12n
0089
(K12n
0089
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 8 3 11 5 12 7 9 10
Solving Sequence
8,11 3,7
6 5 9 12 2 1 4 10
c
7
c
6
c
5
c
8
c
11
c
2
c
1
c
4
c
10
c
3
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h1.87206 × 10
64
u
33
− 2.48020 × 10
64
u
32
+ ··· + 2.02349 × 10
64
b + 8.89696 × 10
65
,
− 6.94907 × 10
63
u
33
+ 9.26153 × 10
63
u
32
+ ··· + 2.89069 × 10
63
a − 3.41795 × 10
65
,
u
34
− 2u
33
+ ··· + 160u − 32i
I
u
2
= h−2u
7
+ u
6
+ 3u
5
− 3u
4
− 4u
3
+ 3u
2
+ b + 2u − 4, 6u
7
− 2u
6
− 8u
5
+ 7u
4
+ 11u
3
− 5u
2
+ a − 4u + 9,
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
I
v
1
= ha, −16v
4
− 47v
3
− 36v
2
+ 29b − 104v + 5, v
5
+ 3v
4
+ 3v
3
+ 8v
2
+ v + 1i
* 3 irreducible components of dim
C
= 0, with total 47 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
= h1.87 × 10
64
u
33
− 2.48 × 10
64
u
32
+ · · · + 2.02 × 10
64
b + 8.90 ×
10
65
, −6.95 × 10
63
u
33
+ 9.26 × 10
63
u
32
+ · · · + 2.89 × 10
63
a − 3.42 ×
10
65
, u
34
− 2u
33
+ · · · + 160u − 32i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
3
=
2.40394u
33
− 3.20391u
32
+ ··· − 401.792u + 118.240
−0.925166u
33
+ 1.22571u
32
+ ··· + 151.603u − 43.9685
a
7
=
1
−u
2
a
6
=
0.0490683u
33
− 0.0407884u
32
+ ··· − 6.92779u + 0.312774
−0.0578563u
33
+ 0.0671443u
32
+ ··· + 7.65767u − 1.83101
a
5
=
−0.00878796u
33
+ 0.0263559u
32
+ ··· + 0.729884u − 1.51823
−0.0578563u
33
+ 0.0671443u
32
+ ··· + 7.65767u − 1.83101
a
9
=
0.135575u
33
− 0.179353u
32
+ ··· − 21.7295u + 6.37504
−0.0869253u
33
+ 0.113172u
32
+ ··· + 14.9258u − 4.00887
a
12
=
−0.159247u
33
+ 0.209304u
32
+ ··· + 26.3062u − 7.44642
0.0515752u
33
− 0.0698375u
32
+ ··· − 7.79789u + 2.42273
a
2
=
2.36320u
33
− 3.16331u
32
+ ··· − 395.428u + 117.533
−0.844124u
33
+ 1.12821u
32
+ ··· + 140.065u − 41.0098
a
1
=
0.222500u
33
− 0.292525u
32
+ ··· − 36.6553u + 10.3839
−0.0181808u
33
+ 0.0260797u
32
+ ··· + 2.35027u − 0.870343
a
4
=
2.25921u
33
− 3.02557u
32
+ ··· − 381.646u + 112.963
−0.870163u
33
+ 1.15421u
32
+ ··· + 144.300u − 42.2250
a
10
=
u
−u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −1.73770u
33
+ 2.33240u
32
+ ··· + 305.829u − 87.7743
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
34
+ 50u
33
+ ··· + 7022u + 1
c
2
, c
4
u
34
− 10u
33
+ ··· − 94u + 1
c
3
, c
6
u
34
+ 6u
33
+ ··· + 1408u + 256
c
5
, c
8
u
34
− 3u
33
+ ··· + 2u − 1
c
7
, c
10
u
34
+ 2u
33
+ ··· − 160u − 32
c
9
, c
11
, c
12
u
34
− 7u
33
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
34
− 122y
33
+ ··· − 49242950y + 1
c
2
, c
4
y
34
− 50y
33
+ ··· − 7022y + 1
c
3
, c
6
y
34
+ 54y
33
+ ··· − 5357568y + 65536
c
5
, c
8
y
34
− y
33
+ ··· − 14y + 1
c
7
, c
10
y
34
− 36y
33
+ ··· − 3584y + 1024
c
9
, c
11
, c
12
y
34
− 41y
33
+ ··· − 152y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.956156 + 0.210490I
a = 0.92043 − 2.41941I
b = −0.297004 + 1.016390I
−3.57437 − 2.68652I −15.9734 + 5.7320I
u = −0.956156 − 0.210490I
a = 0.92043 + 2.41941I
b = −0.297004 − 1.016390I
−3.57437 + 2.68652I −15.9734 − 5.7320I
u = −0.825291 + 0.508770I
a = 0.290634 + 0.392014I
b = 0.215796 + 0.185230I
1.50616 + 2.15286I −1.89528 − 3.55598I
u = −0.825291 − 0.508770I
a = 0.290634 − 0.392014I
b = 0.215796 − 0.185230I
1.50616 − 2.15286I −1.89528 + 3.55598I
u = −0.459276 + 0.600077I
a = 0.34212 − 2.13952I
b = 0.325798 + 0.681195I
−4.37210 + 0.56022I −15.7627 − 4.5815I
u = −0.459276 − 0.600077I
a = 0.34212 + 2.13952I
b = 0.325798 − 0.681195I
−4.37210 − 0.56022I −15.7627 + 4.5815I
u = −1.25779
a = 0.262102
b = −0.999548
−7.19178 −11.0680
u = 0.421643 + 0.589535I
a = 0.76749 − 1.22638I
b = −0.076416 − 0.398409I
−1.23502 + 0.89870I −5.08124 + 0.75731I
u = 0.421643 − 0.589535I
a = 0.76749 + 1.22638I
b = −0.076416 + 0.398409I
−1.23502 − 0.89870I −5.08124 − 0.75731I
u = 0.679857 + 0.008937I
a = 1.75490 − 5.31791I
b = −0.87873 + 2.06096I
−2.48043 + 0.15884I −35.3818 − 0.1674I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.679857 − 0.008937I
a = 1.75490 + 5.31791I
b = −0.87873 − 2.06096I
−2.48043 − 0.15884I −35.3818 + 0.1674I
u = 1.204240 + 0.640025I
a = 0.153762 − 0.187566I
b = 0.460927 + 0.211334I
−3.73420 − 5.65524I −8.00000 + 0.I
u = 1.204240 − 0.640025I
a = 0.153762 + 0.187566I
b = 0.460927 − 0.211334I
−3.73420 + 5.65524I −8.00000 + 0.I
u = 0.610196
a = 0.685401
b = −0.364452
−0.859418 −11.8170
u = −0.021309 + 0.580331I
a = 0.0854012 − 0.0908812I
b = −0.412066 + 1.299410I
−7.07612 − 4.33049I −3.74509 + 2.01968I
u = −0.021309 − 0.580331I
a = 0.0854012 + 0.0908812I
b = −0.412066 − 1.299410I
−7.07612 + 4.33049I −3.74509 − 2.01968I
u = 0.033914 + 0.417650I
a = 0.837170 − 0.008519I
b = 0.336239 − 0.914967I
−0.57544 − 1.50411I −4.52476 + 4.55824I
u = 0.033914 − 0.417650I
a = 0.837170 + 0.008519I
b = 0.336239 + 0.914967I
−0.57544 + 1.50411I −4.52476 − 4.55824I
u = 0.333190
a = 5.02872
b = −1.11629
−2.28474 0.324850
u = −1.71423 + 0.26922I
a = 0.105003 − 1.399260I
b = 0.34011 + 1.96867I
−13.7038 + 7.6996I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.71423 − 0.26922I
a = 0.105003 + 1.399260I
b = 0.34011 − 1.96867I
−13.7038 − 7.6996I 0
u = 1.71822 + 0.31095I
a = −0.469943 − 1.223830I
b = −0.06244 + 1.83419I
−13.63590 + 0.50051I 0
u = 1.71822 − 0.31095I
a = −0.469943 + 1.223830I
b = −0.06244 − 1.83419I
−13.63590 − 0.50051I 0
u = −1.74355 + 0.15186I
a = 0.014975 − 1.244930I
b = 1.07725 + 2.72182I
−10.90540 + 1.31562I 0
u = −1.74355 − 0.15186I
a = 0.014975 + 1.244930I
b = 1.07725 − 2.72182I
−10.90540 − 1.31562I 0
u = −0.01973 + 1.82329I
a = 0.0829691 + 0.0828182I
b = −0.11557 − 1.98219I
−16.1286 + 4.0950I 0
u = −0.01973 − 1.82329I
a = 0.0829691 − 0.0828182I
b = −0.11557 + 1.98219I
−16.1286 − 4.0950I 0
u = 1.85359 + 0.31631I
a = −0.231341 − 1.229070I
b = −0.48186 + 1.53845I
−12.71500 − 5.35446I 0
u = 1.85359 − 0.31631I
a = −0.231341 + 1.229070I
b = −0.48186 − 1.53845I
−12.71500 + 5.35446I 0
u = 1.72228 + 0.86852I
a = 0.576188 + 1.082400I
b = 0.76478 − 2.07350I
18.1726 − 13.4286I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.72228 − 0.86852I
a = 0.576188 − 1.082400I
b = 0.76478 + 2.07350I
18.1726 + 13.4286I 0
u = −1.71660 + 0.90954I
a = −0.667196 + 0.794326I
b = −0.51034 − 1.64958I
18.3538 + 5.3451I 0
u = −1.71660 − 0.90954I
a = −0.667196 − 0.794326I
b = −0.51034 + 1.64958I
18.3538 − 5.3451I 0
u = 1.95923
a = −0.601374
b = −0.892648
−15.4063 0
8
II. I
u
2
= h−2u
7
+ u
6
+ · · · + b − 4, 6u
7
− 2u
6
+ · · · + a + 9, u
8
− u
7
− u
6
+
2u
5
+ u
4
− 2u
3
+ 2u − 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
3
=
−6u
7
+ 2u
6
+ 8u
5
− 7u
4
− 11u
3
+ 5u
2
+ 4u − 9
2u
7
− u
6
− 3u
5
+ 3u
4
+ 4u
3
− 3u
2
− 2u + 4
a
7
=
1
−u
2
a
6
=
1
−u
2
a
5
=
−u
2
+ 1
−u
2
a
9
=
u
4
− u
2
+ 1
u
4
a
12
=
u
6
− u
4
+ 2u
2
− 1
−u
7
+ u
6
+ 2u
5
− u
4
− 2u
3
+ 2u
2
+ 2u − 1
a
2
=
−6u
7
+ 2u
6
+ 8u
5
− 7u
4
− 11u
3
+ 6u
2
+ 4u − 10
2u
7
− u
6
− 3u
5
+ 3u
4
+ 4u
3
− 2u
2
− 2u + 4
a
1
=
u
2
− 1
u
2
a
4
=
−6u
7
+ 2u
6
+ 8u
5
− 7u
4
− 11u
3
+ 5u
2
+ 4u − 9
2u
7
− u
6
− 3u
5
+ 3u
4
+ 4u
3
− 3u
2
− 2u + 4
a
10
=
u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −44u
7
+ 15u
6
+ 58u
5
− 53u
4
− 78u
3
+ 42u
2
+ 28u − 85
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
6
u
8
c
4
(u + 1)
8
c
5
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
7
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
8
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
9
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
10
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
11
, c
12
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
6
y
8
c
5
, c
8
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
7
, c
10
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
9
, c
11
, c
12
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570868 + 0.730671I
a = 1.194470 − 0.635084I
b = −0.281371 − 1.128550I
−2.68559 + 1.13123I −12.74421 + 0.55338I
u = 0.570868 − 0.730671I
a = 1.194470 + 0.635084I
b = −0.281371 + 1.128550I
−2.68559 − 1.13123I −12.74421 − 0.55338I
u = −0.855237 + 0.665892I
a = 0.637416 − 0.344390I
b = 0.208670 + 0.825203I
0.51448 + 2.57849I −9.60894 − 4.72239I
u = −0.855237 − 0.665892I
a = 0.637416 + 0.344390I
b = 0.208670 − 0.825203I
0.51448 − 2.57849I −9.60894 + 4.72239I
u = −1.09818
a = −0.687555
b = 0.829189
−8.14766 −20.4520
u = 1.031810 + 0.655470I
a = 0.286111 + 0.344558I
b = 0.284386 − 0.605794I
−4.02461 − 6.44354I −12.4754 + 9.9976I
u = 1.031810 − 0.655470I
a = 0.286111 − 0.344558I
b = 0.284386 + 0.605794I
−4.02461 + 6.44354I −12.4754 − 9.9976I
u = 0.603304
a = −7.54843
b = 2.74744
−2.48997 −72.8910
12
III.
I
v
1
= ha, −16v
4
− 47v
3
− 36v
2
+ 29b − 104v + 5, v
5
+ 3v
4
+ 3v
3
+ 8v
2
+ v + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
v
0
a
3
=
0
0.551724v
4
+ 1.62069v
3
+ ··· + 3.58621v − 0.172414
a
7
=
1
0
a
6
=
1
−0.344828v
4
− 1.13793v
3
+ ··· − 3.24138v − 1.51724
a
5
=
−0.344828v
4
− 1.13793v
3
+ ··· − 3.24138v − 0.517241
−0.344828v
4
− 1.13793v
3
+ ··· − 3.24138v − 1.51724
a
9
=
0.655172v
4
+ 1.86207v
3
+ ··· + 4.75862v + 0.482759
v
4
+ 3v
3
+ 3v
2
+ 8v + 1
a
12
=
−0.655172v
4
− 1.86207v
3
+ ··· − 3.75862v − 0.482759
−v
4
− 3v
3
− 3v
2
− 8v − 1
a
2
=
−0.655172v
4
− 1.86207v
3
+ ··· − 4.75862v − 0.482759
−0.137931v
4
− 0.655172v
3
+ ··· − 1.89655v − 2.20690
a
1
=
−0.655172v
4
− 1.86207v
3
+ ··· − 4.75862v − 0.482759
−v
4
− 3v
3
− 3v
2
− 8v − 1
a
4
=
0.551724v
4
+ 1.62069v
3
+ ··· + 3.58621v − 0.172414
0.0344828v
4
+ 0.413793v
3
+ ··· + 0.724138v + 1.55172
a
10
=
v
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =
65
29
v
4
+
142
29
v
3
+
81
29
v
2
+
437
29
v −
613
29
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− 5u
4
+ 8u
3
− 3u
2
− u − 1
c
2
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
3
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
4
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
5
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
6
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
7
, c
10
u
5
c
8
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
c
9
(u − 1)
5
c
11
, c
12
(u + 1)
5
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y − 1
c
2
, c
4
y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1
c
3
, c
6
y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1
c
5
, c
8
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1
c
7
, c
10
y
5
c
9
, c
11
, c
12
(y − 1)
5
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.01014 + 1.59703I
a = 0
b = 0.339110 − 0.822375I
−1.97403 − 1.53058I −13.4575 + 4.4032I
v = −0.01014 − 1.59703I
a = 0
b = 0.339110 + 0.822375I
−1.97403 + 1.53058I −13.4575 − 4.4032I
v = −0.043806 + 0.365575I
a = 0
b = −0.455697 + 1.200150I
−7.51750 − 4.40083I −22.0438 + 5.2094I
v = −0.043806 − 0.365575I
a = 0
b = −0.455697 − 1.200150I
−7.51750 + 4.40083I −22.0438 − 5.2094I
v = −2.89210
a = 0
b = −0.766826
−4.04602 −2.99730
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
5
− 5u
4
+ ··· − u − 1)(u
34
+ 50u
33
+ ··· + 7022u + 1)
c
2
((u − 1)
8
)(u
5
+ u
4
+ ··· + u − 1)(u
34
− 10u
33
+ ··· − 94u + 1)
c
3
u
8
(u
5
− u
4
+ ··· + u − 1)(u
34
+ 6u
33
+ ··· + 1408u + 256)
c
4
((u + 1)
8
)(u
5
− u
4
+ ··· + u + 1)(u
34
− 10u
33
+ ··· − 94u + 1)
c
5
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
· (u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
34
− 3u
33
+ ··· + 2u − 1)
c
6
u
8
(u
5
+ u
4
+ ··· + u + 1)(u
34
+ 6u
33
+ ··· + 1408u + 256)
c
7
u
5
(u
8
− u
7
+ ··· + 2u − 1)(u
34
+ 2u
33
+ ··· − 160u − 32)
c
8
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
· (u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
34
− 3u
33
+ ··· + 2u − 1)
c
9
((u − 1)
5
)(u
8
+ u
7
+ ··· + 2u − 1)(u
34
− 7u
33
+ ··· + 2u + 1)
c
10
u
5
(u
8
+ u
7
+ ··· − 2u − 1)(u
34
+ 2u
33
+ ··· − 160u − 32)
c
11
, c
12
((u + 1)
5
)(u
8
− u
7
+ ··· − 2u − 1)(u
34
− 7u
33
+ ··· + 2u + 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y − 1)
8
(y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y − 1)
· (y
34
− 122y
33
+ ··· − 49242950y + 1)
c
2
, c
4
((y − 1)
8
)(y
5
− 5y
4
+ ··· − y − 1)(y
34
− 50y
33
+ ··· − 7022y + 1)
c
3
, c
6
y
8
(y
5
+ 3y
4
+ ··· − y − 1)(y
34
+ 54y
33
+ ··· − 5357568y + 65536)
c
5
, c
8
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
34
− y
33
+ ··· − 14y + 1)
c
7
, c
10
y
5
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
34
− 36y
33
+ ··· − 3584y + 1024)
c
9
, c
11
, c
12
(y − 1)
5
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
34
− 41y
33
+ ··· − 152y + 1)
18
