12a
0824
(K12a
0824
)
A knot diagram
1
Linearized knot diagam
4 5 8 2 10 11 3 12 1 6 7 9
Solving Sequence
5,10
6 11
3,7
8 12 2 4 1 9
c
5
c
10
c
6
c
7
c
11
c
2
c
4
c
1
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h2.98047 × 10
56
u
61
− 7.28716 × 10
56
u
60
+ ··· + 2.15534 × 10
55
b − 3.50669 × 10
57
,
− 3.70578 × 10
56
u
61
+ 9.04815 × 10
56
u
60
+ ··· + 2.15534 × 10
55
a + 4.10393 × 10
57
,
u
62
− 2u
61
+ ··· − 12u − 4i
I
u
2
= hb + 1, u
2
+ a + u − 2, u
3
+ u
2
− 2u − 1i
I
u
3
= hau + b + 2a − 1, 2a
2
+ au − 2a + 2u − 3, u
2
− 2i
I
v
1
= ha, b + v −2, v
2
− 3v + 1i
* 4 irreducible components of dim
C
= 0, with total 71 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
= h2.98 × 10
56
u
61
− 7.29 × 10
56
u
60
+ · · · + 2.16 × 10
55
b − 3.51 ×
10
57
, −3.71 × 10
56
u
61
+ 9.05 × 10
56
u
60
+ · · · + 2.16 × 10
55
a + 4.10 ×
10
57
, u
62
− 2u
61
+ · · · − 12u − 4i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
−u
3
+ u
a
3
=
17.1935u
61
− 41.9802u
60
+ ··· − 59.0803u − 190.408
−13.8283u
61
+ 33.8098u
60
+ ··· + 35.7338u + 162.698
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
8
=
1.21636u
61
− 2.59080u
60
+ ··· − 18.6035u − 0.834246
−22.4729u
61
+ 50.9927u
60
+ ··· + 34.3595u + 239.437
a
12
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
2
=
3.36518u
61
− 8.17039u
60
+ ··· − 23.3465u − 27.7096
−13.8283u
61
+ 33.8098u
60
+ ··· + 35.7338u + 162.698
a
4
=
38.7754u
61
− 92.5739u
60
+ ··· − 86.8745u − 442.022
17.6227u
61
− 43.4791u
60
+ ··· − 45.7098u − 215.395
a
1
=
39.6334u
61
− 94.3136u
60
+ ··· − 95.8498u − 449.301
17.1297u
61
− 42.0208u
60
+ ··· − 39.9181u − 209.663
a
9
=
−8.92692u
61
+ 20.4057u
60
+ ··· − 0.725611u + 105.357
−26.0460u
61
+ 58.7518u
60
+ ··· + 38.2629u + 274.601
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2.64973u
61
+ 4.67865u
60
+ ··· − 82.0084u + 32.7971
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
62
− 7u
61
+ ··· + 29u + 1
c
3
, c
7
u
62
− 2u
61
+ ··· + 108u − 8
c
5
, c
6
, c
10
c
11
u
62
+ 2u
61
+ ··· + 12u − 4
c
8
, c
9
, c
12
u
62
− 4u
61
+ ··· + 81u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
62
− 57y
61
+ ··· − 427y + 1
c
3
, c
7
y
62
− 30y
61
+ ··· − 4688y + 64
c
5
, c
6
, c
10
c
11
y
62
− 74y
61
+ ··· − 624y + 16
c
8
, c
9
, c
12
y
62
− 60y
61
+ ··· − 3519y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.728927 + 0.693480I
a = 0.50031 − 1.40841I
b = 1.41056 + 0.37112I
−0.16183 − 11.03280I 0
u = −0.728927 − 0.693480I
a = 0.50031 + 1.40841I
b = 1.41056 − 0.37112I
−0.16183 + 11.03280I 0
u = 0.979023 + 0.319186I
a = 0.687788 + 0.612198I
b = 0.174252 − 0.593165I
6.85692 + 1.08905I 0
u = 0.979023 − 0.319186I
a = 0.687788 − 0.612198I
b = 0.174252 + 0.593165I
6.85692 − 1.08905I 0
u = −0.869463 + 0.382228I
a = −0.78515 − 1.20030I
b = 1.394320 + 0.085865I
−4.30575 − 1.13280I 0
u = −0.869463 − 0.382228I
a = −0.78515 + 1.20030I
b = 1.394320 − 0.085865I
−4.30575 + 1.13280I 0
u = 0.744058 + 0.573909I
a = 0.05054 + 1.60912I
b = 1.41065 − 0.23730I
−5.93385 + 6.46887I 0. − 6.93804I
u = 0.744058 − 0.573909I
a = 0.05054 − 1.60912I
b = 1.41065 + 0.23730I
−5.93385 − 6.46887I 0. + 6.93804I
u = −0.757731 + 0.545713I
a = −0.418942 + 0.852081I
b = −0.234673 − 0.886336I
5.04847 − 6.51467I 0. + 6.90536I
u = −0.757731 − 0.545713I
a = −0.418942 − 0.852081I
b = −0.234673 + 0.886336I
5.04847 + 6.51467I 0. − 6.90536I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.294207 + 0.839087I
a = 0.678495 − 0.208859I
b = 1.351380 − 0.291230I
−1.47807 + 5.97196I 0.82075 − 3.88105I
u = −0.294207 − 0.839087I
a = 0.678495 + 0.208859I
b = 1.351380 + 0.291230I
−1.47807 − 5.97196I 0.82075 + 3.88105I
u = 0.700864 + 0.454130I
a = −1.08574 − 1.35076I
b = −1.381210 + 0.269324I
1.85807 + 4.31924I 4.65831 − 4.87668I
u = 0.700864 − 0.454130I
a = −1.08574 + 1.35076I
b = −1.381210 − 0.269324I
1.85807 − 4.31924I 4.65831 + 4.87668I
u = −0.720139 + 0.323549I
a = −0.041734 − 0.370738I
b = −0.990063 + 0.574650I
2.74398 − 1.42912I 4.95658 + 3.06315I
u = −0.720139 − 0.323549I
a = −0.041734 + 0.370738I
b = −0.990063 − 0.574650I
2.74398 + 1.42912I 4.95658 − 3.06315I
u = 0.612514 + 0.397650I
a = −0.270980 − 1.353370I
b = −0.326776 + 0.647435I
−0.38950 + 3.27691I 2.40587 − 8.31237I
u = 0.612514 − 0.397650I
a = −0.270980 + 1.353370I
b = −0.326776 − 0.647435I
−0.38950 − 3.27691I 2.40587 + 8.31237I
u = −0.154333 + 0.690721I
a = 0.498196 + 0.076734I
b = −0.141589 + 0.700436I
3.24167 + 2.36306I 4.92638 − 2.57172I
u = −0.154333 − 0.690721I
a = 0.498196 − 0.076734I
b = −0.141589 − 0.700436I
3.24167 − 2.36306I 4.92638 + 2.57172I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.168802 + 0.683958I
a = 0.772109 + 0.104136I
b = 1.43972 + 0.11421I
−7.63058 − 2.23574I −4.95663 + 1.98264I
u = 0.168802 − 0.683958I
a = 0.772109 − 0.104136I
b = 1.43972 − 0.11421I
−7.63058 + 2.23574I −4.95663 − 1.98264I
u = −1.35461
a = −0.662234
b = 1.56599
−3.74962 0
u = −0.472149 + 0.408625I
a = −1.37617 + 1.82546I
b = −1.274780 − 0.089703I
−2.79224 − 1.47778I −0.57816 + 4.38549I
u = −0.472149 − 0.408625I
a = −1.37617 − 1.82546I
b = −1.274780 + 0.089703I
−2.79224 + 1.47778I −0.57816 − 4.38549I
u = 1.37684
a = 0.886850
b = −0.0476686
6.50761 0
u = 1.315790 + 0.425847I
a = −0.397092 + 0.322882I
b = 1.255370 + 0.180857I
3.54224 − 1.50691I 0
u = 1.315790 − 0.425847I
a = −0.397092 − 0.322882I
b = 1.255370 − 0.180857I
3.54224 + 1.50691I 0
u = −1.43056
a = 8.65850
b = −0.979198
4.96917 0
u = 0.150519 + 0.505903I
a = −3.05850 − 1.07646I
b = −1.166570 − 0.207757I
0.273379 − 0.971704I 0.08959 − 3.04439I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.150519 − 0.505903I
a = −3.05850 + 1.07646I
b = −1.166570 + 0.207757I
0.273379 + 0.971704I 0.08959 + 3.04439I
u = −0.523123 + 0.028134I
a = 1.13938 − 0.89193I
b = −0.084530 + 0.244571I
0.906568 − 0.098785I 10.73700 + 0.03939I
u = −0.523123 − 0.028134I
a = 1.13938 + 0.89193I
b = −0.084530 − 0.244571I
0.906568 + 0.098785I 10.73700 − 0.03939I
u = −1.51079 + 0.03677I
a = 1.03861 − 1.20593I
b = −0.962369 + 0.432397I
4.73047 − 0.55582I 0
u = −1.51079 − 0.03677I
a = 1.03861 + 1.20593I
b = −0.962369 − 0.432397I
4.73047 + 0.55582I 0
u = 0.280851 + 0.380725I
a = 0.583491 + 0.202281I
b = −0.543197 − 0.407436I
−1.335370 − 0.411278I −3.93368 + 0.07785I
u = 0.280851 − 0.380725I
a = 0.583491 − 0.202281I
b = −0.543197 + 0.407436I
−1.335370 + 0.411278I −3.93368 − 0.07785I
u = 1.53718 + 0.08815I
a = 0.26347 − 1.53965I
b = −1.318230 + 0.251318I
3.97848 + 3.12179I 0
u = 1.53718 − 0.08815I
a = 0.26347 + 1.53965I
b = −1.318230 − 0.251318I
3.97848 − 3.12179I 0
u = 1.54676
a = −0.552532
b = 1.77372
−0.257952 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.58047 + 0.01988I
a = 0.307453 − 1.365900I
b = 0.028298 + 0.630886I
8.23798 + 0.05802I 0
u = 1.58047 − 0.01988I
a = 0.307453 + 1.365900I
b = 0.028298 − 0.630886I
8.23798 − 0.05802I 0
u = −1.58423 + 0.10551I
a = 0.00098 + 1.59403I
b = −0.206251 − 0.827340I
7.10069 − 5.07913I 0
u = −1.58423 − 0.10551I
a = 0.00098 − 1.59403I
b = −0.206251 + 0.827340I
7.10069 + 5.07913I 0
u = −1.60674 + 0.13048I
a = 0.203247 + 1.201000I
b = −1.51444 − 0.32292I
9.71439 − 6.49643I 0
u = −1.60674 − 0.13048I
a = 0.203247 − 1.201000I
b = −1.51444 + 0.32292I
9.71439 + 6.49643I 0
u = 1.61104 + 0.09528I
a = 0.709620 + 0.974902I
b = −1.082630 − 0.771501I
10.73290 + 3.02061I 0
u = 1.61104 − 0.09528I
a = 0.709620 − 0.974902I
b = −1.082630 + 0.771501I
10.73290 − 3.02061I 0
u = −1.61936 + 0.17707I
a = −0.71301 − 1.47528I
b = 1.38942 + 0.34866I
2.05187 − 9.32176I 0
u = −1.61936 − 0.17707I
a = −0.71301 + 1.47528I
b = 1.38942 − 0.34866I
2.05187 + 9.32176I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.62558 + 0.16107I
a = −0.18148 − 1.48352I
b = −0.265871 + 1.040180I
13.1382 + 9.1870I 0
u = 1.62558 − 0.16107I
a = −0.18148 + 1.48352I
b = −0.265871 − 1.040180I
13.1382 − 9.1870I 0
u = 1.61953 + 0.21948I
a = −0.42080 + 1.54749I
b = 1.45736 − 0.44242I
7.6958 + 14.4824I 0
u = 1.61953 − 0.21948I
a = −0.42080 − 1.54749I
b = 1.45736 + 0.44242I
7.6958 − 14.4824I 0
u = −0.363008
a = 3.16745
b = −0.297973
0.945431 16.1240
u = −0.353073
a = 1.07459
b = 1.65773
−6.97573 16.8660
u = 1.65034 + 0.12971I
a = −0.97563 + 1.03708I
b = 1.278460 − 0.240889I
4.33770 + 3.20222I 0
u = 1.65034 − 0.12971I
a = −0.97563 − 1.03708I
b = 1.278460 + 0.240889I
4.33770 − 3.20222I 0
u = −1.67507 + 0.05882I
a = 0.039583 − 1.084640I
b = 0.468851 + 0.766491I
16.0964 − 2.4237I 0
u = −1.67507 − 0.05882I
a = 0.039583 + 1.084640I
b = 0.468851 − 0.766491I
16.0964 + 2.4237I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.284099
a = 2.60629
b = −0.896150
−1.25065 −13.1380
u = −1.82703
a = −0.674961
b = 1.09260
15.7512 0
11
II. I
u
2
= hb + 1, u
2
+ a + u − 2, u
3
+ u
2
− 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
u
2
− u − 1
a
3
=
−u
2
− u + 2
−1
a
7
=
−u
2
+ 1
u
2
− u − 1
a
8
=
−u
2
+ 1
u
2
− u − 1
a
12
=
u
2
− 1
−u
2
a
2
=
−u
2
− u + 1
−1
a
4
=
−u
2
− u + 2
−1
a
1
=
−1
0
a
9
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
− 4u + 12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
7
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
8
c
9
u
3
+ u
2
− 2u − 1
c
10
, c
11
, c
12
u
3
− u
2
− 2u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
7
y
3
c
5
, c
6
, c
8
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = −0.801938
b = −1.00000
4.69981 8.56700
u = −0.445042
a = 2.24698
b = −1.00000
−0.939962 13.9780
u = −1.80194
a = 0.554958
b = −1.00000
15.9794 22.4550
15
III. I
u
3
= hau + b + 2a − 1, 2a
2
+ au − 2a + 2u − 3, u
2
− 2i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−2
a
11
=
u
−u
a
3
=
a
−au − 2a + 1
a
7
=
−1
0
a
8
=
−
1
2
u
au + 2a − 2
a
12
=
0
−u
a
2
=
−au − a + 1
−au − 2a + 1
a
4
=
au + 2a −
1
2
u
au + 2a − 2
a
1
=
−
1
2
u
au + 2a − 2
a
9
=
−
1
2
u
au + 2a + u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u
2
+ u − 1)
2
c
3
, c
4
(u
2
− u − 1)
2
c
5
, c
6
, c
10
c
11
(u
2
− 2)
2
c
8
, c
9
(u − 1)
4
c
12
(u + 1)
4
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
(y
2
− 3y + 1)
2
c
5
, c
6
, c
10
c
11
(y −2)
4
c
8
, c
9
, c
12
(y −1)
4
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = 0.473911
b = −0.618034
5.59278 4.00000
u = 1.41421
a = −0.181018
b = 1.61803
−2.30291 4.00000
u = −1.41421
a = −1.05505
b = 1.61803
−2.30291 4.00000
u = −1.41421
a = 2.76216
b = −0.618034
5.59278 4.00000
19
IV. I
v
1
= ha, b + v − 2, v
2
− 3v + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
v
0
a
6
=
1
0
a
11
=
v
0
a
3
=
0
−v + 2
a
7
=
1
0
a
8
=
1
−v + 3
a
12
=
v
0
a
2
=
−v + 2
−v + 2
a
4
=
v − 2
v − 3
a
1
=
−1
v − 3
a
9
=
v + 1
−v + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −14
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
u
2
+ u − 1
c
4
, c
7
u
2
− u − 1
c
5
, c
6
, c
10
c
11
u
2
c
8
, c
9
(u + 1)
2
c
12
(u − 1)
2
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
y
2
− 3y + 1
c
5
, c
6
, c
10
c
11
y
2
c
8
, c
9
, c
12
(y − 1)
2
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.381966
a = 0
b = 1.61803
−7.23771 −14.0000
v = 2.61803
a = 0
b = −0.618034
0.657974 −14.0000
23
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
3
)(u
2
+ u − 1)
3
(u
62
− 7u
61
+ ··· + 29u + 1)
c
3
u
3
(u
2
− u − 1)
2
(u
2
+ u − 1)(u
62
− 2u
61
+ ··· + 108u − 8)
c
4
((u + 1)
3
)(u
2
− u − 1)
3
(u
62
− 7u
61
+ ··· + 29u + 1)
c
5
, c
6
u
2
(u
2
− 2)
2
(u
3
+ u
2
− 2u − 1)(u
62
+ 2u
61
+ ··· + 12u − 4)
c
7
u
3
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
62
− 2u
61
+ ··· + 108u − 8)
c
8
, c
9
((u − 1)
4
)(u + 1)
2
(u
3
+ u
2
− 2u − 1)(u
62
− 4u
61
+ ··· + 81u − 9)
c
10
, c
11
u
2
(u
2
− 2)
2
(u
3
− u
2
− 2u + 1)(u
62
+ 2u
61
+ ··· + 12u − 4)
c
12
((u − 1)
2
)(u + 1)
4
(u
3
− u
2
− 2u + 1)(u
62
− 4u
61
+ ··· + 81u − 9)
24
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y − 1)
3
)(y
2
− 3y + 1)
3
(y
62
− 57y
61
+ ··· − 427y + 1)
c
3
, c
7
y
3
(y
2
− 3y + 1)
3
(y
62
− 30y
61
+ ··· − 4688y + 64)
c
5
, c
6
, c
10
c
11
y
2
(y − 2)
4
(y
3
− 5y
2
+ 6y − 1)(y
62
− 74y
61
+ ··· − 624y + 16)
c
8
, c
9
, c
12
((y − 1)
6
)(y
3
− 5y
2
+ 6y − 1)(y
62
− 60y
61
+ ··· − 3519y + 81)
25
