12n
0566
(K12n
0566
)
A knot diagram
1
Linearized knot diagam
3 7 9 8 11 10 2 4 12 6 7 9
Solving Sequence
3,9 4,7
2 1 8 5 12 10 6 11
c
3
c
2
c
1
c
8
c
4
c
12
c
9
c
6
c
11
c
5
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−4.65742 × 10
21
u
31
− 2.20023 × 10
22
u
30
+ ··· + 9.36708 × 10
23
b + 5.12230 × 10
23
,
2.94856 × 10
25
u
31
− 3.10386 × 10
25
u
30
+ ··· + 2.99747 × 10
25
a − 2.87196 × 10
25
, u
32
− u
31
+ ··· − 2u + 1i
I
u
2
= hb − u, a
5
− a
4
+ 2a
3
− a
2
+ a − 1, u
2
+ 1i
I
u
3
= hb − u, a, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
I
u
4
= hb − u, a, u
3
+ u
2
+ 2u + 1i
* 4 irreducible components of dim
C
= 0, with total 51 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−4.66×10
21
u
31
−2.20×10
22
u
30
+· · ·+9.37×10
23
b+5.12×10
23
, 2.95×
10
25
u
31
−3.10×10
25
u
30
+· · ·+3.00×10
25
a−2.87×10
25
, u
32
−u
31
+· · ·−2u+1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
7
=
−0.983683u
31
+ 1.03549u
30
+ ··· + 42.1633u + 0.958130
0.00497211u
31
+ 0.0234890u
30
+ ··· + 1.12532u − 0.546841
a
2
=
0.495029u
31
− 0.483684u
30
+ ··· + 26.1222u − 0.952046
−0.0284611u
31
+ 0.0288036u
30
+ ··· + 0.536896u − 0.995028
a
1
=
0.466568u
31
− 0.454881u
30
+ ··· + 26.6591u − 1.94707
−0.0284611u
31
+ 0.0288036u
30
+ ··· + 0.536896u − 0.995028
a
8
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
12
=
0.466568u
31
− 0.454881u
30
+ ··· + 26.6591u − 1.94707
−0.0000739509u
31
+ 0.00179294u
30
+ ··· + 0.0937032u − 1.00672
a
10
=
0.935244u
31
− 1.05640u
30
+ ··· − 35.9185u + 0.318643
−0.00328364u
31
− 0.0312396u
30
+ ··· + 1.93055u + 0.535840
a
6
=
−0.357364u
31
+ 0.452611u
30
+ ··· − 8.86369u + 3.30364
0.136131u
31
− 0.134930u
30
+ ··· − 6.56253u + 0.588091
a
11
=
0.465451u
31
− 0.514663u
30
+ ··· − 8.17487u + 4.78562
0.151172u
31
− 0.159734u
30
+ ··· − 2.72954u − 0.0680096
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1674544166583428940712685
3746832618160827210734692
u
31
−
3228763469657253763874131
7493665236321654421469384
u
30
+ ··· +
115090649984726751416336801
7493665236321654421469384
u −
4624345191860318935353055
936708154540206802683673
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
+ 3u
31
+ ··· − 20u + 1
c
2
, c
7
u
32
+ u
31
+ ··· − 10u
2
+ 1
c
3
, c
4
, c
8
u
32
+ u
31
+ ··· + 2u + 1
c
5
, c
6
, c
10
u
32
+ 2u
31
+ ··· + 5u + 2
c
9
, c
12
u
32
+ 8u
31
+ ··· + 837u + 136
c
11
u
32
− 2u
31
+ ··· − 96u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
+ 63y
31
+ ··· − 48y + 1
c
2
, c
7
y
32
+ 3y
31
+ ··· − 20y + 1
c
3
, c
4
, c
8
y
32
+ 47y
31
+ ··· − 84y + 1
c
5
, c
6
, c
10
y
32
+ 28y
31
+ ··· + 19y + 4
c
9
, c
12
y
32
+ 44y
30
+ ··· − 169353y + 18496
c
11
y
32
− 4y
31
+ ··· − 256y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.149637 + 1.036980I
a = −0.417547 + 0.954803I
b = 0.302187 − 0.006644I
1.43807 − 1.50420I 9.22727 + 3.53831I
u = −0.149637 − 1.036980I
a = −0.417547 − 0.954803I
b = 0.302187 + 0.006644I
1.43807 + 1.50420I 9.22727 − 3.53831I
u = 0.570569 + 0.926273I
a = 0.652139 + 0.672144I
b = −0.729100 + 0.429975I
0.356823 − 1.078380I 5.98174 + 1.89501I
u = 0.570569 − 0.926273I
a = 0.652139 − 0.672144I
b = −0.729100 − 0.429975I
0.356823 + 1.078380I 5.98174 − 1.89501I
u = −0.759558 + 0.797956I
a = −0.669157 + 0.706117I
b = 0.806103 + 0.667276I
2.84076 + 4.52561I 7.98035 − 6.47723I
u = −0.759558 − 0.797956I
a = −0.669157 − 0.706117I
b = 0.806103 − 0.667276I
2.84076 − 4.52561I 7.98035 + 6.47723I
u = 0.080786 + 1.113860I
a = 0.450290 + 1.273100I
b = −0.240058 − 0.218923I
−4.20210 + 4.53860I 4.95050 − 3.52289I
u = 0.080786 − 1.113860I
a = 0.450290 − 1.273100I
b = −0.240058 + 0.218923I
−4.20210 − 4.53860I 4.95050 + 3.52289I
u = 0.856979 + 0.733531I
a = 0.658177 + 0.744370I
b = −0.839448 + 0.780469I
−2.06044 − 8.15993I 2.47068 + 7.37481I
u = 0.856979 − 0.733531I
a = 0.658177 − 0.744370I
b = −0.839448 − 0.780469I
−2.06044 + 8.15993I 2.47068 − 7.37481I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.388206 + 0.750499I
a = 0.618021 + 0.553690I
b = −0.445761 + 0.493038I
0.37663 − 1.40948I 4.16499 + 4.71779I
u = 0.388206 − 0.750499I
a = 0.618021 − 0.553690I
b = −0.445761 − 0.493038I
0.37663 + 1.40948I 4.16499 − 4.71779I
u = −0.564144 + 0.368266I
a = −1.072100 + 0.757048I
b = 0.426872 + 0.891946I
−4.95649 + 1.03511I −2.62837 − 3.37474I
u = −0.564144 − 0.368266I
a = −1.072100 − 0.757048I
b = 0.426872 − 0.891946I
−4.95649 − 1.03511I −2.62837 + 3.37474I
u = −0.14043 + 1.61047I
a = 1.344160 + 0.225189I
b = −0.909776 − 0.978319I
1.96786 + 3.39058I 0
u = −0.14043 − 1.61047I
a = 1.344160 − 0.225189I
b = −0.909776 + 0.978319I
1.96786 − 3.39058I 0
u = 0.31474 + 1.69085I
a = −1.283680 − 0.037904I
b = 0.93918 − 1.20942I
6.02211 − 12.78960I 0
u = 0.31474 − 1.69085I
a = −1.283680 + 0.037904I
b = 0.93918 + 1.20942I
6.02211 + 12.78960I 0
u = 0.20411 + 1.72625I
a = −1.226140 + 0.091965I
b = 1.02007 − 1.09862I
9.50782 − 4.34458I 0
u = 0.20411 − 1.72625I
a = −1.226140 − 0.091965I
b = 1.02007 + 1.09862I
9.50782 + 4.34458I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.27493 + 1.71725I
a = 1.249820 + 0.010579I
b = −0.98277 − 1.17471I
11.3763 + 8.7540I 0
u = −0.27493 − 1.71725I
a = 1.249820 − 0.010579I
b = −0.98277 + 1.17471I
11.3763 − 8.7540I 0
u = 0.08047 + 1.76969I
a = −1.133210 + 0.199889I
b = 1.12271 − 0.97669I
9.94630 − 3.44071I 0
u = 0.08047 − 1.76969I
a = −1.133210 − 0.199889I
b = 1.12271 + 0.97669I
9.94630 + 3.44071I 0
u = −0.07166 + 1.77723I
a = −1.040780 + 0.308237I
b = 1.20703 − 0.80705I
7.36926 + 5.04799I 0
u = −0.07166 − 1.77723I
a = −1.040780 − 0.308237I
b = 1.20703 + 0.80705I
7.36926 − 5.04799I 0
u = 0.00485 + 1.78607I
a = 1.074930 + 0.259134I
b = −1.18249 − 0.88752I
12.35040 − 0.91754I 0
u = 0.00485 − 1.78607I
a = 1.074930 − 0.259134I
b = −1.18249 + 0.88752I
12.35040 + 0.91754I 0
u = −0.156144 + 0.076843I
a = −4.97166 − 4.58572I
b = 0.069135 − 1.045370I
−7.55319 − 4.33239I −6.63531 + 3.69864I
u = −0.156144 − 0.076843I
a = −4.97166 + 4.58572I
b = 0.069135 + 1.045370I
−7.55319 + 4.33239I −6.63531 − 3.69864I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.1157890 + 0.0692134I
a = 6.26675 − 1.88119I
b = −0.063886 + 0.968637I
−2.01181 − 1.57269I −2.90508 + 4.76814I
u = 0.1157890 − 0.0692134I
a = 6.26675 + 1.88119I
b = −0.063886 − 0.968637I
−2.01181 + 1.57269I −2.90508 − 4.76814I
8
II. I
u
2
= hb − u, a
5
− a
4
+ 2a
3
− a
2
+ a − 1, u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−1
a
7
=
a
u
a
2
=
au + 1
−1
a
1
=
au
−1
a
8
=
u
0
a
5
=
0
−1
a
12
=
au
−au − 1
a
10
=
−a
2
u
a
2
u + a + u
a
6
=
a
4
− a
3
+ a
2
+ 1
a
4
u − a
4
+ a
2
u − a
2
+ u − 1
a
11
=
a
3
u + au
−a
2
− au − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a
3
− 4a
2
+ 4a
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
10
c
2
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
5
c
5
, c
6
, c
10
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
c
9
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
11
u
10
+ u
8
+ 8u
6
+ 3u
4
+ 3u
2
+ 1
c
12
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
10
c
2
, c
3
, c
4
c
7
, c
8
(y + 1)
10
c
5
, c
6
, c
10
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
c
9
, c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
11
(y
5
+ y
4
+ 8y
3
+ 3y
2
+ 3y + 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −0.339110 + 0.822375I
b = 1.000000I
−0.32910 − 1.53058I 3.48489 + 4.43065I
u = 1.000000I
a = −0.339110 − 0.822375I
b = 1.000000I
−0.32910 + 1.53058I 3.48489 − 4.43065I
u = 1.000000I
a = 0.766826
b = 1.000000I
−2.40108 2.51890
u = 1.000000I
a = 0.455697 + 1.200150I
b = 1.000000I
−5.87256 + 4.40083I −0.74431 − 3.49859I
u = 1.000000I
a = 0.455697 − 1.200150I
b = 1.000000I
−5.87256 − 4.40083I −0.74431 + 3.49859I
u = − 1.000000I
a = −0.339110 + 0.822375I
b = − 1.000000I
−0.32910 − 1.53058I 3.48489 + 4.43065I
u = − 1.000000I
a = −0.339110 − 0.822375I
b = − 1.000000I
−0.32910 + 1.53058I 3.48489 − 4.43065I
u = − 1.000000I
a = 0.766826
b = − 1.000000I
−2.40108 2.51890
u = − 1.000000I
a = 0.455697 + 1.200150I
b = − 1.000000I
−5.87256 + 4.40083I −0.74431 − 3.49859I
u = − 1.000000I
a = 0.455697 − 1.200150I
b = − 1.000000I
−5.87256 − 4.40083I −0.74431 + 3.49859I
12
III. I
u
3
= hb − u, a, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
7
=
0
u
a
2
=
1
u
2
a
1
=
u
2
+ 1
u
2
a
8
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
12
=
u
2
+ 1
u
4
+ 2u
2
a
10
=
u
5
+ 2u
3
+ u
2u
5
+ 2u
3
+ 2u − 1
a
6
=
−2u
5
+ u
4
− 4u
3
+ 3u
2
− 3u + 3
−3u
5
+ 2u
4
− 4u
3
+ 3u
2
− u + 3
a
11
=
u
2
+ 1
2u
4
+ 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
− 4u + 6
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
c
2
, c
3
, c
4
c
7
, c
8
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
5
, c
6
, c
10
(u
3
− u
2
+ 2u − 1)
2
c
9
, c
11
, c
12
(u
3
+ u
2
− 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
c
2
, c
3
, c
4
c
7
, c
8
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
5
, c
6
, c
10
(y
3
+ 3y
2
+ 2y −1)
2
c
9
, c
11
, c
12
(y
3
− y
2
+ 2y −1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498832 + 1.001300I
a = 0
b = −0.498832 + 1.001300I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = −0.498832 − 1.001300I
a = 0
b = −0.498832 − 1.001300I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = 0.284920 + 1.115140I
a = 0
b = 0.284920 + 1.115140I
1.11345 9.01951 + 0.I
u = 0.284920 − 1.115140I
a = 0
b = 0.284920 − 1.115140I
1.11345 9.01951 + 0.I
u = 0.713912 + 0.305839I
a = 0
b = 0.713912 + 0.305839I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = 0.713912 − 0.305839I
a = 0
b = 0.713912 − 0.305839I
−3.02413 − 2.82812I 2.49024 + 2.97945I
16
IV. I
u
4
= hb − u, a, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
7
=
0
u
a
2
=
1
u
2
a
1
=
u
2
+ 1
u
2
a
8
=
u
−u
2
− u − 1
a
5
=
u
2
+ 1
u
2
+ u + 1
a
12
=
u
2
+ 1
u
2
+ u + 1
a
10
=
−1
2u
a
6
=
−u
2u
2
+ u
a
11
=
u
2
+ 1
u
2
+ 2u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
2
+ 4u + 10
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ 3u
2
+ 2u − 1
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
10
u
3
− u
2
+ 2u − 1
c
9
, c
11
, c
12
u
3
+ u
2
− 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
3
− 5y
2
+ 10y −1
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
10
y
3
+ 3y
2
+ 2y −1
c
9
, c
11
, c
12
y
3
− y
2
+ 2y −1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0
b = −0.215080 + 1.307140I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = −0.215080 − 1.307140I
a = 0
b = −0.215080 − 1.307140I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = −0.569840
a = 0
b = −0.569840
1.11345 9.01950
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
10
(u
3
+ 3u
2
+ 2u − 1)(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
· (u
32
+ 3u
31
+ ··· − 20u + 1)
c
2
, c
7
(u
2
+ 1)
5
(u
3
− u
2
+ 2u − 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
32
+ u
31
+ ··· − 10u
2
+ 1)
c
3
, c
4
, c
8
(u
2
+ 1)
5
(u
3
− u
2
+ 2u − 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
32
+ u
31
+ ··· + 2u + 1)
c
5
, c
6
, c
10
(u
3
− u
2
+ 2u − 1)
3
(u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1)
· (u
32
+ 2u
31
+ ··· + 5u + 2)
c
9
(u
3
+ u
2
− 1)
3
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
· (u
32
+ 8u
31
+ ··· + 837u + 136)
c
11
(u
3
+ u
2
− 1)
3
(u
10
+ u
8
+ 8u
6
+ 3u
4
+ 3u
2
+ 1)
· (u
32
− 2u
31
+ ··· − 96u + 16)
c
12
(u
3
+ u
2
− 1)
3
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
· (u
32
+ 8u
31
+ ··· + 837u + 136)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
10
(y
3
− 5y
2
+ 10y −1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
32
+ 63y
31
+ ··· − 48y + 1)
c
2
, c
7
(y + 1)
10
(y
3
+ 3y
2
+ 2y −1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
32
+ 3y
31
+ ··· − 20y + 1)
c
3
, c
4
, c
8
(y + 1)
10
(y
3
+ 3y
2
+ 2y −1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
32
+ 47y
31
+ ··· − 84y + 1)
c
5
, c
6
, c
10
(y
3
+ 3y
2
+ 2y −1)
3
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
· (y
32
+ 28y
31
+ ··· + 19y + 4)
c
9
, c
12
(y
3
− y
2
+ 2y −1)
3
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
· (y
32
+ 44y
30
+ ··· − 169353y + 18496)
c
11
(y
3
− y
2
+ 2y −1)
3
(y
5
+ y
4
+ 8y
3
+ 3y
2
+ 3y + 1)
2
· (y
32
− 4y
31
+ ··· − 256y + 256)
22
