12n
0334
(K12n
0334
)
A knot diagram
1
Linearized knot diagam
3 6 8 7 10 2 4 12 11 6 9 8
Solving Sequence
2,7
6
3,10
11 1 5 4 8 9 12
c
6
c
2
c
10
c
1
c
5
c
4
c
7
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.23152 × 10
16
u
29
+ 1.53695 × 10
16
u
28
+ ··· + 5.84790 × 10
16
b − 9.74776 × 10
15
,
− 4.32154 × 10
16
u
29
+ 2.87372 × 10
16
u
28
+ ··· + 1.16958 × 10
17
a − 5.09997 × 10
16
, u
30
− u
29
+ ··· + 7u + 2i
I
u
2
= h9a
3
u + 23a
3
− a
2
u + 11a
2
+ 57au + 61b + 105a − 66u − 6, a
4
− a
3
u + 2a
2
u + 3a
2
− 5au − a + 2u − 3,
u
2
+ 1i
I
u
3
= hu
8
− u
7
+ 2u
6
− 2u
5
+ u
4
− u
3
+ u
2
+ b − u, u
7
+ u
6
+ 2u
5
+ 2u
4
+ u
3
+ u
2
+ a + u + 1,
u
9
+ 3u
7
− u
6
+ 3u
5
− 2u
4
+ 3u
3
− u
2
+ 2u − 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
= h−1.23×10
16
u
29
+1.54×10
16
u
28
+· · ·+5.85×10
16
b−9.75×10
15
, −4.32×
10
16
u
29
+2.87×10
16
u
28
+· · ·+1.17×10
17
a−5.10×10
16
, u
30
−u
29
+· · ·+7u+2i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
3
=
u
u
3
+ u
a
10
=
0.369495u
29
− 0.245705u
28
+ ··· + 14.2746u + 0.436051
0.210593u
29
− 0.262820u
28
+ ··· + 3.53091u + 0.166688
a
11
=
0.672620u
29
− 0.609580u
28
+ ··· + 19.4110u + 0.850319
0.211783u
29
− 0.281668u
28
+ ··· + 3.34990u + 0.288187
a
1
=
u
3
u
5
+ u
3
+ u
a
5
=
−0.688243u
29
+ 0.596482u
28
+ ··· − 22.1445u − 4.75119
−0.0998548u
29
+ 0.00973725u
28
+ ··· − 4.79516u − 0.887435
a
4
=
−0.788098u
29
+ 0.606219u
28
+ ··· − 26.9397u − 5.63863
−0.0998548u
29
+ 0.00973725u
28
+ ··· − 4.79516u − 0.887435
a
8
=
0.625597u
29
− 0.813840u
28
+ ··· + 3.22541u − 3.26533
0.0901176u
29
− 0.131952u
28
+ ··· + 0.188451u − 1.19971
a
9
=
0.772220u
29
− 1.17415u
28
+ ··· − 1.69794u − 6.69657
0.193509u
29
− 0.293937u
28
+ ··· − 0.448992u − 0.953828
a
12
=
−1.07677u
29
+ 0.793142u
28
+ ··· − 38.8926u − 7.27684
−0.188243u
29
+ 0.0964819u
28
+ ··· − 7.64451u − 1.25119
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
6673390003312773
9746501057251736
u
29
−
1203194374045575
2436625264312934
u
28
+ ··· +
172887881292571847
9746501057251736
u +
53741926556471299
4873250528625868
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
30
+ 5u
29
+ ··· + 75u + 4
c
2
, c
6
u
30
− u
29
+ ··· + 7u + 2
c
3
, c
4
, c
7
u
30
− u
29
+ ··· + 13u + 2
c
5
, c
10
u
30
− 2u
29
+ ··· − u + 2
c
8
, c
9
, c
11
c
12
u
30
− 8u
29
+ ··· + 19u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
30
+ 49y
29
+ ··· − 2273y + 16
c
2
, c
6
y
30
+ 5y
29
+ ··· + 75y + 4
c
3
, c
4
, c
7
y
30
+ 41y
29
+ ··· + 283y + 4
c
5
, c
10
y
30
− 8y
29
+ ··· + 19y + 4
c
8
, c
9
, c
11
c
12
y
30
+ 28y
29
+ ··· − 721y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.698998 + 0.613459I
a = 2.39480 + 0.61262I
b = −0.15133 − 1.76319I
2.77724 − 3.85593I 9.07460 + 7.10921I
u = −0.698998 − 0.613459I
a = 2.39480 − 0.61262I
b = −0.15133 + 1.76319I
2.77724 + 3.85593I 9.07460 − 7.10921I
u = 0.011422 + 1.077760I
a = 0.273728 + 1.371180I
b = 0.555566 − 0.741416I
−8.36072 + 3.13944I −5.78917 − 2.58972I
u = 0.011422 − 1.077760I
a = 0.273728 − 1.371180I
b = 0.555566 + 0.741416I
−8.36072 − 3.13944I −5.78917 + 2.58972I
u = 0.108984 + 0.894375I
a = 0.301960 − 0.991096I
b = −0.689592 + 0.428612I
−1.44382 + 1.63203I −3.48932 − 5.49543I
u = 0.108984 − 0.894375I
a = 0.301960 + 0.991096I
b = −0.689592 − 0.428612I
−1.44382 − 1.63203I −3.48932 + 5.49543I
u = 0.682002 + 0.922822I
a = 0.189288 + 0.408857I
b = 0.203062 − 0.735831I
−3.81485 + 2.30509I 0.49031 − 2.71546I
u = 0.682002 − 0.922822I
a = 0.189288 − 0.408857I
b = 0.203062 + 0.735831I
−3.81485 − 2.30509I 0.49031 + 2.71546I
u = −0.755396 + 0.903984I
a = −1.66217 − 1.20277I
b = −0.26664 + 1.80105I
−3.27475 − 8.23910I 1.93994 + 7.72708I
u = −0.755396 − 0.903984I
a = −1.66217 + 1.20277I
b = −0.26664 − 1.80105I
−3.27475 + 8.23910I 1.93994 − 7.72708I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.407081 + 0.637690I
a = −0.274690 − 0.288008I
b = 0.208144 + 0.378027I
0.05686 + 1.46890I 0.92942 − 4.73947I
u = 0.407081 − 0.637690I
a = −0.274690 + 0.288008I
b = 0.208144 − 0.378027I
0.05686 − 1.46890I 0.92942 + 4.73947I
u = −1.029980 + 0.698288I
a = −0.453579 − 0.409036I
b = −0.149512 − 0.441012I
4.59059 − 0.39183I 4.29290 + 1.76865I
u = −1.029980 − 0.698288I
a = −0.453579 + 0.409036I
b = −0.149512 + 0.441012I
4.59059 + 0.39183I 4.29290 − 1.76865I
u = 1.121490 + 0.688518I
a = −0.917790 − 0.402386I
b = 0.92316 − 2.47975I
5.57907 − 5.57951I 5.69966 + 3.23734I
u = 1.121490 − 0.688518I
a = −0.917790 + 0.402386I
b = 0.92316 + 2.47975I
5.57907 + 5.57951I 5.69966 − 3.23734I
u = −0.951194 + 0.966858I
a = 0.344441 + 0.102557I
b = 0.246527 + 0.139595I
7.98345 − 3.49396I 3.58725 + 2.25604I
u = −0.951194 − 0.966858I
a = 0.344441 − 0.102557I
b = 0.246527 − 0.139595I
7.98345 + 3.49396I 3.58725 − 2.25604I
u = 1.08159 + 0.91703I
a = 1.56601 − 0.06152I
b = −0.44838 + 3.31749I
11.86890 + 0.24298I 9.59323 + 0.78680I
u = 1.08159 − 0.91703I
a = 1.56601 + 0.06152I
b = −0.44838 − 3.31749I
11.86890 − 0.24298I 9.59323 − 0.78680I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.83819 + 1.14876I
a = −0.494841 + 0.034426I
b = −0.393342 + 0.029742I
3.17963 − 6.45848I 2.96666 + 2.59326I
u = −0.83819 − 1.14876I
a = −0.494841 − 0.034426I
b = −0.393342 − 0.029742I
3.17963 + 6.45848I 2.96666 − 2.59326I
u = 0.98645 + 1.08169I
a = −1.63870 + 0.94953I
b = −0.62498 − 3.36593I
11.33410 + 7.25716I 8.55357 − 5.59796I
u = 0.98645 − 1.08169I
a = −1.63870 − 0.94953I
b = −0.62498 + 3.36593I
11.33410 − 7.25716I 8.55357 + 5.59796I
u = 0.85706 + 1.18834I
a = 1.22904 − 1.46577I
b = 1.13371 + 2.70027I
3.97939 + 12.74170I 4.00000 − 7.13590I
u = 0.85706 − 1.18834I
a = 1.22904 + 1.46577I
b = 1.13371 − 2.70027I
3.97939 − 12.74170I 4.00000 + 7.13590I
u = −0.441623 + 0.214148I
a = −2.94794 + 1.06535I
b = 0.615118 + 1.075540I
2.07762 + 1.07439I 9.76956 − 2.59156I
u = −0.441623 − 0.214148I
a = −2.94794 − 1.06535I
b = 0.615118 − 1.075540I
2.07762 − 1.07439I 9.76956 + 2.59156I
u = −0.040685 + 0.322181I
a = −1.65956 + 3.26961I
b = −0.661518 + 0.572597I
−5.27889 − 3.20038I 6.37013 + 2.60565I
u = −0.040685 − 0.322181I
a = −1.65956 − 3.26961I
b = −0.661518 − 0.572597I
−5.27889 + 3.20038I 6.37013 − 2.60565I
7
II.
I
u
2
= h9a
3
u−a
2
u+· · ·+105a−6, a
4
−a
3
u+2a
2
u+3a
2
−5au−a+2u−3, u
2
+1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
6
=
1
−1
a
3
=
u
0
a
10
=
a
−0.147541a
3
u + 0.0163934a
2
u + ··· − 1.72131a + 0.0983607
a
11
=
−0.147541a
3
u + 0.0163934a
2
u + ··· + 0.278689a + 0.0983607
−a
a
1
=
−u
u
a
5
=
−0.360656a
3
u + 0.262295a
2
u + ··· + 0.459016a − 0.426230
u
a
4
=
−0.360656a
3
u + 0.262295a
2
u + ··· + 0.459016a − 0.426230
u
a
8
=
−0.0327869a
3
u + 0.114754a
2
u + ··· + 0.950820a − 0.311475
−1
a
9
=
0.0655738a
3
u − 0.229508a
2
u + ··· − 1.90164a + 2.62295
0.0819672a
3
u + 0.213115a
2
u + ··· − 0.377049a − 0.721311
a
12
=
−0.0983607a
3
u + 0.344262a
2
u + ··· − 0.147541a + 0.0655738
0.360656a
3
u − 0.262295a
2
u + ··· − 0.459016a + 0.426230
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
4
61
a
3
u +
44
61
a
3
−
108
61
a
2
u −
32
61
a
2
+
56
61
au +
116
61
a −
296
61
u +
84
61
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
8
c
2
, c
3
, c
4
c
6
, c
7
(u
2
+ 1)
4
c
5
, c
10
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
c
8
, c
9
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
11
, c
12
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
8
c
2
, c
3
, c
4
c
6
, c
7
(y + 1)
8
c
5
, c
10
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 0.947956 + 0.221642I
b = −1.66830 − 0.57345I
0.21101 + 1.41510I 3.82674 − 4.90874I
u = 1.000000I
a = −0.221784 + 0.813580I
b = 1.133080 + 0.038228I
−6.79074 + 3.16396I 0.17326 − 2.56480I
u = 1.000000I
a = 0.14689 − 2.02011I
b = 0.57345 + 1.66830I
0.21101 − 1.41510I 3.82674 + 4.90874I
u = 1.000000I
a = −0.87306 + 1.98488I
b = −0.038228 − 1.133080I
−6.79074 − 3.16396I 0.17326 + 2.56480I
u = − 1.000000I
a = 0.947956 − 0.221642I
b = −1.66830 + 0.57345I
0.21101 − 1.41510I 3.82674 + 4.90874I
u = − 1.000000I
a = −0.221784 − 0.813580I
b = 1.133080 − 0.038228I
−6.79074 − 3.16396I 0.17326 + 2.56480I
u = − 1.000000I
a = 0.14689 + 2.02011I
b = 0.57345 − 1.66830I
0.21101 + 1.41510I 3.82674 − 4.90874I
u = − 1.000000I
a = −0.87306 − 1.98488I
b = −0.038228 + 1.133080I
−6.79074 + 3.16396I 0.17326 − 2.56480I
11
III. I
u
3
= hu
8
− u
7
+ 2u
6
− 2u
5
+ u
4
− u
3
+ u
2
+ b − u, u
7
+ u
6
+ 2u
5
+ 2u
4
+
u
3
+ u
2
+ a + u + 1, u
9
+ 3u
7
− u
6
+ 3u
5
− 2u
4
+ 3u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
3
=
u
u
3
+ u
a
10
=
−u
7
− u
6
− 2u
5
− 2u
4
− u
3
− u
2
− u − 1
−u
8
+ u
7
− 2u
6
+ 2u
5
− u
4
+ u
3
− u
2
+ u
a
11
=
−u
7
− 2u
5
− 2u
3
− 2u
u
7
+ u
5
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
5
=
−u
u
a
4
=
0
u
a
8
=
1
u
2
a
9
=
−u
4
− u
2
− 1
u
4
a
12
=
−u
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
6
+ 8u
4
− 4u
3
+ 4u
2
− 4u + 10
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
+ 6u
8
+ 15u
7
+ 23u
6
+ 27u
5
+ 24u
4
+ 15u
3
+ 7u
2
+ 2u − 1
c
2
, c
3
, c
4
c
6
, c
7
u
9
+ 3u
7
− u
6
+ 3u
5
− 2u
4
+ 3u
3
− u
2
+ 2u − 1
c
5
, c
10
(u
3
+ u
2
− 1)
3
c
8
, c
9
, c
11
c
12
(u
3
− u
2
+ 2u − 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
− 6y
8
+ 3y
7
+ 23y
6
− 5y
5
− 16y
4
+ 43y
3
+ 59y
2
+ 18y −1
c
2
, c
3
, c
4
c
6
, c
7
y
9
+ 6y
8
+ 15y
7
+ 23y
6
+ 27y
5
+ 24y
4
+ 15y
3
+ 7y
2
+ 2y −1
c
5
, c
10
(y
3
− y
2
+ 2y −1)
3
c
8
, c
9
, c
11
c
12
(y
3
+ 3y
2
+ 2y −1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.656619 + 0.765660I
a = 0.657957 + 0.314065I
b = −0.66369 − 1.45514I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = −0.656619 − 0.765660I
a = 0.657957 − 0.314065I
b = −0.66369 + 1.45514I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = 0.701160 + 0.628458I
a = 1.48015 − 0.54026I
b = −0.258224 + 0.507366I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = 0.701160 − 0.628458I
a = 1.48015 + 0.54026I
b = −0.258224 − 0.507366I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = −0.233800 + 1.078880I
a = −1.01500 − 1.42921I
b = 1.15982 + 2.09752I
1.11345 9.01951 + 0.I
u = −0.233800 − 1.078880I
a = −1.01500 + 1.42921I
b = 1.15982 − 2.09752I
1.11345 9.01951 + 0.I
u = −0.044542 + 1.394120I
a = −0.15103 + 1.46064I
b = −0.40281 − 2.07233I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = −0.044542 − 1.394120I
a = −0.15103 − 1.46064I
b = −0.40281 + 2.07233I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = 0.467600
a = −1.94416
b = 0.329789
1.11345 9.01950
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
9
+ 6u
8
+ ··· + 2u − 1)
· (u
30
+ 5u
29
+ ··· + 75u + 4)
c
2
, c
6
(u
2
+ 1)
4
(u
9
+ 3u
7
− u
6
+ 3u
5
− 2u
4
+ 3u
3
− u
2
+ 2u − 1)
· (u
30
− u
29
+ ··· + 7u + 2)
c
3
, c
4
, c
7
(u
2
+ 1)
4
(u
9
+ 3u
7
− u
6
+ 3u
5
− 2u
4
+ 3u
3
− u
2
+ 2u − 1)
· (u
30
− u
29
+ ··· + 13u + 2)
c
5
, c
10
((u
3
+ u
2
− 1)
3
)(u
8
− u
6
+ 3u
4
− 2u
2
+ 1)(u
30
− 2u
29
+ ··· − u + 2)
c
8
, c
9
(u
3
− u
2
+ 2u − 1)
3
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
· (u
30
− 8u
29
+ ··· + 19u + 4)
c
11
, c
12
(u
3
− u
2
+ 2u − 1)
3
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· (u
30
− 8u
29
+ ··· + 19u + 4)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
9
− 6y
8
+ ··· + 18y −1)
· (y
30
+ 49y
29
+ ··· − 2273y + 16)
c
2
, c
6
((y + 1)
8
)(y
9
+ 6y
8
+ ··· + 2y −1)
· (y
30
+ 5y
29
+ ··· + 75y + 4)
c
3
, c
4
, c
7
((y + 1)
8
)(y
9
+ 6y
8
+ ··· + 2y −1)
· (y
30
+ 41y
29
+ ··· + 283y + 4)
c
5
, c
10
(y
3
− y
2
+ 2y −1)
3
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
· (y
30
− 8y
29
+ ··· + 19y + 4)
c
8
, c
9
, c
11
c
12
(y
3
+ 3y
2
+ 2y −1)
3
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
30
+ 28y
29
+ ··· − 721y + 16)
17
