12n
0307
(K12n
0307
)
A knot diagram
1
Linearized knot diagam
3 6 7 9 2 5 11 12 5 1 9 8
Solving Sequence
2,5
6 3
7,11
8 1 10 9 4 12
c
5
c
2
c
6
c
7
c
1
c
10
c
9
c
4
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−28u
46
− 99u
45
+ ··· + 4b + 28, −8u
46
− 13u
45
+ ··· + 4a + 9, u
47
+ 4u
46
+ ··· + 3u − 1i
I
u
2
= h−u
2
a + b + a, u
2
a + a
2
− au + u
2
+ a − u + 1, u
3
− u
2
+ 1i
I
u
3
= h−u
2
+ b + 1, a − 1, u
3
− u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 56 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
= h−28u
46
− 99u
45
+ · · · + 4b + 28, −8u
46
− 13u
45
+ · · · + 4a + 9, u
47
+
4u
46
+ · · · + 3u − 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
u
2
a
11
=
2u
46
+
13
4
u
45
+ ··· −
11
2
u −
9
4
7u
46
+
99
4
u
45
+ ··· +
113
4
u − 7
a
8
=
1
4
u
45
+
3
4
u
44
+ ··· +
13
4
u + 2
−
1
4
u
46
− u
45
+ ··· − 2u +
1
4
a
1
=
u
3
u
5
− u
3
+ u
a
10
=
23
4
u
46
+
39
2
u
45
+ ··· +
41
2
u −
35
4
7
2
u
46
+ 12u
45
+ ··· + 15u −
7
2
a
9
=
9
4
u
46
+
15
2
u
45
+ ··· +
11
2
u −
21
4
7
2
u
46
+ 12u
45
+ ··· + 15u −
7
2
a
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
12
=
−
7
4
u
45
−
19
4
u
44
+ ··· −
31
4
u +
1
2
3u
46
+
43
4
u
45
+ ··· +
49
4
u − 3
(ii) Obstruction class = −1
(iii) Cusp Shapes =
67
4
u
46
+
101
2
u
45
+ ··· +
225
4
u −
3
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
47
+ 18u
46
+ ··· + 35u + 1
c
2
, c
5
u
47
+ 4u
46
+ ··· + 3u − 1
c
3
u
47
− 4u
46
+ ··· + 9u − 1
c
4
, c
9
u
47
− u
46
+ ··· − 1024u − 512
c
7
u
47
− 4u
46
+ ··· − 4441u − 1153
c
8
, c
11
, c
12
u
47
+ 4u
46
+ ··· − 5u − 1
c
10
u
47
+ 6u
46
+ ··· − 31u − 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
47
+ 26y
46
+ ··· + 787y −1
c
2
, c
5
y
47
− 18y
46
+ ··· + 35y −1
c
3
y
47
− 58y
46
+ ··· + 35y −1
c
4
, c
9
y
47
+ 49y
46
+ ··· − 1703936y −262144
c
7
y
47
+ 22y
46
+ ··· − 27059341y −1329409
c
8
, c
11
, c
12
y
47
+ 46y
46
+ ··· − 5y −1
c
10
y
47
+ 50y
46
+ ··· + 475y −9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.542579 + 0.845522I
a = 0.52046 − 1.65523I
b = −0.81563 + 1.38652I
−2.89318 − 4.02038I 2.80109 + 3.23953I
u = −0.542579 − 0.845522I
a = 0.52046 + 1.65523I
b = −0.81563 − 1.38652I
−2.89318 + 4.02038I 2.80109 − 3.23953I
u = −0.849325 + 0.570079I
a = 0.646245 + 0.025782I
b = 0.284968 − 1.290140I
2.17863 + 2.27566I 3.19435 − 3.09284I
u = −0.849325 − 0.570079I
a = 0.646245 − 0.025782I
b = 0.284968 + 1.290140I
2.17863 − 2.27566I 3.19435 + 3.09284I
u = −0.406871 + 0.865991I
a = −0.021631 − 1.238880I
b = 0.731358 + 1.204980I
−9.98450 + 3.38049I −1.32078 − 2.56603I
u = −0.406871 − 0.865991I
a = −0.021631 + 1.238880I
b = 0.731358 − 1.204980I
−9.98450 − 3.38049I −1.32078 + 2.56603I
u = −0.750352 + 0.593025I
a = −1.020220 + 0.373354I
b = 0.367043 + 1.185280I
−1.34099 − 1.39615I −0.069604 + 1.411136I
u = −0.750352 − 0.593025I
a = −1.020220 − 0.373354I
b = 0.367043 − 1.185280I
−1.34099 + 1.39615I −0.069604 − 1.411136I
u = −0.559716 + 0.892329I
a = −0.45129 + 1.94560I
b = 1.07277 − 2.03235I
−9.05378 − 7.60668I −0.58639 + 3.22129I
u = −0.559716 − 0.892329I
a = −0.45129 − 1.94560I
b = 1.07277 + 2.03235I
−9.05378 + 7.60668I −0.58639 − 3.22129I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.035120 + 0.209994I
a = 0.882211 − 0.713191I
b = 0.774400 − 0.897748I
−6.95929 − 0.03774I −5.96271 − 0.75839I
u = −1.035120 − 0.209994I
a = 0.882211 + 0.713191I
b = 0.774400 + 0.897748I
−6.95929 + 0.03774I −5.96271 + 0.75839I
u = 0.766231 + 0.547591I
a = −1.65889 − 0.74494I
b = 1.179640 − 0.111554I
1.49256 − 0.66743I 4.54233 − 0.09202I
u = 0.766231 − 0.547591I
a = −1.65889 + 0.74494I
b = 1.179640 + 0.111554I
1.49256 + 0.66743I 4.54233 + 0.09202I
u = −0.472512 + 0.810458I
a = −0.352292 + 1.319250I
b = 0.091933 − 0.956029I
−3.35426 + 0.41641I 1.81917 − 2.68288I
u = −0.472512 − 0.810458I
a = −0.352292 − 1.319250I
b = 0.091933 + 0.956029I
−3.35426 − 0.41641I 1.81917 + 2.68288I
u = 0.916719 + 0.584389I
a = 1.43793 + 1.12449I
b = −1.185440 + 0.406825I
0.99495 − 3.89558I 2.00000 + 7.22930I
u = 0.916719 − 0.584389I
a = 1.43793 − 1.12449I
b = −1.185440 − 0.406825I
0.99495 + 3.89558I 2.00000 − 7.22930I
u = −0.919108 + 0.591025I
a = −0.158662 − 0.317671I
b = −0.820356 + 1.073340I
−1.86367 + 6.09961I −1.78165 − 6.44137I
u = −0.919108 − 0.591025I
a = −0.158662 + 0.317671I
b = −0.820356 − 1.073340I
−1.86367 − 6.09961I −1.78165 + 6.44137I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.807896 + 0.791990I
a = 1.46759 − 0.27866I
b = −1.13467 + 1.37041I
−0.27990 − 1.40024I 0. + 3.69287I
u = 0.807896 − 0.791990I
a = 1.46759 + 0.27866I
b = −1.13467 − 1.37041I
−0.27990 + 1.40024I 0. − 3.69287I
u = 1.005160 + 0.543180I
a = −1.74932 − 1.37109I
b = 1.071900 − 0.860916I
−5.01656 − 6.25958I −2.44147 + 6.00005I
u = 1.005160 − 0.543180I
a = −1.74932 + 1.37109I
b = 1.071900 + 0.860916I
−5.01656 + 6.25958I −2.44147 − 6.00005I
u = 0.880391 + 0.769089I
a = −0.562518 + 0.643186I
b = −0.135747 − 0.962284I
3.63076 − 2.90147I −4.60953 + 3.97403I
u = 0.880391 − 0.769089I
a = −0.562518 − 0.643186I
b = −0.135747 + 0.962284I
3.63076 + 2.90147I −4.60953 − 3.97403I
u = 1.175580 + 0.026203I
a = 0.226154 − 0.429398I
b = 0.18576 − 1.60517I
−9.13930 − 2.39543I −3.30646 + 2.93927I
u = 1.175580 − 0.026203I
a = 0.226154 + 0.429398I
b = 0.18576 + 1.60517I
−9.13930 + 2.39543I −3.30646 − 2.93927I
u = −0.797096 + 0.147471I
a = −0.470761 + 0.404647I
b = −0.305079 + 0.399977I
−1.341120 + 0.348467I −5.61848 − 0.75974I
u = −0.797096 − 0.147471I
a = −0.470761 − 0.404647I
b = −0.305079 − 0.399977I
−1.341120 − 0.348467I −5.61848 + 0.75974I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.710745 + 0.353826I
a = 1.99025 + 0.84733I
b = −0.945738 − 0.231377I
−3.72397 + 2.21391I 0.64719 + 2.18592I
u = 0.710745 − 0.353826I
a = 1.99025 − 0.84733I
b = −0.945738 + 0.231377I
−3.72397 − 2.21391I 0.64719 − 2.18592I
u = 1.210150 + 0.061384I
a = −0.556915 + 0.710918I
b = −0.53977 + 1.79323I
−15.7778 − 5.9089I −6.14260 + 0.I
u = 1.210150 − 0.061384I
a = −0.556915 − 0.710918I
b = −0.53977 − 1.79323I
−15.7778 + 5.9089I −6.14260 + 0.I
u = 0.941818 + 0.771815I
a = 0.18006 − 1.54737I
b = 1.40667 + 1.15010I
−0.67750 − 4.47945I 0
u = 0.941818 − 0.771815I
a = 0.18006 + 1.54737I
b = 1.40667 − 1.15010I
−0.67750 + 4.47945I 0
u = −1.083300 + 0.641319I
a = 1.62476 + 0.25908I
b = −0.486943 − 1.100850I
−5.15574 + 5.00343I 0
u = −1.083300 − 0.641319I
a = 1.62476 − 0.25908I
b = −0.486943 + 1.100850I
−5.15574 − 5.00343I 0
u = −1.082000 + 0.677407I
a = −2.00058 + 0.02197I
b = 0.94583 + 1.66881I
−4.51999 + 9.69823I 0
u = −1.082000 − 0.677407I
a = −2.00058 − 0.02197I
b = 0.94583 − 1.66881I
−4.51999 − 9.69823I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.122540 + 0.615470I
a = −1.60372 − 0.82721I
b = −0.358676 + 1.009600I
−12.16410 + 2.06990I 0
u = −1.122540 − 0.615470I
a = −1.60372 + 0.82721I
b = −0.358676 − 1.009600I
−12.16410 − 2.06990I 0
u = −1.095820 + 0.699570I
a = 2.36060 − 0.03371I
b = −1.00797 − 2.27284I
−10.6948 + 13.4951I 0
u = −1.095820 − 0.699570I
a = 2.36060 + 0.03371I
b = −1.00797 + 2.27284I
−10.6948 − 13.4951I 0
u = 0.207923 + 0.417127I
a = 1.69044 − 0.06549I
b = −0.631998 − 0.720191I
−3.46548 + 2.21318I 2.85688 − 2.29805I
u = 0.207923 − 0.417127I
a = 1.69044 + 0.06549I
b = −0.631998 + 0.720191I
−3.46548 − 2.21318I 2.85688 + 2.29805I
u = 0.187442
a = −2.83979
b = 0.511458
0.826035 12.4720
9
II. I
u
2
= h−u
2
a + b + a, u
2
a + a
2
− au + u
2
+ a − u + 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
11
=
a
u
2
a − a
a
8
=
au − u
2
− a + u
−au + u
2
− u
a
1
=
u
2
− 1
−u
2
a
10
=
u
2
a − au
0
a
9
=
u
2
a − au
0
a
4
=
1
0
a
12
=
−u
2
a + au − u
2
+ 2u − 2
u
2
a − a
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
a − 7au + a + 3u + 5
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
11
c
12
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
, c
7
, c
10
(u
3
− u
2
+ 1)
2
c
6
, c
8
(u
3
+ u
2
+ 2u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
, c
7
c
10
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.162359 − 0.986732I
b = 1.16236 + 0.98673I
− 5.65624I 2.97732 + 6.46189I
u = 0.877439 + 0.744862I
a = −0.500000 + 0.424452I
b = −0.162359 − 0.986732I
4.13758 − 2.82812I 11.75410 + 2.09676I
u = 0.877439 − 0.744862I
a = 0.162359 + 0.986732I
b = 1.16236 − 0.98673I
5.65624I 2.97732 − 6.46189I
u = 0.877439 − 0.744862I
a = −0.500000 − 0.424452I
b = −0.162359 + 0.986732I
4.13758 + 2.82812I 11.75410 − 2.09676I
u = −0.754878
a = −1.16236 + 0.98673I
b = 0.500000 − 0.424452I
−4.13758 − 2.82812I −5.23142 + 6.76304I
u = −0.754878
a = −1.16236 − 0.98673I
b = 0.500000 + 0.424452I
−4.13758 + 2.82812I −5.23142 − 6.76304I
13
III. I
u
3
= h−u
2
+ b + 1, a − 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
11
=
1
u
2
− 1
a
8
=
−u
2
− u + 1
u + 1
a
1
=
u
2
− 1
−u
2
a
10
=
u
2
− u
0
a
9
=
u
2
− u
0
a
4
=
1
0
a
12
=
u + 1
u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
+ u + 1
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
11
c
12
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
, c
9
u
3
c
5
, c
7
, c
10
u
3
− u
2
+ 1
c
6
, c
8
u
3
+ u
2
+ 2u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
, c
7
c
10
y
3
− y
2
+ 2y −1
c
4
, c
9
y
3
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 1.00000
b = −0.78492 + 1.30714I
0 1.66236 − 0.56228I
u = 0.877439 − 0.744862I
a = 1.00000
b = −0.78492 − 1.30714I
0 1.66236 + 0.56228I
u = −0.754878
a = 1.00000
b = −0.430160
0 −0.324720
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
3
)(u
47
+ 18u
46
+ ··· + 35u + 1)
c
2
((u
3
+ u
2
− 1)
3
)(u
47
+ 4u
46
+ ··· + 3u − 1)
c
3
((u
3
− u
2
+ 2u − 1)
3
)(u
47
− 4u
46
+ ··· + 9u − 1)
c
4
, c
9
u
9
(u
47
− u
46
+ ··· − 1024u − 512)
c
5
((u
3
− u
2
+ 1)
3
)(u
47
+ 4u
46
+ ··· + 3u − 1)
c
6
((u
3
+ u
2
+ 2u + 1)
3
)(u
47
+ 18u
46
+ ··· + 35u + 1)
c
7
((u
3
− u
2
+ 1)
3
)(u
47
− 4u
46
+ ··· − 4441u − 1153)
c
8
((u
3
+ u
2
+ 2u + 1)
3
)(u
47
+ 4u
46
+ ··· − 5u − 1)
c
10
((u
3
− u
2
+ 1)
3
)(u
47
+ 6u
46
+ ··· − 31u − 3)
c
11
, c
12
((u
3
− u
2
+ 2u − 1)
3
)(u
47
+ 4u
46
+ ··· − 5u − 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y
3
+ 3y
2
+ 2y −1)
3
)(y
47
+ 26y
46
+ ··· + 787y −1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
3
)(y
47
− 18y
46
+ ··· + 35y −1)
c
3
((y
3
+ 3y
2
+ 2y −1)
3
)(y
47
− 58y
46
+ ··· + 35y −1)
c
4
, c
9
y
9
(y
47
+ 49y
46
+ ··· − 1703936y −262144)
c
7
((y
3
− y
2
+ 2y −1)
3
)(y
47
+ 22y
46
+ ··· − 2.70593 × 10
7
y −1329409)
c
8
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
47
+ 46y
46
+ ··· − 5y −1)
c
10
((y
3
− y
2
+ 2y −1)
3
)(y
47
+ 50y
46
+ ··· + 475y −9)
19
