12n
0670
(K12n
0670
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 8 3 11 5 12 7 9 10
Solving Sequence
8,11 3,7
4 6 5 9 12 2 1 10
c
7
c
3
c
6
c
5
c
8
c
11
c
2
c
1
c
10
c
4
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h1.08226 × 10
53
u
26
+ 3.46651 × 10
53
u
25
+ ··· + 6.35629 × 10
53
b − 9.30418 × 10
53
,
1.28555 × 10
53
u
26
+ 3.81262 × 10
52
u
25
+ ··· + 1.27126 × 10
54
a − 1.73709 × 10
55
, u
27
+ 4u
26
+ ··· − 36u − 8i
I
u
2
= h2u
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
u
3
= hb + 2u + 1, a − u + 3, u
2
− u − 1i
I
u
4
= hb − u, a, u
2
− u − 1i
I
v
1
= ha, −5v
2
+ 7b − 49v −11, v
3
+ 10v
2
+ 5v + 1i
* 5 irreducible components of dim
C
= 0, with total 42 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
=
h1.08×10
53
u
26
+3.47×10
53
u
25
+· · ·+6.36×10
53
b−9.30×10
53
, 1.29×10
53
u
26
+
3.81 × 10
52
u
25
+ · · · + 1.27 × 10
54
a − 1.74 × 10
55
, u
27
+ 4u
26
+ · · · − 36u − 8i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
3
=
−0.101124u
26
− 0.0299909u
25
+ ··· + 208.958u + 13.6643
−0.170266u
26
− 0.545367u
25
+ ··· + 9.62422u + 1.46377
a
7
=
1
u
2
a
4
=
−0.180542u
26
− 0.294451u
25
+ ··· + 212.007u + 15.1966
−0.206040u
26
− 0.663628u
25
+ ··· + 10.9044u + 1.88945
a
6
=
0.426084u
26
+ 1.57401u
25
+ ··· + 106.382u + 3.92285
0.102378u
26
+ 0.343812u
25
+ ··· − 1.38646u − 1.36007
a
5
=
0.323706u
26
+ 1.23020u
25
+ ··· + 107.769u + 5.28292
0.102378u
26
+ 0.343812u
25
+ ··· − 1.38646u − 1.36007
a
9
=
0.0171155u
26
+ 0.0283407u
25
+ ··· − 26.6157u − 2.71004
0.0302735u
26
+ 0.0988551u
25
+ ··· − 0.938445u − 0.463848
a
12
=
−0.0251997u
26
− 0.0548227u
25
+ ··· + 26.2467u + 2.85292
−0.0188792u
26
− 0.0620300u
25
+ ··· + 1.82255u + 0.224933
a
2
=
−0.336468u
26
− 0.923526u
25
+ ··· + 132.165u + 10.0199
−0.251652u
26
− 0.818179u
25
+ ··· + 10.8630u + 2.53569
a
1
=
0.0473890u
26
+ 0.127196u
25
+ ··· − 27.5541u − 3.17389
0.00858260u
26
+ 0.0283575u
25
+ ··· − 0.927413u − 0.0350346
a
10
=
−u
−u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.0379585u
26
+ 1.16314u
25
+ ··· + 780.278u + 65.8410
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
27
− 12u
26
+ ··· − 82u − 1
c
3
, c
6
u
27
+ 4u
26
+ ··· + 640u − 256
c
5
, c
8
u
27
+ 3u
26
+ ··· − 112u + 16
c
7
, c
10
u
27
− 4u
26
+ ··· − 36u + 8
c
9
, c
11
, c
12
u
27
+ 7u
26
+ ··· − 65u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
27
− 46y
26
+ ··· + 6314y −1
c
3
, c
6
y
27
− 54y
26
+ ··· + 5095424y −65536
c
5
, c
8
y
27
+ 25y
26
+ ··· + 12928y −256
c
7
, c
10
y
27
+ 12y
26
+ ··· + 7696y −64
c
9
, c
11
, c
12
y
27
− 15y
26
+ ··· + 4023y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.598885 + 1.018170I
a = −0.527451 − 0.308767I
b = 0.147510 + 0.585443I
−2.46734 + 1.28188I 0.019660 − 0.966602I
u = 0.598885 − 1.018170I
a = −0.527451 + 0.308767I
b = 0.147510 − 0.585443I
−2.46734 − 1.28188I 0.019660 + 0.966602I
u = −0.975452 + 0.786268I
a = −0.196879 + 0.002027I
b = 0.073018 − 0.478334I
1.16826 − 5.86191I 6.68570 + 1.60407I
u = −0.975452 − 0.786268I
a = −0.196879 − 0.002027I
b = 0.073018 + 0.478334I
1.16826 + 5.86191I 6.68570 − 1.60407I
u = −0.538379 + 0.455019I
a = 1.62953 + 0.35861I
b = −0.061105 − 0.490914I
1.156740 + 0.801856I 6.71973 + 0.16728I
u = −0.538379 − 0.455019I
a = 1.62953 − 0.35861I
b = −0.061105 + 0.490914I
1.156740 − 0.801856I 6.71973 − 0.16728I
u = −0.702983 + 0.023598I
a = −2.18915 + 4.62670I
b = −0.99845 + 1.93414I
−0.834252 − 0.150815I 17.5464 + 6.6365I
u = −0.702983 − 0.023598I
a = −2.18915 − 4.62670I
b = −0.99845 − 1.93414I
−0.834252 + 0.150815I 17.5464 − 6.6365I
u = 0.442597 + 0.499853I
a = −0.078287 − 0.292577I
b = −0.33969 − 1.38997I
4.69512 − 2.40532I 3.41247 − 6.32084I
u = 0.442597 − 0.499853I
a = −0.078287 + 0.292577I
b = −0.33969 + 1.38997I
4.69512 + 2.40532I 3.41247 + 6.32084I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.610551
a = 0.710713
b = 0.376982
0.859867 11.9670
u = 0.594482
a = −4.50707
b = 0.0513464
−7.81649 57.8300
u = 1.58256
a = 1.18182
b = 3.57318
7.98804 43.0640
u = 0.34038 + 1.67519I
a = 1.057830 + 0.148663I
b = −0.340400 + 0.098336I
−13.16990 + 3.53439I 0. − 2.09406I
u = 0.34038 − 1.67519I
a = 1.057830 − 0.148663I
b = −0.340400 − 0.098336I
−13.16990 − 3.53439I 0. + 2.09406I
u = −0.60820 + 1.69659I
a = 0.877793 − 0.221512I
b = −0.35535 − 1.77081I
−6.34499 − 6.08931I 0. + 3.73020I
u = −0.60820 − 1.69659I
a = 0.877793 + 0.221512I
b = −0.35535 + 1.77081I
−6.34499 + 6.08931I 0. − 3.73020I
u = 0.03133 + 1.80621I
a = 0.524823 + 0.076099I
b = −0.25035 + 1.73182I
−7.44159 − 1.29405I −1.37770 + 1.27497I
u = 0.03133 − 1.80621I
a = 0.524823 − 0.076099I
b = −0.25035 − 1.73182I
−7.44159 + 1.29405I −1.37770 − 1.27497I
u = 0.142980
a = 49.1336
b = 0.408460
0.561362 203.500
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.125648
a = −6.99415
b = −0.661597
−1.12664 −9.59770
u = −1.42516 + 1.34942I
a = −0.835437 + 0.737623I
b = 0.57720 + 2.23959I
−15.7574 − 13.5083I 0
u = −1.42516 − 1.34942I
a = −0.835437 − 0.737623I
b = 0.57720 − 2.23959I
−15.7574 + 13.5083I 0
u = 1.66937 + 1.50035I
a = −0.579428 − 0.676345I
b = 0.28102 − 2.41058I
18.5782 + 5.9421I 0
u = 1.66937 − 1.50035I
a = −0.579428 + 0.676345I
b = 0.28102 + 2.41058I
18.5782 − 5.9421I 0
u = −1.62430 + 1.65382I
a = −0.445794 + 0.495251I
b = −0.10758 + 2.58327I
−16.0044 + 2.3695I 0
u = −1.62430 − 1.65382I
a = −0.445794 − 0.495251I
b = −0.10758 − 2.58327I
−16.0044 − 2.3695I 0
7
II. I
u
2
= h2u
7
+ u
6
− 3u
5
− 3u
4
+ 4u
3
+ 3u
2
+ b − 2u − 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
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
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
− 4u
2
+ 4u + 8
−2u
7
− u
6
+ 3u
5
+ 3u
4
− 4u
3
− 4u
2
+ 2u + 4
a
1
=
u
2
− 1
−u
2
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 + 73
8
(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
9
(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
10
(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
−0.604279 + 1.131230I 0.744211 + 0.553382I
u = −0.570868 − 0.730671I
a = −1.194470 + 0.635084I
b = −0.281371 − 1.128550I
−0.604279 − 1.131230I 0.744211 − 0.553382I
u = 0.855237 + 0.665892I
a = −0.637416 − 0.344390I
b = 0.208670 − 0.825203I
−3.80435 + 2.57849I −2.39106 − 4.72239I
u = 0.855237 − 0.665892I
a = −0.637416 + 0.344390I
b = 0.208670 + 0.825203I
−3.80435 − 2.57849I −2.39106 + 4.72239I
u = 1.09818
a = 0.687555
b = 0.829189
4.85780 8.45210
u = −1.031810 + 0.655470I
a = −0.286111 + 0.344558I
b = 0.284386 + 0.605794I
0.73474 − 6.44354I 0.47538 + 9.99765I
u = −1.031810 − 0.655470I
a = −0.286111 − 0.344558I
b = 0.284386 − 0.605794I
0.73474 + 6.44354I 0.47538 − 9.99765I
u = −0.603304
a = 7.54843
b = 2.74744
−0.799899 60.8910
11
III. I
u
3
= hb + 2u + 1, a − u + 3, u
2
− u − 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
3
=
u − 3
−2u − 1
a
7
=
1
u + 1
a
4
=
2u − 4
−u − 1
a
6
=
−3u + 5
0
a
5
=
−3u + 5
0
a
9
=
1
0
a
12
=
u
u
a
2
=
−4u + 5
−2u − 1
a
1
=
−1
−u − 1
a
10
=
−u
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −45
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
, c
11
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
9
u
2
− u − 1
c
5
, c
8
u
2
13
(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
9
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
8
y
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −3.61803
b = 0.236068
−7.89568 −45.0000
u = 1.61803
a = −1.38197
b = −4.23607
7.89568 −45.0000
15
IV. I
u
4
= hb − u, a, u
2
− u − 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
3
=
0
u
a
7
=
1
u + 1
a
4
=
−u
−u − 1
a
6
=
1
0
a
5
=
1
0
a
9
=
1
0
a
12
=
u
u
a
2
=
u
u
a
1
=
−1
−u − 1
a
10
=
−u
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
, c
11
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
9
u
2
− u − 1
c
5
, c
8
u
2
17
(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
9
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
8
y
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0
b = −0.618034
0 0
u = 1.61803
a = 0
b = 1.61803
0 0
19
V. I
v
1
= ha, −5v
2
+ 7b − 49v − 11, v
3
+ 10v
2
+ 5v + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
v
0
a
3
=
0
5
7
v
2
+ 7v +
11
7
a
7
=
1
0
a
4
=
−
5
7
v
2
− 7v −
11
7
5
7
v
2
+ 7v +
11
7
a
6
=
1
2
7
v
2
+ 3v +
17
7
a
5
=
−
2
7
v
2
− 3v −
10
7
2
7
v
2
+ 3v +
17
7
a
9
=
5
7
v
2
+ 7v +
25
7
−v
2
− 10v − 5
a
12
=
−
5
7
v
2
− 6v −
25
7
v
2
+ 10v + 5
a
2
=
−1
2
7
v
2
+ 3v +
17
7
a
1
=
−
5
7
v
2
− 7v −
25
7
v
2
+ 10v + 5
a
10
=
v
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
54
7
v
2
− 65v −
95
7
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
u
3
+ u
2
− 1
c
3
u
3
− u
2
+ 2u − 1
c
4
u
3
− u
2
+ 1
c
5
u
3
+ 3u
2
+ 2u − 1
c
6
u
3
+ u
2
+ 2u + 1
c
7
, c
10
u
3
c
8
u
3
− 3u
2
+ 2u + 1
c
9
(u + 1)
3
c
11
, c
12
(u − 1)
3
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
3
− y
2
+ 2y − 1
c
3
, c
6
y
3
+ 3y
2
+ 2y − 1
c
5
, c
8
y
3
− 5y
2
+ 10y − 1
c
7
, c
10
y
3
c
9
, c
11
, c
12
(y − 1)
3
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.258045 + 0.197115I
a = 0
b = −0.215080 + 1.307140I
4.66906 + 2.82812I 2.98758 − 12.02771I
v = −0.258045 − 0.197115I
a = 0
b = −0.215080 − 1.307140I
4.66906 − 2.82812I 2.98758 + 12.02771I
v = −9.48391
a = 0
b = −0.569840
0.531480 −90.9750
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
8
)(u
2
+ u − 1)
2
(u
3
+ u
2
− 1)(u
27
− 12u
26
+ ··· − 82u − 1)
c
3
u
8
(u
2
+ u − 1)
2
(u
3
− u
2
+ 2u − 1)(u
27
+ 4u
26
+ ··· + 640u − 256)
c
4
((u + 1)
8
)(u
2
− u − 1)
2
(u
3
− u
2
+ 1)(u
27
− 12u
26
+ ··· − 82u − 1)
c
5
u
4
(u
3
+ 3u
2
+ 2u − 1)
· (u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
27
+ 3u
26
+ ··· − 112u + 16)
c
6
u
8
(u
2
− u − 1)
2
(u
3
+ u
2
+ 2u + 1)(u
27
+ 4u
26
+ ··· + 640u − 256)
c
7
u
3
(u
2
− u − 1)
2
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
· (u
27
− 4u
26
+ ··· − 36u + 8)
c
8
u
4
(u
3
− 3u
2
+ 2u + 1)
· (u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
27
+ 3u
26
+ ··· − 112u + 16)
c
9
(u + 1)
3
(u
2
− u − 1)
2
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)
· (u
27
+ 7u
26
+ ··· − 65u + 1)
c
10
u
3
(u
2
+ u − 1)
2
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
· (u
27
− 4u
26
+ ··· − 36u + 8)
c
11
, c
12
(u − 1)
3
(u
2
+ u − 1)
2
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
· (u
27
+ 7u
26
+ ··· − 65u + 1)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
8
(y
2
− 3y + 1)
2
(y
3
− y
2
+ 2y − 1)
· (y
27
− 46y
26
+ ··· + 6314y − 1)
c
3
, c
6
y
8
(y
2
− 3y + 1)
2
(y
3
+ 3y
2
+ 2y − 1)
· (y
27
− 54y
26
+ ··· + 5095424y − 65536)
c
5
, c
8
y
4
(y
3
− 5y
2
+ 10y − 1)
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
27
+ 25y
26
+ ··· + 12928y − 256)
c
7
, c
10
y
3
(y
2
− 3y + 1)
2
(y
8
− 3y
7
+ ··· − 4y + 1)
· (y
27
+ 12y
26
+ ··· + 7696y − 64)
c
9
, c
11
, c
12
(y − 1)
3
(y
2
− 3y + 1)
2
· (y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
27
− 15y
26
+ ··· + 4023y − 1)
25
