12a
1256
(K12a
1256
)
A knot diagram
1
Linearized knot diagam
5 7 8 1 11 10 3 12 2 6 4 9
Solving Sequence
5,11 2,6
1 4 12 10 7 3 9 8
c
5
c
1
c
4
c
11
c
10
c
6
c
2
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−5.64042 × 10
124
u
74
+ 9.11620 × 10
125
u
73
+ ··· + 2.70123 × 10
127
b + 2.73247 × 10
127
,
4.87882 × 10
130
u
74
− 9.04411 × 10
130
u
73
+ ··· + 1.18314 × 10
130
a + 2.77838 × 10
131
, u
75
− 2u
74
+ ··· + 2u − 1i
I
u
2
= hb − 1, 3a
2
− 4au − 2a + 2u − 1, u
2
+ u + 1i
I
u
3
= hb + 1, 3a − 2u + 1, u
2
− u + 1i
* 3 irreducible components of dim
C
= 0, with total 81 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−5.64 × 10
124
u
74
+ 9.12 × 10
125
u
73
+ · · · + 2.70 × 10
127
b + 2.73 ×
10
127
, 4.88 × 10
130
u
74
− 9.04 × 10
130
u
73
+ · · · + 1.18 × 10
130
a + 2.78 ×
10
131
, u
75
− 2u
74
+ · · · + 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
2
=
−4.12363u
74
+ 7.64417u
73
+ ··· − 122.730u − 23.4832
0.00208809u
74
− 0.0337483u
73
+ ··· − 0.242348u − 1.01157
a
6
=
1
u
2
a
1
=
−4.12154u
74
+ 7.61042u
73
+ ··· − 122.972u − 24.4947
0.00208809u
74
− 0.0337483u
73
+ ··· − 0.242348u − 1.01157
a
4
=
−4.08530u
74
+ 7.78212u
73
+ ··· − 132.075u − 25.7741
0.258927u
74
− 0.465895u
73
+ ··· − 1.90127u − 0.653622
a
12
=
−36.6756u
74
+ 67.7990u
73
+ ··· − 1097.47u − 238.095
−0.354825u
74
+ 0.662723u
73
+ ··· − 20.6852u − 4.84533
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
3
=
−3.85208u
74
+ 7.11086u
73
+ ··· − 116.744u − 23.2694
−0.0278066u
74
+ 0.0198137u
73
+ ··· + 0.251746u − 0.982130
a
9
=
−32.4337u
74
+ 60.1196u
73
+ ··· − 960.246u − 207.404
−0.475413u
74
+ 0.935445u
73
+ ··· − 15.5725u − 4.66742
a
8
=
5.31272u
74
− 9.69537u
73
+ ··· + 154.183u + 37.8595
0.280173u
74
− 0.556071u
73
+ ··· + 5.64826u + 0.243813
(ii) Obstruction class = −1
(iii) Cusp Shapes = −7.72579u
74
+ 14.3690u
73
+ ··· − 174.823u − 40.1034
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
75
+ 3u
74
+ ··· − 25u − 63
c
2
, c
3
, c
7
u
75
+ 3u
74
+ ··· − 180u + 36
c
5
, c
6
, c
10
u
75
+ 2u
74
+ ··· + 2u + 1
c
8
, c
12
u
75
− 4u
74
+ ··· − 12u − 1
c
9
219(219u
75
+ 2561u
74
+ ··· − 4.11026 × 10
7
u + 9603649)
c
11
219(219u
75
− 3206u
74
+ ··· + 2099664u − 1065445)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
75
− 39y
74
+ ··· − 12479y −3969
c
2
, c
3
, c
7
y
75
− 85y
74
+ ··· + 21456y −1296
c
5
, c
6
, c
10
y
75
+ 74y
74
+ ··· + 48y −1
c
8
, c
12
y
75
− 46y
74
+ ··· − 16y −1
c
9
47961(47961y
75
− 5944207y
74
+ ··· + 1.71498 × 10
15
y −9.22301 × 10
13
)
c
11
47961
· (47961y
75
− 1905190y
74
+ ··· − 29363839817524y −1135173048025)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.876275 + 0.562290I
a = 0.705559 − 0.938222I
b = 1.235900 + 0.607876I
8.0251 − 11.7833I 0
u = 0.876275 − 0.562290I
a = 0.705559 + 0.938222I
b = 1.235900 − 0.607876I
8.0251 + 11.7833I 0
u = 0.645654 + 0.681122I
a = −0.141459 + 0.379704I
b = 0.211314 − 0.993029I
11.12440 − 6.04518I 0
u = 0.645654 − 0.681122I
a = −0.141459 − 0.379704I
b = 0.211314 + 0.993029I
11.12440 + 6.04518I 0
u = 0.869381 + 0.333279I
a = 0.989488 − 0.311346I
b = 0.475379 + 0.683609I
9.93704 + 1.09383I 0
u = 0.869381 − 0.333279I
a = 0.989488 + 0.311346I
b = 0.475379 − 0.683609I
9.93704 − 1.09383I 0
u = 0.887065 + 0.671795I
a = 0.080791 − 0.173949I
b = 1.043970 − 0.564127I
8.26968 + 5.91564I 0
u = 0.887065 − 0.671795I
a = 0.080791 + 0.173949I
b = 1.043970 + 0.564127I
8.26968 − 5.91564I 0
u = 0.617922 + 0.932156I
a = −0.091814 + 0.364728I
b = −0.900452 + 0.105348I
−1.38488 − 1.76088I 0
u = 0.617922 − 0.932156I
a = −0.091814 − 0.364728I
b = −0.900452 − 0.105348I
−1.38488 + 1.76088I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.630870 + 0.610023I
a = 0.209312 − 0.433808I
b = −1.020630 − 0.440018I
1.00908 − 3.90722I 0
u = −0.630870 − 0.610023I
a = 0.209312 + 0.433808I
b = −1.020630 + 0.440018I
1.00908 + 3.90722I 0
u = 0.781439 + 0.396315I
a = −0.681940 + 0.719996I
b = −1.063160 − 0.318737I
−2.82400 − 3.24021I 0
u = 0.781439 − 0.396315I
a = −0.681940 − 0.719996I
b = −1.063160 + 0.318737I
−2.82400 + 3.24021I 0
u = −0.200166 + 1.112240I
a = −0.555735 + 0.472886I
b = 1.303290 − 0.017947I
0.00341 + 2.25841I 0
u = −0.200166 − 1.112240I
a = −0.555735 − 0.472886I
b = 1.303290 + 0.017947I
0.00341 − 2.25841I 0
u = −1.008550 + 0.522926I
a = 0.622760 + 0.595151I
b = 0.988198 − 0.455399I
3.78624 + 4.97496I 0
u = −1.008550 − 0.522926I
a = 0.622760 − 0.595151I
b = 0.988198 + 0.455399I
3.78624 − 4.97496I 0
u = −0.715363 + 0.463662I
a = −0.662169 − 0.968737I
b = −1.207050 + 0.599270I
0.55943 + 8.51493I 0
u = −0.715363 − 0.463662I
a = −0.662169 + 0.968737I
b = −1.207050 − 0.599270I
0.55943 − 8.51493I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.295177 + 0.790335I
a = 1.085820 − 0.610170I
b = −0.174335 + 0.231695I
5.11741 + 2.09681I 0
u = −0.295177 − 0.790335I
a = 1.085820 + 0.610170I
b = −0.174335 − 0.231695I
5.11741 − 2.09681I 0
u = −0.820615 + 0.920549I
a = 0.221481 + 0.060826I
b = 0.669207 + 0.351108I
4.93623 + 1.38385I 0
u = −0.820615 − 0.920549I
a = 0.221481 − 0.060826I
b = 0.669207 − 0.351108I
4.93623 − 1.38385I 0
u = 0.476482 + 0.349271I
a = 0.536263 − 0.913748I
b = 1.179300 + 0.646970I
−0.37836 − 3.63250I 0.18820 + 7.11754I
u = 0.476482 − 0.349271I
a = 0.536263 + 0.913748I
b = 1.179300 − 0.646970I
−0.37836 + 3.63250I 0.18820 − 7.11754I
u = −0.12416 + 1.40957I
a = −0.49322 + 1.49192I
b = 1.156720 − 0.514868I
2.45398 + 2.51607I 0
u = −0.12416 − 1.40957I
a = −0.49322 − 1.49192I
b = 1.156720 + 0.514868I
2.45398 − 2.51607I 0
u = 0.06595 + 1.41816I
a = −1.06518 − 1.97261I
b = 0.726970 + 0.104746I
5.61190 − 0.24157I 0
u = 0.06595 − 1.41816I
a = −1.06518 + 1.97261I
b = 0.726970 − 0.104746I
5.61190 + 0.24157I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.21401 + 1.41679I
a = −0.32012 − 1.44823I
b = −0.818643 + 0.496806I
7.77805 + 2.57329I 0
u = −0.21401 − 1.41679I
a = −0.32012 + 1.44823I
b = −0.818643 − 0.496806I
7.77805 − 2.57329I 0
u = −0.318983 + 0.457075I
a = 0.073646 + 0.417549I
b = −0.203568 − 1.059760I
3.56627 + 2.77566I 7.45932 − 8.59352I
u = −0.318983 − 0.457075I
a = 0.073646 − 0.417549I
b = −0.203568 + 1.059760I
3.56627 − 2.77566I 7.45932 + 8.59352I
u = −0.02663 + 1.44552I
a = 0.33553 + 1.40138I
b = −1.237260 − 0.349849I
8.55739 + 0.60577I 0
u = −0.02663 − 1.44552I
a = 0.33553 − 1.40138I
b = −1.237260 + 0.349849I
8.55739 − 0.60577I 0
u = −0.509960 + 0.186306I
a = −1.86666 − 0.33094I
b = −0.427581 + 0.394554I
2.65696 − 0.21953I 3.66100 − 2.90140I
u = −0.509960 − 0.186306I
a = −1.86666 + 0.33094I
b = −0.427581 − 0.394554I
2.65696 + 0.21953I 3.66100 + 2.90140I
u = 0.13294 + 1.45442I
a = −0.70369 − 1.75188I
b = 1.30725 + 1.04690I
5.48724 − 5.77129I 0
u = 0.13294 − 1.45442I
a = −0.70369 + 1.75188I
b = 1.30725 − 1.04690I
5.48724 + 5.77129I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.501413 + 0.173375I
a = 1.18080 + 0.88385I
b = 1.178950 − 0.178137I
−2.66084 + 0.41205I −3.72410 + 2.82659I
u = −0.501413 − 0.173375I
a = 1.18080 − 0.88385I
b = 1.178950 + 0.178137I
−2.66084 − 0.41205I −3.72410 − 2.82659I
u = 0.07244 + 1.46773I
a = 0.129197 − 1.212420I
b = −0.164854 + 0.790449I
6.08949 − 1.95344I 0
u = 0.07244 − 1.46773I
a = 0.129197 + 1.212420I
b = −0.164854 − 0.790449I
6.08949 + 1.95344I 0
u = −0.07739 + 1.48364I
a = 0.47777 + 1.88720I
b = −0.38939 − 1.65589I
9.90796 + 4.12833I 0
u = −0.07739 − 1.48364I
a = 0.47777 − 1.88720I
b = −0.38939 + 1.65589I
9.90796 − 4.12833I 0
u = 0.02616 + 1.48660I
a = 1.37108 − 0.93149I
b = −2.00675 + 0.80274I
12.23390 − 1.45071I 0
u = 0.02616 − 1.48660I
a = 1.37108 + 0.93149I
b = −2.00675 − 0.80274I
12.23390 + 1.45071I 0
u = 0.25514 + 1.48012I
a = 0.25389 + 1.43353I
b = −1.160740 − 0.553557I
3.26790 − 6.92996I 0
u = 0.25514 − 1.48012I
a = 0.25389 − 1.43353I
b = −1.160740 + 0.553557I
3.26790 + 6.92996I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.11892 + 1.50576I
a = 0.824177 + 0.454392I
b = −0.704332 − 0.473961I
8.08900 − 1.43331I 0
u = −0.11892 − 1.50576I
a = 0.824177 − 0.454392I
b = −0.704332 + 0.473961I
8.08900 + 1.43331I 0
u = −0.24148 + 1.50171I
a = 0.38547 − 1.69930I
b = −1.30554 + 0.78742I
6.96349 + 11.98440I 0
u = −0.24148 − 1.50171I
a = 0.38547 + 1.69930I
b = −1.30554 − 0.78742I
6.96349 − 11.98440I 0
u = 0.290488 + 0.380055I
a = −1.87771 − 2.07115I
b = 0.958564 − 0.242001I
0.033322 + 0.939980I −0.28548 + 8.35060I
u = 0.290488 − 0.380055I
a = −1.87771 + 2.07115I
b = 0.958564 + 0.242001I
0.033322 − 0.939980I −0.28548 − 8.35060I
u = 0.246631 + 0.362441I
a = −0.412210 − 0.560123I
b = 0.095378 + 0.306837I
0.022305 − 0.818920I 0.60380 + 8.33349I
u = 0.246631 − 0.362441I
a = −0.412210 + 0.560123I
b = 0.095378 − 0.306837I
0.022305 + 0.818920I 0.60380 − 8.33349I
u = 0.38353 + 1.51514I
a = 0.343106 − 1.142940I
b = 0.865449 + 0.692400I
15.8330 − 3.6081I 0
u = 0.38353 − 1.51514I
a = 0.343106 + 1.142940I
b = 0.865449 − 0.692400I
15.8330 + 3.6081I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.129313 + 0.412192I
a = −0.761012 − 0.474573I
b = −1.48075 + 0.50912I
5.94006 − 0.95777I 13.8115 + 6.3144I
u = 0.129313 − 0.412192I
a = −0.761012 + 0.474573I
b = −1.48075 − 0.50912I
5.94006 + 0.95777I 13.8115 − 6.3144I
u = 0.19790 + 1.57054I
a = −0.45227 + 1.42247I
b = 0.201938 − 1.324450I
18.5743 − 9.1429I 0
u = 0.19790 − 1.57054I
a = −0.45227 − 1.42247I
b = 0.201938 + 1.324450I
18.5743 + 9.1429I 0
u = 0.30448 + 1.56248I
a = −0.18734 − 1.61409I
b = 1.35622 + 0.69432I
14.9505 − 16.1112I 0
u = 0.30448 − 1.56248I
a = −0.18734 + 1.61409I
b = 1.35622 − 0.69432I
14.9505 + 16.1112I 0
u = −0.17402 + 1.59685I
a = −0.163007 − 1.070100I
b = 0.276738 + 0.882519I
13.38520 + 4.43309I 0
u = −0.17402 − 1.59685I
a = −0.163007 + 1.070100I
b = 0.276738 − 0.882519I
13.38520 − 4.43309I 0
u = −0.33867 + 1.58027I
a = −0.107663 + 1.311530I
b = 1.184290 − 0.597745I
10.6747 + 9.8638I 0
u = −0.33867 − 1.58027I
a = −0.107663 − 1.311530I
b = 1.184290 + 0.597745I
10.6747 − 9.8638I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.22442 + 1.64310I
a = −0.436152 + 0.582597I
b = 0.746431 − 0.718512I
16.1817 + 1.7243I 0
u = 0.22442 − 1.64310I
a = −0.436152 − 0.582597I
b = 0.746431 + 0.718512I
16.1817 − 1.7243I 0
u = −0.243105 + 0.095142I
a = 2.59239 + 2.02032I
b = −0.861746 − 0.111814I
3.19638 + 0.01813I 0.93086 + 1.13023I
u = −0.243105 − 0.095142I
a = 2.59239 − 2.02032I
b = −0.861746 + 0.111814I
3.19638 − 0.01813I 0.93086 − 1.13023I
u = 0.151778
a = −54.3761
b = −1.06936
4.98695 −85.6430
12
II. I
u
2
= hb − 1, 3a
2
− 4au − 2a + 2u − 1, u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
2
=
a
1
a
6
=
1
−u − 1
a
1
=
a + 1
1
a
4
=
−a
−1
a
12
=
2
3
au +
4
3
a − u −
2
3
−au + u
a
10
=
u
u + 1
a
7
=
−u
−u − 2
a
3
=
−au + 2a − u − 1
−3au − 3a + u
a
9
=
1
3
au +
2
3
a +
2
3
u − 1
−au − a + 2u + 1
a
8
=
−au +
4
3
u −
4
3
−au − 2a + 2u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u + 1)
4
c
2
, c
3
, c
7
(u
2
− 2)
2
c
4
(u − 1)
4
c
5
, c
6
, c
8
(u
2
+ u + 1)
2
c
9
9(9u
4
+ 12u + 7)
c
10
, c
12
(u
2
− u + 1)
2
c
11
9(9u
4
+ 18u
3
+ 27u
2
+ 18u + 7)
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y −1)
4
c
2
, c
3
, c
7
(y −2)
4
c
5
, c
6
, c
8
c
10
, c
12
(y
2
+ y + 1)
2
c
9
81(81y
4
+ 126y
2
− 144y + 49)
c
11
81(81y
4
+ 162y
3
+ 207y
2
+ 54y + 49)
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.707107 + 0.985599I
b = 1.00000
3.28987 + 2.02988I 2.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = 0.707107 + 0.169102I
b = 1.00000
3.28987 + 2.02988I 2.00000 − 3.46410I
u = −0.500000 − 0.866025I
a = −0.707107 − 0.985599I
b = 1.00000
3.28987 − 2.02988I 2.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.707107 − 0.169102I
b = 1.00000
3.28987 − 2.02988I 2.00000 + 3.46410I
16
III. I
u
3
= hb + 1, 3a − 2u + 1, u
2
− u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
2
=
2
3
u −
1
3
−1
a
6
=
1
u − 1
a
1
=
2
3
u −
4
3
−1
a
4
=
2
3
u −
1
3
−1
a
12
=
1
3
u
4
3
u −
2
3
a
10
=
u
u − 1
a
7
=
u
u − 2
a
3
=
2
3
u −
1
3
−1
a
9
=
u +
1
3
5
3
u −
4
3
a
8
=
u
u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes =
28
3
u − 10
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
c
2
, c
3
, c
7
u
2
c
4
(u + 1)
2
c
5
, c
6
, c
12
u
2
− u + 1
c
8
, c
10
u
2
+ u + 1
c
9
3(3u
2
+ 1)
c
11
3(3u
2
− 3u + 1)
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y −1)
2
c
2
, c
3
, c
7
y
2
c
5
, c
6
, c
8
c
10
, c
12
y
2
+ y + 1
c
9
9(3y + 1)
2
c
11
9(9y
2
− 3y + 1)
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.577350I
b = −1.00000
−1.64493 − 2.02988I −5.33333 + 8.08290I
u = 0.500000 − 0.866025I
a = − 0.577350I
b = −1.00000
−1.64493 + 2.02988I −5.33333 − 8.08290I
20
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u + 1)
4
(u
75
+ 3u
74
+ ··· − 25u − 63)
c
2
, c
3
, c
7
u
2
(u
2
− 2)
2
(u
75
+ 3u
74
+ ··· − 180u + 36)
c
4
((u − 1)
4
)(u + 1)
2
(u
75
+ 3u
74
+ ··· − 25u − 63)
c
5
, c
6
(u
2
− u + 1)(u
2
+ u + 1)
2
(u
75
+ 2u
74
+ ··· + 2u + 1)
c
8
((u
2
+ u + 1)
3
)(u
75
− 4u
74
+ ··· − 12u − 1)
c
9
5913(3u
2
+ 1)(9u
4
+ 12u + 7)
· (219u
75
+ 2561u
74
+ ··· − 41102589u + 9603649)
c
10
((u
2
− u + 1)
2
)(u
2
+ u + 1)(u
75
+ 2u
74
+ ··· + 2u + 1)
c
11
5913(3u
2
− 3u + 1)(9u
4
+ 18u
3
+ 27u
2
+ 18u + 7)
· (219u
75
− 3206u
74
+ ··· + 2099664u − 1065445)
c
12
((u
2
− u + 1)
3
)(u
75
− 4u
74
+ ··· − 12u − 1)
21
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
6
)(y
75
− 39y
74
+ ··· − 12479y −3969)
c
2
, c
3
, c
7
y
2
(y −2)
4
(y
75
− 85y
74
+ ··· + 21456y −1296)
c
5
, c
6
, c
10
((y
2
+ y + 1)
3
)(y
75
+ 74y
74
+ ··· + 48y −1)
c
8
, c
12
((y
2
+ y + 1)
3
)(y
75
− 46y
74
+ ··· − 16y −1)
c
9
34963569(3y + 1)
2
(81y
4
+ 126y
2
− 144y + 49)
· (4.80 × 10
4
y
75
− 5.94 × 10
6
y
74
+ ··· + 1.71 × 10
15
y −9.22 × 10
13
)
c
11
34963569(9y
2
− 3y + 1)(81y
4
+ 162y
3
+ 207y
2
+ 54y + 49)
· (47961y
75
− 1905190y
74
+ ··· − 29363839817524y −1135173048025)
22
