 5u
6
+ 4u
5
+ 7u
4
+ 4u
3
+ 2u
2
− 1
a
6
=
1
2
u
12
+
3
2
u
11
+ ··· − 2u − 2
u
8
+ u
7
+ 5u
6
+ 4u
5
+ 7u
4
+ 4u
3
+ 2u
2
− 1
a
5
=
1
2
u
12
+
3
2
u
11
+ ··· − 2u − 2
u
9
+ u
8
+ 6u
7
+ 5u
6
+ 11u
5
+ 7u
4
+ 6u
3
+ 2u
2
+ u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
12
+6u
11
+24u
10
+52u
9
+100u
8
+154u
7
+174u
6
+174u
5
+108u
4
+46u
3
−4u
2
−14u−16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
c
10
u
13
+ u
12
+ ··· + u + 1
c
2
, c
3
, c
8
c
9
u
13
− 3u
12
+ ··· − 4u + 2
c
4
, c
6
u
13
+ 5u
12
+ ··· + 9u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
10
y
13
− 5y
12
+ ··· + 9y −1
c
2
, c
3
, c
8
c
9
y
13
+ 15y
12
+ ··· + 16y −4
c
4
, c
6
y
13
+ 11y
12
+ ··· + 25y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.138146 + 0.948701I
a = 0.317222 + 0.611463I
b = −0.644264 − 0.592137I
2.65637 − 1.35876I −4.47319 + 3.17078I
u = −0.138146 − 0.948701I
a = 0.317222 − 0.611463I
b = −0.644264 + 0.592137I
2.65637 + 1.35876I −4.47319 − 3.17078I
u = −0.578420 + 0.729059I
a = −0.85431 − 1.51986I
b = −1.089570 + 0.623417I
−0.00714 + 8.67404I −9.53036 − 8.43648I
u = −0.578420 − 0.729059I
a = −0.85431 + 1.51986I
b = −1.089570 − 0.623417I
−0.00714 − 8.67404I −9.53036 + 8.43648I
u = −0.694065 + 0.222366I
a = 0.835992 + 0.144863I
b = 0.982157 + 0.559210I
−1.52198 − 4.38846I −11.77625 + 4.32757I
u = −0.694065 − 0.222366I
a = 0.835992 − 0.144863I
b = 0.982157 − 0.559210I
−1.52198 + 4.38846I −11.77625 − 4.32757I
u = −0.063059 + 1.278080I
a = −0.069487 + 0.291937I
b = −0.750183 − 0.366139I
2.83101 − 1.40076I −6.04773 + 4.90140I
u = −0.063059 − 1.278080I
a = −0.069487 − 0.291937I
b = −0.750183 + 0.366139I
2.83101 + 1.40076I −6.04773 − 4.90140I
u = 0.400549
a = 0.898581
b = 0.421510
−0.714503 −13.6630
u = −0.17430 + 1.61896I
a = −0.03628 + 1.72509I
b = 1.168160 − 0.683587I
7.93590 + 11.51170I −7.17210 − 6.84034I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.17430 − 1.61896I
a = −0.03628 − 1.72509I
b = 1.168160 + 0.683587I
7.93590 − 11.51170I −7.17210 + 6.84034I
u = −0.05229 + 1.64838I
a = −0.642426 − 1.259340I
b = 0.622947 + 0.904317I
11.49220 − 0.51506I −3.16885 + 2.03529I
u = −0.05229 − 1.64838I
a = −0.642426 + 1.259340I
b = 0.622947 − 0.904317I
11.49220 + 0.51506I −3.16885 − 2.03529I
6
II.
I
u
2
= h−2u
8
a + 2u
8
+· · · −4a +3, −u
7
+u
6
+· · · −2a +2, u
9
−u
8
+· · · +u −1i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
3
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
a
2u
8
a − 2u
8
+ ··· + 4a − 3
a
2
=
u
3
+ 2u
u
5
+ 3u
3
+ u
a
7
=
−2u
8
a + 2u
8
+ ··· − 3a + 3
−3u
8
a + 3u
8
+ ··· − 5a + 4
a
6
=
−5u
8
a + 5u
8
+ ··· − 8a + 7
−3u
8
a + 3u
8
+ ··· − 5a + 4
a
5
=
2u
8
a − 2u
8
+ ··· + 3a − 2
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
7
+ 4u
6
− 20u
5
+ 16u
4
− 28u
3
+ 16u
2
− 8u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
c
10
u
18
+ u
17
+ ··· + 4u + 3
c
2
, c
3
, c
8
c
9
(u
9
+ u
8
+ 6u
7
+ 5u
6
+ 11u
5
+ 7u
4
+ 6u
3
+ 2u
2
+ u + 1)
2
c
4
, c
6
u
18
+ 9u
17
+ ··· + 40u + 9
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
10
y
18
− 9y
17
+ ··· − 40y + 9
c
2
, c
3
, c
8
c
9
(y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
− 6y
2
− 3y −1)
2
c
4
, c
6
y
18
− y
17
+ ··· + 524y + 81
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.429032 + 0.787939I
a = 0.559116 − 0.339074I
b = −0.444651 + 0.766223I
1.87293 − 3.41073I −6.11762 + 4.39642I
u = 0.429032 + 0.787939I
a = −0.47019 + 1.53024I
b = −0.935577 − 0.603792I
1.87293 − 3.41073I −6.11762 + 4.39642I
u = 0.429032 − 0.787939I
a = 0.559116 + 0.339074I
b = −0.444651 − 0.766223I
1.87293 + 3.41073I −6.11762 − 4.39642I
u = 0.429032 − 0.787939I
a = −0.47019 − 1.53024I
b = −0.935577 + 0.603792I
1.87293 + 3.41073I −6.11762 − 4.39642I
u = 0.590618
a = 0.834260 + 0.039950I
b = 0.640279 + 0.479450I
−0.453072 −10.3330
u = 0.590618
a = 0.834260 − 0.039950I
b = 0.640279 − 0.479450I
−0.453072 −10.3330
u = −0.290170 + 0.487341I
a = 1.066630 + 0.144171I
b = 1.174710 + 0.153689I
−3.25448 + 1.10969I −11.44626 − 6.23947I
u = −0.290170 + 0.487341I
a = 0.06769 − 3.10644I
b = −0.943806 + 0.303030I
−3.25448 + 1.10969I −11.44626 − 6.23947I
u = −0.290170 − 0.487341I
a = 1.066630 − 0.144171I
b = 1.174710 − 0.153689I
−3.25448 − 1.10969I −11.44626 + 6.23947I
u = −0.290170 − 0.487341I
a = 0.06769 + 3.10644I
b = −0.943806 − 0.303030I
−3.25448 − 1.10969I −11.44626 + 6.23947I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.05587 + 1.55975I
a = 0.1256620 + 0.0280657I
b = −1.339950 − 0.113954I
3.77376 + 2.21388I −7.75885 − 3.04598I
u = −0.05587 + 1.55975I
a = −0.77131 + 1.94759I
b = 0.857711 − 0.553032I
3.77376 + 2.21388I −7.75885 − 3.04598I
u = −0.05587 − 1.55975I
a = 0.1256620 − 0.0280657I
b = −1.339950 + 0.113954I
3.77376 − 2.21388I −7.75885 + 3.04598I
u = −0.05587 − 1.55975I
a = −0.77131 − 1.94759I
b = 0.857711 + 0.553032I
3.77376 − 2.21388I −7.75885 + 3.04598I
u = 0.12170 + 1.63384I
a = −0.664164 + 1.104630I
b = 0.437217 − 0.966793I
10.17130 − 5.50049I −4.51063 + 2.97298I
u = 0.12170 + 1.63384I
a = −0.24771 − 1.68585I
b = 1.054070 + 0.732497I
10.17130 − 5.50049I −4.51063 + 2.97298I
u = 0.12170 − 1.63384I
a = −0.664164 − 1.104630I
b = 0.437217 + 0.966793I
10.17130 + 5.50049I −4.51063 − 2.97298I
u = 0.12170 − 1.63384I
a = −0.24771 + 1.68585I
b = 1.054070 − 0.732497I
10.17130 + 5.50049I −4.51063 − 2.97298I
11
III. I
u
3
= hb + 1, 2a − u, u
2
+ 2i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
−2
a
3
=
u
−u
a
10
=
−1
0
a
1
=
1
2
u
−1
a
2
=
0
−u
a
7
=
1
2
u + 1
−1
a
6
=
1
2
u
−1
a
5
=
1
2
u
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u + 1)
2
c
2
, c
3
, c
8
c
9
u
2
+ 2
c
4
, c
5
, c
6
c
10
(u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
7
, c
10
(y − 1)
2
c
2
, c
3
, c
8
c
9
(y + 2)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = 0.707107I
b = −1.00000
1.64493 −12.0000
u = − 1.414210I
a = − 0.707107I
b = −1.00000
1.64493 −12.0000
15
IV. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
4
=
−1
0
a
8
=
1
0
a
9
=
1
0
a
3
=
−1
0
a
10
=
1
0
a
1
=
0
1
a
2
=
−1
0
a
7
=
1
−1
a
6
=
0
−1
a
5
=
−1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
7
u − 1
c
2
, c
3
, c
8
c
9
u
c
5
, c
10
u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
7
, c
10
y − 1
c
2
, c
3
, c
8
c
9
y
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
(u − 1)(u + 1)
2
(u
13
+ u
12
+ ··· + u + 1)(u
18
+ u
17
+ ··· + 4u + 3)
c
2
, c
3
, c
8
c
9
u(u
2
+ 2)(u
9
+ u
8
+ 6u
7
+ 5u
6
+ 11u
5
+ 7u
4
+ 6u
3
+ 2u
2
+ u + 1)
2
· (u
13
− 3u
12
+ ··· − 4u + 2)
c
4
, c
6
((u − 1)
3
)(u
13
+ 5u
12
+ ··· + 9u + 1)(u
18
+ 9u
17
+ ··· + 40u + 9)
c
5
, c
10
((u − 1)
2
)(u + 1)(u
13
+ u
12
+ ··· + u + 1)(u
18
+ u
17
+ ··· + 4u + 3)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
10
((y − 1)
3
)(y
13
− 5y
12
+ ··· + 9y − 1)(y
18
− 9y
17
+ ··· − 40y + 9)
c
2
, c
3
, c
8
c
9
y(y + 2)
2
· (y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
− 6y
2
− 3y − 1)
2
· (y
13
+ 15y
12
+ ··· + 16y − 4)
c
4
, c
6
((y − 1)
3
)(y
13
+ 11y
12
+ ··· + 25y − 1)(y
18
− y
17
+ ··· + 524y + 81)
21
