12a
0453
(K12a
0453
)
A knot diagram
1
Linearized knot diagam
3 6 10 11 7 2 12 1 5 4 9 8
Solving Sequence
7,12
8 1
2,9
6 3 5 10 11 4
c
7
c
12
c
8
c
6
c
2
c
5
c
9
c
11
c
4
c
1
, c
3
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−5.18498 × 10
53
u
79
+ 9.21176 × 10
53
u
78
+ ··· + 1.01955 × 10
53
b + 2.95698 × 10
54
,
− 5.89948 × 10
53
u
79
+ 1.14051 × 10
54
u
78
+ ··· + 3.56844 × 10
53
a + 2.68572 × 10
54
,
u
80
− 3u
79
+ ··· + 39u + 7i
I
u
2
= h−a
2
+ 2b + 2a − 1, a
4
− 4a
3
+ 8a
2
− 8a + 7, u − 1i
I
u
3
= hb
2
− b + 1, a + 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 86 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−5.18 × 10
53
u
79
+ 9.21 × 10
53
u
78
+ · · · + 1.02 × 10
53
b + 2.96 ×
10
54
, −5.90 × 10
53
u
79
+ 1.14 × 10
54
u
78
+ · · · + 3.57 × 10
53
a + 2.69 ×
10
54
, u
80
− 3u
79
+ · · · + 39u + 7i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
−u
2
a
1
=
u
−u
3
+ u
a
2
=
1.65324u
79
− 3.19611u
78
+ ··· − 25.8162u − 7.52631
5.08555u
79
− 9.03510u
78
+ ··· − 179.732u − 29.0027
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
6
=
3.74801u
79
− 5.88500u
78
+ ··· − 176.187u − 27.7131
−2.52283u
79
+ 4.91016u
78
+ ··· + 84.4363u + 12.8759
a
3
=
3.51962u
79
− 6.36792u
78
+ ··· − 79.7796u − 13.5372
−0.545223u
79
+ 0.560292u
78
+ ··· + 14.1983u + 3.10047
a
5
=
1.22518u
79
− 0.974834u
78
+ ··· − 91.7506u − 14.8372
−2.52283u
79
+ 4.91016u
78
+ ··· + 84.4363u + 12.8759
a
10
=
−1.04068u
79
+ 1.20381u
78
+ ··· + 93.2953u + 14.9388
−2.67930u
79
+ 4.50705u
78
+ ··· + 118.727u + 18.8216
a
11
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
4
=
5.22259u
79
− 7.87498u
78
+ ··· − 229.649u − 37.4148
−5.32240u
79
+ 9.42963u
78
+ ··· + 159.124u + 25.3193
(ii) Obstruction class = −1
(iii) Cusp Shapes = 32.2654u
79
− 56.5988u
78
+ ··· − 1064.29u − 181.543
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
80
+ 28u
79
+ ··· − 16u + 1
c
2
, c
6
u
80
− 2u
79
+ ··· + 6u + 1
c
3
, c
4
, c
10
u
80
− u
79
+ ··· + 12u + 4
c
7
, c
8
, c
12
u
80
− 3u
79
+ ··· + 39u + 7
c
9
u
80
+ 3u
79
+ ··· − 1980u + 44
c
11
u
80
+ 15u
79
+ ··· + 29696u + 1792
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
80
+ 52y
79
+ ··· − 232y + 1
c
2
, c
6
y
80
+ 28y
79
+ ··· − 16y + 1
c
3
, c
4
, c
10
y
80
− 75y
79
+ ··· + 112y + 16
c
7
, c
8
, c
12
y
80
− 69y
79
+ ··· + 887y + 49
c
9
y
80
− 15y
79
+ ··· − 3730320y + 1936
c
11
y
80
+ 29y
79
+ ··· − 17334272y + 3211264
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.935308 + 0.435305I
a = 0.630371 − 0.162057I
b = −0.660075 + 0.935719I
2.61436 − 3.12647I 0
u = 0.935308 − 0.435305I
a = 0.630371 + 0.162057I
b = −0.660075 − 0.935719I
2.61436 + 3.12647I 0
u = 0.796363 + 0.441945I
a = 1.309720 − 0.265298I
b = −0.680200 − 0.780701I
3.09872 + 2.06011I 0
u = 0.796363 − 0.441945I
a = 1.309720 + 0.265298I
b = −0.680200 + 0.780701I
3.09872 − 2.06011I 0
u = −0.189528 + 0.882751I
a = −1.09473 − 1.20457I
b = 0.690554 − 1.030640I
−5.17726 − 11.04350I 0
u = −0.189528 − 0.882751I
a = −1.09473 + 1.20457I
b = 0.690554 + 1.030640I
−5.17726 + 11.04350I 0
u = −1.011710 + 0.480638I
a = −1.133710 − 0.245043I
b = 0.725723 − 0.676763I
−1.53219 + 0.73845I 0
u = −1.011710 − 0.480638I
a = −1.133710 + 0.245043I
b = 0.725723 + 0.676763I
−1.53219 − 0.73845I 0
u = −0.212777 + 0.844785I
a = −0.020548 − 0.158541I
b = 0.789018 + 0.625628I
−3.96644 − 5.45425I 0
u = −0.212777 − 0.844785I
a = −0.020548 + 0.158541I
b = 0.789018 − 0.625628I
−3.96644 + 5.45425I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.230270 + 0.805111I
a = 1.23350 − 1.18135I
b = −0.683422 − 1.002370I
0.43105 + 7.58116I 0. − 8.40866I
u = 0.230270 − 0.805111I
a = 1.23350 + 1.18135I
b = −0.683422 + 1.002370I
0.43105 − 7.58116I 0. + 8.40866I
u = −0.618703 + 0.562447I
a = −0.403548 − 0.367487I
b = 0.705780 + 0.826705I
0.100507 + 0.533058I 0
u = −0.618703 − 0.562447I
a = −0.403548 + 0.367487I
b = 0.705780 − 0.826705I
0.100507 − 0.533058I 0
u = −0.569488 + 0.611464I
a = −1.42307 − 0.59607I
b = 0.707294 − 0.877786I
−0.04800 − 4.87101I 0. + 6.91200I
u = −0.569488 − 0.611464I
a = −1.42307 + 0.59607I
b = 0.707294 + 0.877786I
−0.04800 + 4.87101I 0. − 6.91200I
u = −1.074410 + 0.502940I
a = −0.588936 − 0.064477I
b = 0.678298 + 0.993914I
−2.48323 + 6.12770I 0
u = −1.074410 − 0.502940I
a = −0.588936 + 0.064477I
b = 0.678298 − 0.993914I
−2.48323 − 6.12770I 0
u = −0.103875 + 0.802998I
a = 0.00658 + 1.73987I
b = −0.067956 + 1.091330I
−10.00530 − 4.77646I −9.45758 + 3.89445I
u = −0.103875 − 0.802998I
a = 0.00658 − 1.73987I
b = −0.067956 − 1.091330I
−10.00530 + 4.77646I −9.45758 − 3.89445I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.787931 + 0.048490I
a = −0.705941 + 0.065873I
b = −0.281416 − 0.032919I
−3.20361 + 0.00081I −1.068722 + 0.283561I
u = −0.787931 − 0.048490I
a = −0.705941 − 0.065873I
b = −0.281416 + 0.032919I
−3.20361 − 0.00081I −1.068722 − 0.283561I
u = 0.266680 + 0.740721I
a = 0.020003 − 0.258774I
b = −0.741730 + 0.664409I
1.44053 + 2.13306I 2.30999 − 3.70879I
u = 0.266680 − 0.740721I
a = 0.020003 + 0.258774I
b = −0.741730 − 0.664409I
1.44053 − 2.13306I 2.30999 + 3.70879I
u = −1.159110 + 0.364671I
a = −1.146160 − 0.714549I
b = −0.019630 − 1.019000I
−6.79020 + 0.55303I 0
u = −1.159110 − 0.364671I
a = −1.146160 + 0.714549I
b = −0.019630 + 1.019000I
−6.79020 − 0.55303I 0
u = −1.225370 + 0.147900I
a = −0.780791 + 0.104181I
b = 0.498165 + 1.025250I
2.35150 + 1.12565I 0
u = −1.225370 − 0.147900I
a = −0.780791 − 0.104181I
b = 0.498165 − 1.025250I
2.35150 − 1.12565I 0
u = 1.204440 + 0.280604I
a = 1.085160 − 0.510984I
b = −0.091913 − 0.991633I
−0.58135 + 1.79490I 0
u = 1.204440 − 0.280604I
a = 1.085160 + 0.510984I
b = −0.091913 + 0.991633I
−0.58135 − 1.79490I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24271
a = 1.03607
b = −0.509667
2.60848 0
u = −1.253110 + 0.182081I
a = 0.73377 + 2.17860I
b = −0.682535 − 0.698735I
−2.04164 + 0.28284I 0
u = −1.253110 − 0.182081I
a = 0.73377 − 2.17860I
b = −0.682535 + 0.698735I
−2.04164 − 0.28284I 0
u = 0.057893 + 0.724540I
a = 0.02131 + 1.71614I
b = 0.042565 + 1.034450I
−4.07583 + 1.86281I −6.38175 − 4.14946I
u = 0.057893 − 0.724540I
a = 0.02131 − 1.71614I
b = 0.042565 − 1.034450I
−4.07583 − 1.86281I −6.38175 + 4.14946I
u = 1.301140 + 0.047924I
a = 0.851162 − 0.262258I
b = −0.353950 − 1.049460I
−0.52296 + 3.40294I 0
u = 1.301140 − 0.047924I
a = 0.851162 + 0.262258I
b = −0.353950 + 1.049460I
−0.52296 − 3.40294I 0
u = −1.278660 + 0.248936I
a = 3.06357 + 0.27921I
b = −0.663651 + 0.979925I
−2.89654 − 4.94761I 0
u = −1.278660 − 0.248936I
a = 3.06357 − 0.27921I
b = −0.663651 − 0.979925I
−2.89654 + 4.94761I 0
u = 1.291440 + 0.263515I
a = 0.690783 + 0.098744I
b = −0.552614 + 1.074470I
−2.71721 + 1.66465I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.291440 − 0.263515I
a = 0.690783 − 0.098744I
b = −0.552614 − 1.074470I
−2.71721 − 1.66465I 0
u = −0.150748 + 0.661001I
a = −0.239366 + 0.050399I
b = −0.636156 + 0.406125I
−5.24145 − 3.04676I −3.68290 + 3.88325I
u = −0.150748 − 0.661001I
a = −0.239366 − 0.050399I
b = −0.636156 − 0.406125I
−5.24145 + 3.04676I −3.68290 − 3.88325I
u = 1.32459
a = 1.00826
b = −0.636706
2.55061 0
u = −0.192008 + 0.640756I
a = −1.55726 − 1.32579I
b = 0.639720 − 0.972486I
−0.54467 − 3.78067I −2.80718 + 2.65962I
u = −0.192008 − 0.640756I
a = −1.55726 + 1.32579I
b = 0.639720 + 0.972486I
−0.54467 + 3.78067I −2.80718 − 2.65962I
u = −1.301240 + 0.293419I
a = −0.927140 − 0.527970I
b = 0.131001 − 1.090690I
0.16791 − 5.53918I 0
u = −1.301240 − 0.293419I
a = −0.927140 + 0.527970I
b = 0.131001 + 1.090690I
0.16791 + 5.53918I 0
u = −1.325140 + 0.190326I
a = −1.008040 − 0.071508I
b = 0.691635 − 0.250001I
4.56651 − 3.15280I 0
u = −1.325140 − 0.190326I
a = −1.008040 + 0.071508I
b = 0.691635 + 0.250001I
4.56651 + 3.15280I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.018029 + 0.660145I
a = 1.26111 − 1.82685I
b = −0.594114 − 1.016650I
−6.81361 + 1.68208I −6.85681 − 2.22241I
u = −0.018029 − 0.660145I
a = 1.26111 + 1.82685I
b = −0.594114 + 1.016650I
−6.81361 − 1.68208I −6.85681 + 2.22241I
u = 1.345210 + 0.277688I
a = 0.996087 − 0.105982I
b = −0.759916 − 0.328235I
−0.53214 + 6.49202I 0
u = 1.345210 − 0.277688I
a = 0.996087 + 0.105982I
b = −0.759916 + 0.328235I
−0.53214 − 6.49202I 0
u = 1.333900 + 0.342685I
a = 0.875466 − 0.582857I
b = −0.106119 − 1.138050I
−5.48737 + 8.90152I 0
u = 1.333900 − 0.342685I
a = 0.875466 + 0.582857I
b = −0.106119 + 1.138050I
−5.48737 − 8.90152I 0
u = 1.373660 + 0.225693I
a = −0.91497 + 1.44154I
b = 0.784938 − 0.698652I
5.34197 + 1.50444I 0
u = 1.373660 − 0.225693I
a = −0.91497 − 1.44154I
b = 0.784938 + 0.698652I
5.34197 − 1.50444I 0
u = 1.373070 + 0.272297I
a = −2.49390 + 0.15890I
b = 0.709636 + 0.998983I
4.42932 + 7.15478I 0
u = 1.373070 − 0.272297I
a = −2.49390 − 0.15890I
b = 0.709636 − 0.998983I
4.42932 − 7.15478I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.40021 + 0.29892I
a = 0.73966 + 1.22831I
b = −0.826565 − 0.658485I
6.72479 − 5.90072I 0
u = −1.40021 − 0.29892I
a = 0.73966 − 1.22831I
b = −0.826565 + 0.658485I
6.72479 + 5.90072I 0
u = −1.39785 + 0.33105I
a = 2.28662 + 0.34323I
b = −0.716310 + 1.030440I
5.59576 − 11.68240I 0
u = −1.39785 − 0.33105I
a = 2.28662 − 0.34323I
b = −0.716310 − 1.030440I
5.59576 + 11.68240I 0
u = 1.39606 + 0.35207I
a = −0.618370 + 1.142030I
b = 0.843709 − 0.622934I
1.13138 + 9.76187I 0
u = 1.39606 − 0.35207I
a = −0.618370 − 1.142030I
b = 0.843709 + 0.622934I
1.13138 − 9.76187I 0
u = 1.39337 + 0.37541I
a = −2.20827 + 0.49155I
b = 0.709809 + 1.051400I
−0.1684 + 15.5626I 0
u = 1.39337 − 0.37541I
a = −2.20827 − 0.49155I
b = 0.709809 − 1.051400I
−0.1684 − 15.5626I 0
u = 1.45673 + 0.06706I
a = −1.57725 + 1.11214I
b = 0.800472 − 0.820840I
6.99812 + 1.19577I 0
u = 1.45673 − 0.06706I
a = −1.57725 − 1.11214I
b = 0.800472 + 0.820840I
6.99812 − 1.19577I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.46037 + 0.10688I
a = −2.08437 − 0.58760I
b = 0.774088 + 0.921625I
6.69269 + 7.09251I 0
u = 1.46037 − 0.10688I
a = −2.08437 + 0.58760I
b = 0.774088 − 0.921625I
6.69269 − 7.09251I 0
u = −1.46462 + 0.02186I
a = 1.86638 − 0.87686I
b = −0.789181 + 0.874587I
10.49570 − 2.95503I 0
u = −1.46462 − 0.02186I
a = 1.86638 + 0.87686I
b = −0.789181 − 0.874587I
10.49570 + 2.95503I 0
u = −0.228484 + 0.437922I
a = 0.219585 − 0.751726I
b = 0.600073 + 0.715905I
0.276256 + 1.175000I −2.28134 − 3.06853I
u = −0.228484 − 0.437922I
a = 0.219585 + 0.751726I
b = 0.600073 − 0.715905I
0.276256 − 1.175000I −2.28134 + 3.06853I
u = 0.164199 + 0.364320I
a = 0.313161 + 0.253667I
b = 0.314435 + 0.339122I
0.038038 + 0.896508I 0.98315 − 7.41629I
u = 0.164199 − 0.364320I
a = 0.313161 − 0.253667I
b = 0.314435 − 0.339122I
0.038038 − 0.896508I 0.98315 + 7.41629I
u = −0.200756 + 0.298050I
a = −1.72839 + 2.08553I
b = −0.356274 + 0.854401I
−5.18017 − 2.38847I −8.45347 + 4.78125I
u = −0.200756 − 0.298050I
a = −1.72839 − 2.08553I
b = −0.356274 − 0.854401I
−5.18017 + 2.38847I −8.45347 − 4.78125I
12
II. I
u
2
= h−a
2
+ 2b + 2a − 1, a
4
− 4a
3
+ 8a
2
− 8a + 7, u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
1
a
8
=
1
−1
a
1
=
1
0
a
2
=
a
1
2
a
2
− a +
1
2
a
9
=
0
−1
a
6
=
1
2
a
3
− a
2
+
1
2
a + 1
−
1
2
a
2
+ a −
3
2
a
3
=
−
1
2
a
3
+
3
2
a
2
−
3
2
a +
1
2
1
2
a
2
− a +
3
2
a
5
=
1
2
a
3
−
3
2
a
2
+
3
2
a −
1
2
−
1
2
a
2
+ a −
3
2
a
10
=
2
a − 2
a
11
=
0
1
a
4
=
1
2
a
3
−
3
2
a
2
+
3
2
a −
1
2
−
1
2
a
3
+ a
2
−
1
2
a − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2a
2
− 4a + 2
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
− u + 1)
2
c
3
, c
4
, c
9
c
10
(u
2
− 2)
2
c
6
(u
2
+ u + 1)
2
c
7
, c
8
(u − 1)
4
c
11
u
4
c
12
(u + 1)
4
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
(y
2
+ y + 1)
2
c
3
, c
4
, c
9
c
10
(y −2)
4
c
7
, c
8
, c
12
(y −1)
4
c
11
y
4
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.292893 + 1.224750I
b = −0.500000 − 0.866025I
−3.28987 + 2.02988I −2.00000 − 3.46410I
u = 1.00000
a = 0.292893 − 1.224750I
b = −0.500000 + 0.866025I
−3.28987 − 2.02988I −2.00000 + 3.46410I
u = 1.00000
a = 1.70711 + 1.22474I
b = −0.500000 + 0.866025I
−3.28987 − 2.02988I −2.00000 + 3.46410I
u = 1.00000
a = 1.70711 − 1.22474I
b = −0.500000 − 0.866025I
−3.28987 + 2.02988I −2.00000 − 3.46410I
16
III. I
u
3
= hb
2
− b + 1, a + 1, u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
−1
a
8
=
1
−1
a
1
=
−1
0
a
2
=
−1
b
a
9
=
0
−1
a
6
=
−b + 1
b − 1
a
3
=
0
b − 1
a
5
=
0
b − 1
a
10
=
0
−1
a
11
=
0
−1
a
4
=
0
b − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4b + 2
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u
2
− u + 1
c
2
u
2
+ u + 1
c
3
, c
4
, c
9
c
10
, c
11
u
2
c
7
, c
8
(u + 1)
2
c
12
(u − 1)
2
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
2
+ y + 1
c
3
, c
4
, c
9
c
10
, c
11
y
2
c
7
, c
8
, c
12
(y −1)
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 0. − 3.46410I
u = −1.00000
a = −1.00000
b = 0.500000 − 0.866025I
1.64493 − 2.02988I 0. + 3.46410I
20
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
((u
2
− u + 1)
3
)(u
80
+ 28u
79
+ ··· − 16u + 1)
c
2
((u
2
− u + 1)
2
)(u
2
+ u + 1)(u
80
− 2u
79
+ ··· + 6u + 1)
c
3
, c
4
, c
10
u
2
(u
2
− 2)
2
(u
80
− u
79
+ ··· + 12u + 4)
c
6
(u
2
− u + 1)(u
2
+ u + 1)
2
(u
80
− 2u
79
+ ··· + 6u + 1)
c
7
, c
8
((u − 1)
4
)(u + 1)
2
(u
80
− 3u
79
+ ··· + 39u + 7)
c
9
u
2
(u
2
− 2)
2
(u
80
+ 3u
79
+ ··· − 1980u + 44)
c
11
u
6
(u
80
+ 15u
79
+ ··· + 29696u + 1792)
c
12
((u − 1)
2
)(u + 1)
4
(u
80
− 3u
79
+ ··· + 39u + 7)
21
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y
2
+ y + 1)
3
)(y
80
+ 52y
79
+ ··· − 232y + 1)
c
2
, c
6
((y
2
+ y + 1)
3
)(y
80
+ 28y
79
+ ··· − 16y + 1)
c
3
, c
4
, c
10
y
2
(y −2)
4
(y
80
− 75y
79
+ ··· + 112y + 16)
c
7
, c
8
, c
12
((y −1)
6
)(y
80
− 69y
79
+ ··· + 887y + 49)
c
9
y
2
(y −2)
4
(y
80
− 15y
79
+ ··· − 3730320y + 1936)
c
11
y
6
(y
80
+ 29y
79
+ ··· − 1.73343 × 10
7
y + 3211264)
22
