12n
0674
(K12n
0674
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 9 3 11 12 5 1 8 9
Solving Sequence
2,5 3,9
6 7 10 4 1 11 12 8
c
2
c
5
c
6
c
9
c
4
c
1
c
10
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−8.95545 × 10
24
u
41
− 2.72305 × 10
25
u
40
+ ··· + 2.32080 × 10
24
b + 1.04846 × 10
24
,
− 2.76332 × 10
24
u
41
− 4.41119 × 10
24
u
40
+ ··· + 2.32080 × 10
24
a + 2.12642 × 10
25
,
u
42
+ 4u
41
+ ··· + 14u − 1i
I
u
2
= hb + 1, a, u
2
+ u − 1i
I
u
3
= hb − u − 2, a, u
2
+ u − 1i
I
u
4
= hb, a + 1, u − 1i
* 4 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
= h−8.96 × 10
24
u
41
− 2.72 × 10
25
u
40
+ · · · + 2.32 × 10
24
b + 1.05 ×
10
24
, −2.76 × 10
24
u
41
− 4.41 × 10
24
u
40
+ · · · + 2.32 × 10
24
a + 2.13 ×
10
25
, u
42
+ 4u
41
+ · · · + 14u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
9
=
1.19067u
41
+ 1.90072u
40
+ ··· + 60.6010u − 9.16243
3.85878u
41
+ 11.7332u
40
+ ··· + 22.3744u − 0.451767
a
6
=
−0.105132u
41
− 0.0128494u
40
+ ··· − 34.8388u + 4.84834
−0.848783u
41
− 2.84893u
40
+ ··· − 15.2500u + 0.685419
a
7
=
0.978378u
41
+ 2.65739u
40
+ ··· − 25.4014u + 4.57060
1.25612u
41
+ 2.68484u
40
+ ··· + 9.12669u − 0.978378
a
10
=
1.19067u
41
+ 1.90072u
40
+ ··· + 60.6010u − 9.16243
0.716340u
41
+ 3.10175u
40
+ ··· − 18.8840u + 2.41021
a
4
=
u
u
a
1
=
−u
2
+ 1
−u
2
a
11
=
2.55115u
41
+ 4.97020u
40
+ ··· + 92.6692u − 12.2894
5.70141u
41
+ 16.8545u
40
+ ··· + 38.1516u − 1.40486
a
12
=
−0.546898u
41
− 1.27331u
40
+ ··· − 34.6001u + 6.91046
−1.81884u
41
− 4.93743u
40
+ ··· − 37.5544u + 2.14072
a
8
=
−0.0791429u
41
+ 0.265882u
40
+ ··· − 13.4298u − 0.240862
−0.00645021u
41
− 0.654296u
40
+ ··· + 17.6519u − 1.13841
(ii) Obstruction class = −1
(iii) Cusp Shapes =
8061029839060072009079083
2320801929915461688487812
u
41
+
4209450828705721840877893
580200482478865422121953
u
40
+ ··· +
55742079073151734903802548
580200482478865422121953
u −
31967218726697708827064461
2320801929915461688487812
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
42
− 4u
41
+ ··· − 14u − 1
c
3
, c
6
u
42
+ 3u
41
+ ··· − 15u
2
+ 2
c
5
, c
9
u
42
− 2u
41
+ ··· − 32u − 16
c
7
, c
8
, c
11
c
12
u
42
+ 4u
41
+ ··· − 10u + 1
c
10
u
42
− 8u
41
+ ··· − 11932u − 167
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
42
− 36y
41
+ ··· − 78y + 1
c
3
, c
6
y
42
− 15y
41
+ ··· − 60y + 4
c
5
, c
9
y
42
− 26y
41
+ ··· − 7296y + 256
c
7
, c
8
, c
11
c
12
y
42
− 48y
41
+ ··· − 154y + 1
c
10
y
42
+ 12y
41
+ ··· − 132872662y + 27889
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.238287 + 0.993330I
a = −0.256688 − 1.363750I
b = 0.448503 − 0.389957I
−3.87928 − 8.09823I −11.51544 + 5.48666I
u = 0.238287 − 0.993330I
a = −0.256688 + 1.363750I
b = 0.448503 + 0.389957I
−3.87928 + 8.09823I −11.51544 − 5.48666I
u = 1.060390 + 0.045339I
a = 0.258751 + 0.376269I
b = −0.47464 + 2.91568I
−2.71639 − 0.33816I −4.0081 − 13.7250I
u = 1.060390 − 0.045339I
a = 0.258751 − 0.376269I
b = −0.47464 − 2.91568I
−2.71639 + 0.33816I −4.0081 + 13.7250I
u = 0.106704 + 0.918803I
a = 0.26496 + 1.45259I
b = −0.086830 + 0.297294I
3.36143 − 5.24537I −7.74689 + 6.20199I
u = 0.106704 − 0.918803I
a = 0.26496 − 1.45259I
b = −0.086830 − 0.297294I
3.36143 + 5.24537I −7.74689 − 6.20199I
u = 1.147150 + 0.210471I
a = −0.302824 − 0.583834I
b = −0.04054 − 3.07018I
−10.17350 − 0.96398I −18.1472 − 3.3317I
u = 1.147150 − 0.210471I
a = −0.302824 + 0.583834I
b = −0.04054 + 3.07018I
−10.17350 + 0.96398I −18.1472 + 3.3317I
u = −0.049648 + 0.825086I
a = −0.32016 − 1.57337I
b = −0.281826 − 0.254570I
4.07908 − 1.05002I −5.39403 − 0.20283I
u = −0.049648 − 0.825086I
a = −0.32016 + 1.57337I
b = −0.281826 + 0.254570I
4.07908 + 1.05002I −5.39403 + 0.20283I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.168990 + 0.199735I
a = 1.249760 + 0.642269I
b = 1.006910 + 0.970924I
−3.90037 + 1.20976I −15.7241 − 4.9251I
u = −1.168990 − 0.199735I
a = 1.249760 − 0.642269I
b = 1.006910 − 0.970924I
−3.90037 − 1.20976I −15.7241 + 4.9251I
u = −0.314243 + 0.696899I
a = 0.56188 + 1.69109I
b = 0.733268 + 0.302512I
−1.53621 + 1.78916I −8.56962 − 1.59900I
u = −0.314243 − 0.696899I
a = 0.56188 − 1.69109I
b = 0.733268 − 0.302512I
−1.53621 − 1.78916I −8.56962 + 1.59900I
u = −1.244640 + 0.090513I
a = 0.277107 − 1.196450I
b = 0.18513 − 2.18299I
−4.74961 + 2.33690I −17.2595 − 5.0904I
u = −1.244640 − 0.090513I
a = 0.277107 + 1.196450I
b = 0.18513 + 2.18299I
−4.74961 − 2.33690I −17.2595 + 5.0904I
u = 1.157940 + 0.481959I
a = −0.776705 + 0.303294I
b = −0.979087 + 0.599725I
0.121226 + 0.269727I −8.00000 − 3.11579I
u = 1.157940 − 0.481959I
a = −0.776705 − 0.303294I
b = −0.979087 − 0.599725I
0.121226 − 0.269727I −8.00000 + 3.11579I
u = 1.057900 + 0.692054I
a = 0.907159 − 0.280920I
b = 0.898757 + 0.137083I
−6.34245 + 2.32209I −13.28779 + 0.I
u = 1.057900 − 0.692054I
a = 0.907159 + 0.280920I
b = 0.898757 − 0.137083I
−6.34245 − 2.32209I −13.28779 + 0.I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.722474
a = −0.399724
b = 0.258120
−1.09552 −8.48520
u = −1.245000 + 0.375694I
a = −0.928202 − 0.816670I
b = −1.17456 − 1.47199I
0.37946 + 5.36737I 0
u = −1.245000 − 0.375694I
a = −0.928202 + 0.816670I
b = −1.17456 + 1.47199I
0.37946 − 5.36737I 0
u = −1.31935
a = −1.01441
b = −0.0413566
−14.8036 −18.1030
u = −1.308760 + 0.256721I
a = −0.597154 + 0.929246I
b = −0.45147 + 2.20887I
−11.72140 + 5.51488I 0
u = −1.308760 − 0.256721I
a = −0.597154 − 0.929246I
b = −0.45147 − 2.20887I
−11.72140 − 5.51488I 0
u = 0.126480 + 0.653902I
a = 1.60072 − 0.06394I
b = −0.579441 − 0.327004I
−7.23822 − 2.23215I −12.66894 + 2.87063I
u = 0.126480 − 0.653902I
a = 1.60072 + 0.06394I
b = −0.579441 + 0.327004I
−7.23822 + 2.23215I −12.66894 − 2.87063I
u = 1.329240 + 0.371704I
a = 0.714985 − 0.404132I
b = 1.36849 − 1.26529I
−0.24891 − 3.25176I 0
u = 1.329240 − 0.371704I
a = 0.714985 + 0.404132I
b = 1.36849 + 1.26529I
−0.24891 + 3.25176I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.35150 + 0.42227I
a = 0.777238 + 0.800492I
b = 1.23840 + 1.87181I
−1.20516 + 10.04620I 0
u = −1.35150 − 0.42227I
a = 0.777238 − 0.800492I
b = 1.23840 − 1.87181I
−1.20516 − 10.04620I 0
u = 1.46265 + 0.31490I
a = −0.703570 + 0.478725I
b = −1.75849 + 1.73373I
−7.27437 − 5.59691I 0
u = 1.46265 − 0.31490I
a = −0.703570 − 0.478725I
b = −1.75849 − 1.73373I
−7.27437 + 5.59691I 0
u = −1.43580 + 0.43130I
a = −0.682703 − 0.780467I
b = −1.24611 − 2.24244I
−9.1607 + 13.1986I 0
u = −1.43580 − 0.43130I
a = −0.682703 + 0.780467I
b = −1.24611 + 2.24244I
−9.1607 − 13.1986I 0
u = 0.403841
a = 1.04866
b = −2.15536
−9.66299 4.31750
u = −1.64023
a = 0.220147
b = 0.661174
−9.70143 0
u = 0.152045 + 0.289588I
a = −1.95648 − 0.86091I
b = 0.166595 + 0.401559I
−0.703904 − 0.992275I −8.59202 + 6.89400I
u = 0.152045 − 0.289588I
a = −1.95648 + 0.86091I
b = 0.166595 − 0.401559I
−0.703904 + 0.992275I −8.59202 − 6.89400I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.71883
a = −0.450881
b = −1.47918
−16.8863 0
u = 0.111686
a = −4.57994
b = 0.810470
−1.32929 −5.96870
9
II. I
u
2
= hb + 1, a, u
2
+ u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
−u + 1
a
9
=
0
−1
a
6
=
0
u
a
7
=
−u
−u + 1
a
10
=
0
−1
a
4
=
u
u
a
1
=
u
u − 1
a
11
=
−u + 1
−2u
a
12
=
u
−1
a
8
=
u − 1
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −21
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
10
u
2
+ u − 1
c
4
, c
6
, c
11
c
12
u
2
− u − 1
c
5
, c
9
u
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
9
y
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0
b = −1.00000
−1.97392 −21.0000
u = −1.61803
a = 0
b = −1.00000
−17.7653 −21.0000
13
III. I
u
3
= hb − u − 2, a, u
2
+ u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
−u + 1
a
9
=
0
u + 2
a
6
=
0
u
a
7
=
−u
−u + 1
a
10
=
0
u + 2
a
4
=
u
u
a
1
=
u
u − 1
a
11
=
−1
2u + 2
a
12
=
u
−u − 4
a
8
=
1
−3u − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −36
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
10
u
2
+ u − 1
c
4
, c
6
, c
11
c
12
u
2
− u − 1
c
5
, c
9
u
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
9
y
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0
b = 2.61803
−9.86960 −36.0000
u = −1.61803
a = 0
b = 0.381966
−9.86960 −36.0000
17
IV. I
u
4
= hb, a + 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
9
=
−1
0
a
6
=
1
1
a
7
=
1
1
a
10
=
−1
1
a
4
=
1
1
a
1
=
0
−1
a
11
=
−1
0
a
12
=
−1
−1
a
8
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
8
, c
10
u − 1
c
3
, c
6
u
c
4
, c
9
, c
11
c
12
u + 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y −1
c
3
, c
6
y
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
21
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)(u
2
+ u − 1)
2
(u
42
− 4u
41
+ ··· − 14u − 1)
c
3
u(u
2
+ u − 1)
2
(u
42
+ 3u
41
+ ··· − 15u
2
+ 2)
c
4
(u + 1)(u
2
− u − 1)
2
(u
42
− 4u
41
+ ··· − 14u − 1)
c
5
u
4
(u − 1)(u
42
− 2u
41
+ ··· − 32u − 16)
c
6
u(u
2
− u − 1)
2
(u
42
+ 3u
41
+ ··· − 15u
2
+ 2)
c
7
, c
8
(u − 1)(u
2
+ u − 1)
2
(u
42
+ 4u
41
+ ··· − 10u + 1)
c
9
u
4
(u + 1)(u
42
− 2u
41
+ ··· − 32u − 16)
c
10
(u − 1)(u
2
+ u − 1)
2
(u
42
− 8u
41
+ ··· − 11932u − 167)
c
11
, c
12
(u + 1)(u
2
− u − 1)
2
(u
42
+ 4u
41
+ ··· − 10u + 1)
22
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)(y
2
− 3y + 1)
2
(y
42
− 36y
41
+ ··· − 78y + 1)
c
3
, c
6
y(y
2
− 3y + 1)
2
(y
42
− 15y
41
+ ··· − 60y + 4)
c
5
, c
9
y
4
(y −1)(y
42
− 26y
41
+ ··· − 7296y + 256)
c
7
, c
8
, c
11
c
12
(y −1)(y
2
− 3y + 1)
2
(y
42
− 48y
41
+ ··· − 154y + 1)
c
10
(y −1)(y
2
− 3y + 1)
2
(y
42
+ 12y
41
+ ··· − 1.32873 × 10
8
y + 27889)
23
