12a
1286
(K12a
1286
)
A knot diagram
1
Linearized knot diagam
5 10 8 9 1 12 11 4 2 3 7 6
Solving Sequence
3,8
4 9
5,11
7 12 6 10 2 1
c
3
c
8
c
4
c
7
c
11
c
6
c
10
c
2
c
1
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, −u
12
+ u
11
+ 8u
10
− 7u
9
− 23u
8
+ 15u
7
+ 26u
6
− 6u
5
− 8u
4
− 4u
3
+ 2u
2
+ 4a − 11u,
u
13
− u
12
− 9u
11
+ 8u
10
+ 31u
9
− 22u
8
− 49u
7
+ 21u
6
+ 34u
5
− 2u
4
− 10u
3
+ 3u
2
+ 2u + 1i
I
u
2
= h5u
13
+ u
12
− 28u
11
− 14u
10
+ 50u
9
+ 49u
8
− 41u
7
− 68u
6
+ 68u
5
+ 70u
4
− 87u
3
− 79u
2
+ 6b + 18u + 30,
17u
13
+ 7u
12
+ ··· + 24a + 105,
u
14
− u
13
− 6u
12
+ 4u
11
+ 14u
10
− 2u
9
− 21u
8
− 5u
7
+ 31u
6
− u
5
− 35u
4
+ 3u
3
+ 23u
2
+ 5u − 8i
I
u
3
= hb − 1, a
2
+ 3, u + 1i
I
u
4
= hb + 1, a
2
+ 1, u − 1i
I
u
5
= hb − 1, a, u + 1i
* 5 irreducible components of dim
C
= 0, with total 32 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, −u
12
+ u
11
+ · · · + 4a − 11u, u
13
− u
12
+ · · · + 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
9
=
u
−u
3
+ u
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
11
=
1
4
u
12
−
1
4
u
11
+ ··· −
1
2
u
2
+
11
4
u
u
a
7
=
1
2
u
12
−
3
4
u
11
+ ··· −
1
2
u +
1
4
1
4
u
12
−
1
4
u
11
+ ··· −
1
2
u
2
+
3
4
u
a
12
=
1
2
u
12
−
1
2
u
11
+ ··· + 2u +
1
2
1
4
u
12
−
1
2
u
11
+ ··· +
3
4
u +
1
4
a
6
=
−
1
4
u
12
+
3
4
u
11
+ ··· − 2u
2
−
7
4
u
1
4
u
12
−
9
4
u
10
+ ··· +
1
4
u +
1
4
a
10
=
1
4
u
12
−
1
4
u
11
+ ··· −
1
2
u
2
+
7
4
u
u
a
2
=
1
4
u
11
−
1
4
u
10
+ ··· −
1
2
u +
3
4
u
2
a
1
=
1
4
u
11
−
1
4
u
10
+ ··· −
1
2
u +
3
4
−
1
4
u
11
+
1
4
u
10
+ ··· +
1
2
u +
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
5
2
u
12
−
9
2
u
11
−21u
10
+
75
2
u
9
+
131
2
u
8
−
225
2
u
7
−91u
6
+136u
5
+59u
4
−55u
3
−26u
2
+
45
2
u+7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
u
13
+ 3u
12
+ ··· + 12u + 2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
u
13
+ u
12
+ ··· + 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
y
13
+ 19y
12
+ ··· − 36y −4
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y
13
− 19y
12
+ ··· − 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.640373 + 0.415565I
a = −1.25824 + 1.93661I
b = −0.640373 + 0.415565I
15.1852 − 1.4688I 6.47409 + 4.73042I
u = −0.640373 − 0.415565I
a = −1.25824 − 1.93661I
b = −0.640373 − 0.415565I
15.1852 + 1.4688I 6.47409 − 4.73042I
u = 0.481196 + 0.382474I
a = 1.09706 + 1.46655I
b = 0.481196 + 0.382474I
4.66648 + 1.38672I 6.13236 − 5.06598I
u = 0.481196 − 0.382474I
a = 1.09706 − 1.46655I
b = 0.481196 − 0.382474I
4.66648 − 1.38672I 6.13236 + 5.06598I
u = −1.60418
a = 0.434953
b = −1.60418
10.2165 6.80950
u = 1.61970 + 0.12491I
a = −0.311845 + 0.455641I
b = 1.61970 + 0.12491I
12.32310 + 4.09027I 9.54540 − 4.06441I
u = 1.61970 − 0.12491I
a = −0.311845 − 0.455641I
b = 1.61970 − 0.12491I
12.32310 − 4.09027I 9.54540 + 4.06441I
u = −0.187914 + 0.306655I
a = −0.476417 + 0.917350I
b = −0.187914 + 0.306655I
0.020555 − 0.726577I 0.71558 + 9.62371I
u = −0.187914 − 0.306655I
a = −0.476417 − 0.917350I
b = −0.187914 − 0.306655I
0.020555 + 0.726577I 0.71558 − 9.62371I
u = −1.65131 + 0.26273I
a = 0.042681 + 0.832421I
b = −1.65131 + 0.26273I
19.0260 − 7.1684I 11.29845 + 3.79891I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.65131 − 0.26273I
a = 0.042681 − 0.832421I
b = −1.65131 − 0.26273I
19.0260 + 7.1684I 11.29845 − 3.79891I
u = 1.68079 + 0.37590I
a = 0.189278 + 1.048750I
b = 1.68079 + 0.37590I
−8.62649 + 8.95936I 11.42936 − 3.31793I
u = 1.68079 − 0.37590I
a = 0.189278 − 1.048750I
b = 1.68079 − 0.37590I
−8.62649 − 8.95936I 11.42936 + 3.31793I
6
II. I
u
2
=
h5u
13
+u
12
+· · ·+6b+30, 17u
13
+7u
12
+· · ·+24a+105, u
14
−u
13
+· · ·+5u−8i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
−u
2
a
9
=
u
−u
3
+ u
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
11
=
−0.708333u
13
− 0.291667u
12
+ ··· − 0.125000u − 4.37500
−
5
6
u
13
−
1
6
u
12
+ ··· − 3u − 5
a
7
=
−0.208333u
13
− 0.125000u
12
+ ··· − 0.458333u − 1.37500
1
2
u
13
+
1
6
u
12
+ ··· +
2
3
u + 3
a
12
=
−
5
12
u
13
−
1
4
u
12
+ ··· +
1
12
u −
11
4
−
5
3
u
13
−
1
3
u
12
+ ··· − 3u − 10
a
6
=
−0.458333u
13
− 0.208333u
12
+ ··· − 0.541667u − 2.12500
1
3
u
13
+
1
6
u
12
+ ··· +
1
2
u +
7
3
a
10
=
1
8
u
13
−
1
8
u
12
+ ··· +
23
8
u +
5
8
−
5
6
u
13
−
1
6
u
12
+ ··· − 3u − 5
a
2
=
−0.625000u
13
− 0.208333u
12
+ ··· − 1.20833u − 5.12500
u
13
+
1
3
u
12
+ ··· +
5
6
u +
23
3
a
1
=
1
24
u
13
+
1
8
u
12
+ ··· −
17
24
u −
1
8
3
2
u
13
+
1
3
u
12
+ ··· +
11
6
u +
31
3
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
26
3
u
13
−
8
3
u
12
+ 50u
11
+ 28u
10
−90u
9
−92u
8
+
208
3
u
7
+
392
3
u
6
−
344
3
u
5
− 136u
4
+
446
3
u
3
+ 146u
2
−
46
3
u −
158
3
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
(u
7
− u
6
+ 6u
5
− 5u
4
+ 10u
3
− 6u
2
+ 4u − 1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
u
14
+ u
13
+ ··· − 5u − 8
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
(y
7
+ 11y
6
+ 46y
5
+ 91y
4
+ 86y
3
+ 34y
2
+ 4y −1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y
14
− 13y
13
+ ··· − 393y + 64
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.648339 + 0.868507I
a = −0.946848 − 0.814097I
b = −1.49879 − 0.07472I
11.28970 + 2.92126I 9.79653 − 2.94858I
u = 0.648339 − 0.868507I
a = −0.946848 + 0.814097I
b = −1.49879 + 0.07472I
11.28970 − 2.92126I 9.79653 + 2.94858I
u = −1.15470
a = −0.288944
b = 0.577082
2.54463 −1.98880
u = −0.613438 + 0.507408I
a = 0.322269 − 0.816953I
b = 1.290190 − 0.016333I
4.55769 − 1.83261I 8.22558 + 5.43914I
u = −0.613438 − 0.507408I
a = 0.322269 + 0.816953I
b = 1.290190 + 0.016333I
4.55769 + 1.83261I 8.22558 − 5.43914I
u = −0.674237 + 1.068950I
a = 1.21462 − 0.78780I
b = 1.63675 − 0.11855I
−16.2972 − 3.4867I 9.97231 + 2.18600I
u = −0.674237 − 1.068950I
a = 1.21462 + 0.78780I
b = 1.63675 + 0.11855I
−16.2972 + 3.4867I 9.97231 − 2.18600I
u = 1.290190 + 0.016333I
a = 0.161518 − 0.517219I
b = −0.613438 − 0.507408I
4.55769 + 1.83261I 8.22558 − 5.43914I
u = 1.290190 − 0.016333I
a = 0.161518 + 0.517219I
b = −0.613438 + 0.507408I
4.55769 − 1.83261I 8.22558 + 5.43914I
u = 0.577082
a = 0.578156
b = −1.15470
2.54463 −1.98880
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.49879 + 0.07472I
a = −0.017211 − 0.901687I
b = 0.648339 − 0.868507I
11.28970 − 2.92126I 9.79653 + 2.94858I
u = −1.49879 − 0.07472I
a = −0.017211 + 0.901687I
b = 0.648339 + 0.868507I
11.28970 + 2.92126I 9.79653 − 2.94858I
u = 1.63675 + 0.11855I
a = −0.066458 − 1.112970I
b = −0.674237 − 1.068950I
−16.2972 + 3.4867I 9.97231 − 2.18600I
u = 1.63675 − 0.11855I
a = −0.066458 + 1.112970I
b = −0.674237 + 1.068950I
−16.2972 − 3.4867I 9.97231 + 2.18600I
11
III. I
u
3
= hb − 1, a
2
+ 3, u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
−1
a
4
=
1
−1
a
9
=
−1
0
a
5
=
0
−1
a
11
=
a
1
a
7
=
3
−a − 1
a
12
=
−2a
a − 2
a
6
=
−3
a + 2
a
10
=
a − 1
1
a
2
=
a
1
a
1
=
a
−a + 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
5
, c
6
c
7
, c
11
, c
12
u
2
+ 3
c
2
, c
8
(u − 1)
2
c
3
, c
4
, c
9
c
10
(u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
(y + 3)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.73205I
b = 1.00000
16.4493 12.0000
u = −1.00000
a = − 1.73205I
b = 1.00000
16.4493 12.0000
15
IV. I
u
4
= hb + 1, a
2
+ 1, u − 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
1
a
4
=
1
−1
a
9
=
1
0
a
5
=
0
−1
a
11
=
a
−1
a
7
=
−1
−a + 1
a
12
=
0
a
a
6
=
−1
−a
a
10
=
a + 1
−1
a
2
=
−a
1
a
1
=
−a
a + 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
5
, c
6
c
7
, c
11
, c
12
u
2
+ 1
c
2
, c
8
(u + 1)
2
c
3
, c
4
, c
9
c
10
(u − 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
(y + 1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(y −1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.000000I
b = −1.00000
6.57974 12.0000
u = 1.00000
a = − 1.000000I
b = −1.00000
6.57974 12.0000
19
V. I
u
5
= hb − 1, a, u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
−1
a
4
=
1
−1
a
9
=
−1
0
a
5
=
0
−1
a
11
=
0
1
a
7
=
0
−1
a
12
=
0
1
a
6
=
0
−1
a
10
=
−1
1
a
2
=
0
1
a
1
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
u
c
2
, c
8
u − 1
c
3
, c
4
, c
9
c
10
u + 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
y
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y −1
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.00000
3.28987 12.0000
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
u(u
2
+ 1)(u
2
+ 3)(u
7
− u
6
+ 6u
5
− 5u
4
+ 10u
3
− 6u
2
+ 4u − 1)
2
· (u
13
+ 3u
12
+ ··· + 12u + 2)
c
2
, c
8
((u − 1)
3
)(u + 1)
2
(u
13
+ u
12
+ ··· + 2u − 1)(u
14
+ u
13
+ ··· − 5u − 8)
c
3
, c
4
, c
9
c
10
((u − 1)
2
)(u + 1)
3
(u
13
+ u
12
+ ··· + 2u − 1)(u
14
+ u
13
+ ··· − 5u − 8)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
, c
12
y(y + 1)
2
(y + 3)
2
· (y
7
+ 11y
6
+ 46y
5
+ 91y
4
+ 86y
3
+ 34y
2
+ 4y −1)
2
· (y
13
+ 19y
12
+ ··· − 36y −4)
c
2
, c
3
, c
4
c
8
, c
9
, c
10
((y −1)
5
)(y
13
− 19y
12
+ ··· − 2y −1)(y
14
− 13y
13
+ ··· − 393y + 64)
25
