12n
0569
(K12n
0569
)
A knot diagram
1
Linearized knot diagam
3 7 12 10 11 2 6 12 3 5 8 10
Solving Sequence
2,6
7 3
8,11
12 1 5 10 4 9
c
6
c
2
c
7
c
11
c
1
c
5
c
10
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h8.11108 × 10
37
u
57
+ 4.91027 × 10
37
u
56
+ ··· + 5.66410 × 10
38
b − 1.34973 × 10
39
,
1.29757 × 10
39
u
57
+ 3.25753 × 10
39
u
56
+ ··· + 5.66410 × 10
38
a − 5.91852 × 10
37
, u
58
+ 2u
57
+ ··· − 9u − 1i
I
u
2
= hu
15
+ u
14
− 3u
13
− 4u
12
+ 7u
11
+ 10u
10
− 10u
9
− 16u
8
+ 11u
7
+ 19u
6
− 8u
5
− 15u
4
+ 4u
3
+ 7u
2
+ b − 2u − 2,
− 2u
16
− u
15
+ ··· + a + 1, u
17
+ u
16
+ ··· − 5u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 75 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
= h8.11 × 10
37
u
57
+ 4.91 × 10
37
u
56
+ · · · + 5.66 × 10
38
b − 1.35 × 10
39
, 1.30 ×
10
39
u
57
+3.26×10
39
u
56
+· · ·+5.66×10
38
a−5.92×10
37
, u
58
+2u
57
+· · ·−9u−1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
u
2
a
11
=
−2.29087u
57
− 5.75119u
56
+ ··· − 44.7958u + 0.104492
−0.143202u
57
− 0.0866910u
56
+ ··· + 6.22932u + 2.38295
a
12
=
−1.52712u
57
− 4.39667u
56
+ ··· − 31.5613u + 3.08463
−0.103380u
57
+ 0.538492u
56
+ ··· + 10.3860u + 2.75551
a
1
=
u
3
u
5
− u
3
+ u
a
5
=
−1.60930u
57
− 3.86858u
56
+ ··· − 1.74961u + 7.98100
1.09318u
57
+ 2.32144u
56
+ ··· + 8.84503u + 3.47131
a
10
=
−1.14552u
57
− 4.13737u
56
+ ··· − 72.7081u − 15.0427
−0.129885u
57
+ 0.333506u
56
+ ··· − 3.41712u − 2.96099
a
4
=
−2.16693u
57
− 5.96581u
56
+ ··· − 57.6969u − 9.85977
0.892300u
57
+ 1.86393u
56
+ ··· + 9.09728u − 0.642831
a
9
=
−0.697310u
57
− 3.28572u
56
+ ··· − 62.1886u − 13.9750
−0.544392u
57
− 0.342782u
56
+ ··· − 13.8913u − 4.07349
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4.70530u
57
− 10.9734u
56
+ ··· − 119.055u + 7.76720
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
58
+ 20u
57
+ ··· + 111u + 1
c
2
, c
6
u
58
− 2u
57
+ ··· + 9u − 1
c
3
u
58
− 2u
57
+ ··· − 4u + 1
c
4
, c
5
, c
10
u
58
+ u
57
+ ··· − 148u − 43
c
8
, c
11
u
58
− 4u
57
+ ··· + 12848u − 2119
c
9
u
58
− u
57
+ ··· + 2150u + 293
c
12
u
58
+ 2u
57
+ ··· + 44u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
58
+ 44y
57
+ ··· − 6891y + 1
c
2
, c
6
y
58
− 20y
57
+ ··· − 111y + 1
c
3
y
58
− 56y
57
+ ··· + 254y + 1
c
4
, c
5
, c
10
y
58
− 57y
57
+ ··· + 10690y + 1849
c
8
, c
11
y
58
− 34y
57
+ ··· − 42707330y + 4490161
c
9
y
58
+ 55y
57
+ ··· − 1142832y + 85849
c
12
y
58
+ 50y
57
+ ··· + 2238y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.649208 + 0.777185I
a = −1.311830 − 0.015356I
b = 1.41367 − 0.24080I
8.99829 − 0.85807I 7.14206 + 0.I
u = −0.649208 − 0.777185I
a = −1.311830 + 0.015356I
b = 1.41367 + 0.24080I
8.99829 + 0.85807I 7.14206 + 0.I
u = −0.192605 + 0.952386I
a = 1.59370 + 0.08023I
b = −1.374660 − 0.179161I
1.95932 + 3.47319I 7.38337 − 3.16681I
u = −0.192605 − 0.952386I
a = 1.59370 − 0.08023I
b = −1.374660 + 0.179161I
1.95932 − 3.47319I 7.38337 + 3.16681I
u = 0.704542 + 0.759694I
a = −2.20492 − 0.44914I
b = 1.66369 + 0.08603I
9.70913 + 0.22811I 5.29825 + 0.I
u = 0.704542 − 0.759694I
a = −2.20492 + 0.44914I
b = 1.66369 − 0.08603I
9.70913 − 0.22811I 5.29825 + 0.I
u = −0.844133 + 0.459005I
a = −0.88197 + 1.28310I
b = 0.794410 + 0.348082I
0.18156 + 3.43068I 3.97292 − 8.38855I
u = −0.844133 − 0.459005I
a = −0.88197 − 1.28310I
b = 0.794410 − 0.348082I
0.18156 − 3.43068I 3.97292 + 8.38855I
u = −0.785206 + 0.690856I
a = −1.51273 + 0.86981I
b = 0.940956 + 0.796608I
0.04805 + 2.72947I 6.00584 − 3.68019I
u = −0.785206 − 0.690856I
a = −1.51273 − 0.86981I
b = 0.940956 − 0.796608I
0.04805 − 2.72947I 6.00584 + 3.68019I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.675145 + 0.804414I
a = 0.143408 − 0.529642I
b = 0.449289 + 0.991573I
−1.36150 + 3.46891I 4.09542 − 2.58287I
u = 0.675145 − 0.804414I
a = 0.143408 + 0.529642I
b = 0.449289 − 0.991573I
−1.36150 − 3.46891I 4.09542 + 2.58287I
u = −1.06579
a = −0.471638
b = −1.49871
4.21383 −2.91260
u = −0.797269 + 0.746574I
a = −2.49041 + 1.51484I
b = 1.313240 − 0.095807I
0.640941 − 0.368761I 0
u = −0.797269 − 0.746574I
a = −2.49041 − 1.51484I
b = 1.313240 + 0.095807I
0.640941 + 0.368761I 0
u = −1.093130 + 0.112916I
a = −0.372147 − 0.718737I
b = −0.097432 − 0.849324I
−7.71465 + 3.17136I 0. − 3.24101I
u = −1.093130 − 0.112916I
a = −0.372147 + 0.718737I
b = −0.097432 + 0.849324I
−7.71465 − 3.17136I 0. + 3.24101I
u = 0.897438 + 0.025837I
a = 0.83649 + 1.36225I
b = −1.246550 + 0.521890I
−4.22900 − 1.73682I −1.19489 + 1.04559I
u = 0.897438 − 0.025837I
a = 0.83649 − 1.36225I
b = −1.246550 − 0.521890I
−4.22900 + 1.73682I −1.19489 − 1.04559I
u = 0.865118 + 0.692108I
a = −0.329927 − 0.483809I
b = −0.099466 − 0.733477I
3.97645 − 2.66265I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.865118 − 0.692108I
a = −0.329927 + 0.483809I
b = −0.099466 + 0.733477I
3.97645 + 2.66265I 0
u = 1.12004
a = −0.852945
b = −1.22513
3.22886 2.00000
u = −0.927826 + 0.684508I
a = 0.367169 − 0.609308I
b = −1.072260 + 0.836652I
−0.38663 + 2.57641I 0
u = −0.927826 − 0.684508I
a = 0.367169 + 0.609308I
b = −1.072260 − 0.836652I
−0.38663 − 2.57641I 0
u = −0.671464 + 0.950623I
a = 1.75527 − 0.30187I
b = −1.53135 + 0.36764I
5.01980 − 8.36674I 0
u = −0.671464 − 0.950623I
a = 1.75527 + 0.30187I
b = −1.53135 − 0.36764I
5.01980 + 8.36674I 0
u = 0.813233 + 0.191671I
a = 0.451115 + 0.585998I
b = 0.221008 + 0.395351I
−1.37591 − 0.60280I −3.98558 + 1.14738I
u = 0.813233 − 0.191671I
a = 0.451115 − 0.585998I
b = 0.221008 − 0.395351I
−1.37591 + 0.60280I −3.98558 − 1.14738I
u = −0.938275 + 0.718133I
a = 2.51455 − 0.78526I
b = −1.40735 − 0.18115I
0.20384 + 5.94515I 0
u = −0.938275 − 0.718133I
a = 2.51455 + 0.78526I
b = −1.40735 + 0.18115I
0.20384 − 5.94515I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.400647 + 0.687623I
a = 0.654871 + 0.604463I
b = 0.179950 − 0.417028I
−3.03619 − 1.14742I 1.89413 + 2.71155I
u = 0.400647 − 0.687623I
a = 0.654871 − 0.604463I
b = 0.179950 + 0.417028I
−3.03619 + 1.14742I 1.89413 − 2.71155I
u = 1.068030 + 0.569719I
a = −1.042390 + 0.124560I
b = 0.140212 − 0.417001I
−4.94353 − 3.70127I 0
u = 1.068030 − 0.569719I
a = −1.042390 − 0.124560I
b = 0.140212 + 0.417001I
−4.94353 + 3.70127I 0
u = −0.899742 + 0.819342I
a = 0.385454 − 0.035512I
b = 0.037648 − 0.622247I
4.54646 + 3.06184I 0
u = −0.899742 − 0.819342I
a = 0.385454 + 0.035512I
b = 0.037648 + 0.622247I
4.54646 − 3.06184I 0
u = 0.998592 + 0.706419I
a = 1.64774 + 1.70115I
b = −1.63453 + 0.16359I
8.81746 − 5.81212I 0
u = 0.998592 − 0.706419I
a = 1.64774 − 1.70115I
b = −1.63453 − 0.16359I
8.81746 + 5.81212I 0
u = −1.029770 + 0.700716I
a = 1.01570 − 1.60259I
b = −1.311010 − 0.312734I
7.85730 + 6.47303I 0
u = −1.029770 − 0.700716I
a = 1.01570 + 1.60259I
b = −1.311010 + 0.312734I
7.85730 − 6.47303I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.018850 + 0.717395I
a = 1.035050 + 0.243093I
b = −0.380419 + 1.097530I
−2.39802 − 9.20313I 0
u = 1.018850 − 0.717395I
a = 1.035050 − 0.243093I
b = −0.380419 − 1.097530I
−2.39802 + 9.20313I 0
u = 0.853820 + 0.915715I
a = 1.68680 + 0.32381I
b = −1.367730 − 0.197646I
9.17939 − 0.17615I 0
u = 0.853820 − 0.915715I
a = 1.68680 − 0.32381I
b = −1.367730 + 0.197646I
9.17939 + 0.17615I 0
u = 1.245010 + 0.220439I
a = −0.109370 − 0.713094I
b = 1.355710 − 0.323925I
−3.11278 − 7.32391I 0
u = 1.245010 − 0.220439I
a = −0.109370 + 0.713094I
b = 1.355710 + 0.323925I
−3.11278 + 7.32391I 0
u = 0.979010 + 0.852448I
a = −1.77380 − 1.22586I
b = 1.340370 − 0.264461I
8.77545 − 6.33818I 0
u = 0.979010 − 0.852448I
a = −1.77380 + 1.22586I
b = 1.340370 + 0.264461I
8.77545 + 6.33818I 0
u = −1.212290 + 0.502784I
a = −0.308235 + 0.936072I
b = 1.265380 − 0.139334I
−1.33691 + 1.73049I 0
u = −1.212290 − 0.502784I
a = −0.308235 − 0.936072I
b = 1.265380 + 0.139334I
−1.33691 − 1.73049I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.606543 + 0.302471I
a = 1.286880 − 0.357784I
b = −0.620734 − 0.064910I
0.958498 + 0.031162I 9.31326 − 1.44144I
u = −0.606543 − 0.302471I
a = 1.286880 + 0.357784I
b = −0.620734 + 0.064910I
0.958498 − 0.031162I 9.31326 + 1.44144I
u = −1.079880 + 0.772455I
a = −1.65559 + 1.44153I
b = 1.53699 + 0.42805I
3.7392 + 14.6910I 0
u = −1.079880 − 0.772455I
a = −1.65559 − 1.44153I
b = 1.53699 − 0.42805I
3.7392 − 14.6910I 0
u = 0.453560 + 0.034976I
a = 3.14159 + 1.95342I
b = 0.810649 − 0.339207I
−2.55565 − 1.59256I −2.14433 + 3.75237I
u = 0.453560 − 0.034976I
a = 3.14159 − 1.95342I
b = 0.810649 + 0.339207I
−2.55565 + 1.59256I −2.14433 − 3.75237I
u = −0.435557
a = 1.52560
b = −0.476437
0.883998 12.6720
u = −0.109994
a = 3.75403
b = 1.56091
7.69349 17.3580
10
II.
I
u
2
= hu
15
+u
14
+· · ·+b −2, −2u
16
−u
15
+· · ·+a +1, u
17
+u
16
+· · ·−5u
2
+1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
u
2
a
11
=
2u
16
+ u
15
+ ··· − 2u − 1
−u
15
− u
14
+ ··· + 2u + 2
a
12
=
u
16
− 3u
14
+ ··· + u − 1
−u
15
− u
14
+ ··· + 3u + 2
a
1
=
u
3
u
5
− u
3
+ u
a
5
=
−u
16
+ 4u
14
+ ··· − 3u − 1
−u
16
− u
15
+ ··· + 4u + 1
a
10
=
−u
16
− u
15
+ ··· + u − 1
u
16
+ u
15
+ ··· − 2u − 2
a
4
=
−u
16
+ 3u
14
+ ··· − 3u + 2
u
15
+ u
14
+ ··· − 2u − 2
a
9
=
−u
16
− u
15
+ ··· + 5u
2
+ u
u
16
+ u
15
+ ··· − 2u − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
16
+ 4u
15
+ 8u
14
− 9u
13
− 25u
12
+ 22u
11
+ 46u
10
− 26u
9
−
69u
8
+ 31u
7
+ 73u
6
− 19u
5
− 62u
4
+ 15u
3
+ 32u
2
− 5u − 10
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
− 7u
16
+ ··· + 10u − 1
c
2
u
17
− u
16
+ ··· + 5u
2
− 1
c
3
u
17
+ 3u
16
+ ··· + 5u − 1
c
4
, c
5
u
17
− 10u
15
+ ··· − 5u − 1
c
6
u
17
+ u
16
+ ··· − 5u
2
+ 1
c
7
u
17
+ 7u
16
+ ··· + 10u + 1
c
8
u
17
− 3u
16
+ ··· + u + 1
c
9
u
17
+ 4u
15
+ ··· − 5u − 1
c
10
u
17
− 10u
15
+ ··· − 5u + 1
c
11
u
17
+ 3u
16
+ ··· + u − 1
c
12
u
17
− u
16
+ ··· − 3u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
17
+ 13y
16
+ ··· + 6y −1
c
2
, c
6
y
17
− 7y
16
+ ··· + 10y −1
c
3
y
17
− 11y
16
+ ··· + 85y −1
c
4
, c
5
, c
10
y
17
− 20y
16
+ ··· + 21y −1
c
8
, c
11
y
17
− 9y
16
+ ··· − 7y −1
c
9
y
17
+ 8y
16
+ ··· + 7y −1
c
12
y
17
+ 7y
16
+ ··· + 9y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.06231
a = −0.790966
b = −1.44148
4.98345 8.53260
u = −1.041520 + 0.456686I
a = −0.162038 + 0.210070I
b = 0.925549 + 0.313937I
−3.48692 + 4.50075I 2.82410 − 4.82928I
u = −1.041520 − 0.456686I
a = −0.162038 − 0.210070I
b = 0.925549 − 0.313937I
−3.48692 − 4.50075I 2.82410 + 4.82928I
u = −0.781537 + 0.841435I
a = −1.83272 + 0.28651I
b = 1.54958 − 0.17210I
11.09800 + 0.42854I 10.90013 − 2.04793I
u = −0.781537 − 0.841435I
a = −1.83272 − 0.28651I
b = 1.54958 + 0.17210I
11.09800 − 0.42854I 10.90013 + 2.04793I
u = 0.625681 + 0.562777I
a = 2.71785 + 1.10655I
b = −1.041450 + 0.419962I
−1.55933 − 2.45596I 2.34776 + 4.55656I
u = 0.625681 − 0.562777I
a = 2.71785 − 1.10655I
b = −1.041450 − 0.419962I
−1.55933 + 2.45596I 2.34776 − 4.55656I
u = 1.054550 + 0.525536I
a = −0.812965 − 1.099150I
b = 1.056020 + 0.333361I
−3.00027 − 1.96632I 0.48389 + 2.00939I
u = 1.054550 − 0.525536I
a = −0.812965 + 1.099150I
b = 1.056020 − 0.333361I
−3.00027 + 1.96632I 0.48389 − 2.00939I
u = −0.817244
a = 1.31700
b = −0.218613
0.403753 −6.40080
14
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.894942 + 0.792445I
a = −0.222477 − 0.373187I
b = −0.060430 − 0.491411I
5.24194 − 2.98033I 10.80588 + 2.40227I
u = 0.894942 − 0.792445I
a = −0.222477 + 0.373187I
b = −0.060430 + 0.491411I
5.24194 + 2.98033I 10.80588 − 2.40227I
u = −0.628631 + 0.430544I
a = −1.289760 − 0.576771I
b = −0.903071 + 0.410573I
−2.04052 − 0.75125I 3.45772 − 1.80188I
u = −0.628631 − 0.430544I
a = −1.289760 + 0.576771I
b = −0.903071 − 0.410573I
−2.04052 + 0.75125I 3.45772 + 1.80188I
u = −1.002690 + 0.779560I
a = 1.57786 − 1.54150I
b = −1.49772 − 0.20521I
10.41440 + 5.64550I 10.01176 − 3.44000I
u = −1.002690 − 0.779560I
a = 1.57786 + 1.54150I
b = −1.49772 + 0.20521I
10.41440 − 5.64550I 10.01176 + 3.44000I
u = 0.513320
a = −1.47754
b = 1.60313
7.33620 −5.79430
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
17
− 7u
16
+ ··· + 10u − 1)(u
58
+ 20u
57
+ ··· + 111u + 1)
c
2
(u
17
− u
16
+ ··· + 5u
2
− 1)(u
58
− 2u
57
+ ··· + 9u − 1)
c
3
(u
17
+ 3u
16
+ ··· + 5u − 1)(u
58
− 2u
57
+ ··· − 4u + 1)
c
4
, c
5
(u
17
− 10u
15
+ ··· − 5u − 1)(u
58
+ u
57
+ ··· − 148u − 43)
c
6
(u
17
+ u
16
+ ··· − 5u
2
+ 1)(u
58
− 2u
57
+ ··· + 9u − 1)
c
7
(u
17
+ 7u
16
+ ··· + 10u + 1)(u
58
+ 20u
57
+ ··· + 111u + 1)
c
8
(u
17
− 3u
16
+ ··· + u + 1)(u
58
− 4u
57
+ ··· + 12848u − 2119)
c
9
(u
17
+ 4u
15
+ ··· − 5u − 1)(u
58
− u
57
+ ··· + 2150u + 293)
c
10
(u
17
− 10u
15
+ ··· − 5u + 1)(u
58
+ u
57
+ ··· − 148u − 43)
c
11
(u
17
+ 3u
16
+ ··· + u − 1)(u
58
− 4u
57
+ ··· + 12848u − 2119)
c
12
(u
17
− u
16
+ ··· − 3u − 1)(u
58
+ 2u
57
+ ··· + 44u + 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
17
+ 13y
16
+ ··· + 6y −1)(y
58
+ 44y
57
+ ··· − 6891y + 1)
c
2
, c
6
(y
17
− 7y
16
+ ··· + 10y −1)(y
58
− 20y
57
+ ··· − 111y + 1)
c
3
(y
17
− 11y
16
+ ··· + 85y −1)(y
58
− 56y
57
+ ··· + 254y + 1)
c
4
, c
5
, c
10
(y
17
− 20y
16
+ ··· + 21y −1)(y
58
− 57y
57
+ ··· + 10690y + 1849)
c
8
, c
11
(y
17
− 9y
16
+ ··· − 7y −1)
· (y
58
− 34y
57
+ ··· − 42707330y + 4490161)
c
9
(y
17
+ 8y
16
+ ··· + 7y −1)(y
58
+ 55y
57
+ ··· − 1142832y + 85849)
c
12
(y
17
+ 7y
16
+ ··· + 9y −1)(y
58
+ 50y
57
+ ··· + 2238y + 1)
17
