12n
0084
(K12n
0084
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 8 4 11 6 7 12 8 10
Solving Sequence
7,11
8
4,12
3 6 5 2 10 1 9
c
7
c
11
c
3
c
6
c
5
c
2
c
10
c
12
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−16510225313889u
40
− 73379619742180u
39
+ ··· + 7015565688188b + 16563398292357,
− 39282009802613u
40
− 170243595424532u
39
+ ··· + 7015565688188a + 73325296186875,
u
41
+ 5u
40
+ ··· + 5u − 1i
I
u
2
= h−u
2
a − 2u
2
+ b + a + u + 2, a
2
+ 2au + 4u
2
+ a − 4u + 4, u
3
− u
2
+ 1i
I
u
3
= hu
2
+ b, a − u, u
3
− u
2
+ 1i
I
u
4
= hb, a + 1, u + 1i
* 4 irreducible components of dim
C
= 0, with total 51 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−1.65×10
13
u
40
−7.34×10
13
u
39
+· · ·+7.02×10
12
b+1.66×10
13
, −3.93×
10
13
u
40
−1.70×10
14
u
39
+· · ·+7.02×10
12
a+7.33×10
13
, u
41
+5u
40
+· · ·+5u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
4
=
5.59926u
40
+ 24.2666u
39
+ ··· + 36.6437u − 10.4518
2.35337u
40
+ 10.4595u
39
+ ··· + 18.2641u − 2.36095
a
12
=
u
−u
3
+ u
a
3
=
7.95264u
40
+ 34.7261u
39
+ ··· + 54.9078u − 12.8128
2.35337u
40
+ 10.4595u
39
+ ··· + 18.2641u − 2.36095
a
6
=
−0.679973u
40
− 2.74952u
39
+ ··· − 2.24406u + 4.15673
0.448072u
40
+ 2.09737u
39
+ ··· − 2.27213u − 0.275451
a
5
=
−0.0387129u
40
− 1.20772u
39
+ ··· − 8.44788u + 4.53163
−1.39663u
40
− 6.29046u
39
+ ··· − 11.2359u + 1.38905
a
2
=
5.78006u
40
+ 24.1415u
39
+ ··· + 36.8035u − 7.87601
1.39663u
40
+ 6.29046u
39
+ ··· + 11.2359u − 1.38905
a
10
=
−u
3
u
5
− u
3
+ u
a
1
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
−u
5
− u
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
93247797228429
1753891422047
u
40
+
1555830189736099
7015565688188
u
39
+ ··· +
2808560039807687
7015565688188
u −
263248389242609
3507782844094
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
41
+ 29u
40
+ ··· + 71u + 1
c
2
, c
4
u
41
− 5u
40
+ ··· + 7u − 1
c
3
, c
6
u
41
− 4u
40
+ ··· − 10u + 2
c
5
, c
8
u
41
− 4u
40
+ ··· + 512u − 512
c
7
, c
11
u
41
− 5u
40
+ ··· + 5u + 1
c
9
u
41
+ 3u
40
+ ··· − 19597u + 2017
c
10
, c
12
u
41
− 9u
40
+ ··· + 95u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
41
− 29y
40
+ ··· + 1223y −1
c
2
, c
4
y
41
− 29y
40
+ ··· + 71y −1
c
3
, c
6
y
41
+ 42y
39
+ ··· + 56y −4
c
5
, c
8
y
41
− 50y
40
+ ··· + 6160384y −262144
c
7
, c
11
y
41
− 9y
40
+ ··· + 95y −1
c
9
y
41
+ 135y
40
+ ··· + 36517343y −4068289
c
10
, c
12
y
41
+ 51y
40
+ ··· + 6127y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.716889 + 0.703853I
a = −0.43425 − 2.19682I
b = −0.336317 + 0.798192I
−4.60661 + 1.52902I −5.50706 − 4.66307I
u = 0.716889 − 0.703853I
a = −0.43425 + 2.19682I
b = −0.336317 − 0.798192I
−4.60661 − 1.52902I −5.50706 + 4.66307I
u = −0.870049 + 0.467634I
a = −1.164530 + 0.454430I
b = 0.135790 − 0.918035I
1.79473 + 0.06262I 2.65743 − 0.68980I
u = −0.870049 − 0.467634I
a = −1.164530 − 0.454430I
b = 0.135790 + 0.918035I
1.79473 − 0.06262I 2.65743 + 0.68980I
u = 0.356456 + 0.980218I
a = 0.282408 + 0.117731I
b = −1.039320 − 0.421414I
−7.17972 − 2.87745I −9.28722 + 2.69256I
u = 0.356456 − 0.980218I
a = 0.282408 − 0.117731I
b = −1.039320 + 0.421414I
−7.17972 + 2.87745I −9.28722 − 2.69256I
u = −0.670401 + 0.652476I
a = 1.010200 − 0.553099I
b = 0.446012 + 1.120950I
0.97725 − 4.44119I −1.27945 + 4.71656I
u = −0.670401 − 0.652476I
a = 1.010200 + 0.553099I
b = 0.446012 − 1.120950I
0.97725 + 4.44119I −1.27945 − 4.71656I
u = 0.940640 + 0.501809I
a = −0.02791 + 1.96139I
b = 0.687266 − 0.758448I
−0.70118 + 4.43572I −0.34738 − 6.39371I
u = 0.940640 − 0.501809I
a = −0.02791 − 1.96139I
b = 0.687266 + 0.758448I
−0.70118 − 4.43572I −0.34738 + 6.39371I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877966 + 0.643589I
a = 0.016521 + 1.234530I
b = −0.609480 − 0.518945I
−4.07708 + 3.51500I −8.28852 − 2.95507I
u = 0.877966 − 0.643589I
a = 0.016521 − 1.234530I
b = −0.609480 + 0.518945I
−4.07708 − 3.51500I −8.28852 + 2.95507I
u = −0.854623 + 0.159200I
a = −0.978695 + 0.858024I
b = 0.041696 − 0.419660I
1.46993 − 0.34552I 6.06026 + 0.56755I
u = −0.854623 − 0.159200I
a = −0.978695 − 0.858024I
b = 0.041696 + 0.419660I
1.46993 + 0.34552I 6.06026 − 0.56755I
u = 0.877954 + 0.775956I
a = −0.968702 − 0.351197I
b = −0.598603 + 0.031599I
−3.83803 + 2.91878I 4.79738 − 5.00057I
u = 0.877954 − 0.775956I
a = −0.968702 + 0.351197I
b = −0.598603 − 0.031599I
−3.83803 − 2.91878I 4.79738 + 5.00057I
u = 0.537453 + 0.625371I
a = −0.426437 − 0.272772I
b = 0.891755 + 0.389086I
−2.04074 − 0.11127I −4.60725 + 0.35706I
u = 0.537453 − 0.625371I
a = −0.426437 + 0.272772I
b = 0.891755 − 0.389086I
−2.04074 + 0.11127I −4.60725 − 0.35706I
u = −0.802702
a = −5.09181
b = −0.275758
−0.345711 −64.7260
u = 1.139710 + 0.532282I
a = 0.29262 − 1.64354I
b = −0.893901 + 0.701047I
−4.48025 + 8.31246I 0. − 7.85034I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.139710 − 0.532282I
a = 0.29262 + 1.64354I
b = −0.893901 − 0.701047I
−4.48025 − 8.31246I 0. + 7.85034I
u = −1.26772
a = 0.964605
b = −0.640215
−0.787444 −12.1570
u = −0.891473 + 0.933329I
a = −0.528554 + 0.154006I
b = 1.25025 − 1.10628I
−10.38260 + 1.04574I 0
u = −0.891473 − 0.933329I
a = −0.528554 − 0.154006I
b = 1.25025 + 1.10628I
−10.38260 − 1.04574I 0
u = −0.841724 + 0.990823I
a = 0.377207 − 0.291737I
b = −1.21490 + 1.06493I
−15.1663 + 7.0683I 0
u = −0.841724 − 0.990823I
a = 0.377207 + 0.291737I
b = −1.21490 − 1.06493I
−15.1663 − 7.0683I 0
u = −0.934438 + 0.935551I
a = −0.61279 + 1.51752I
b = −1.12955 − 1.22573I
−14.6673 − 1.5212I 0
u = −0.934438 − 0.935551I
a = −0.61279 − 1.51752I
b = −1.12955 + 1.22573I
−14.6673 + 1.5212I 0
u = −0.982434 + 0.885456I
a = 0.67122 − 1.65206I
b = 1.17691 + 1.18848I
−10.08510 − 7.74293I 0
u = −0.982434 − 0.885456I
a = 0.67122 + 1.65206I
b = 1.17691 − 1.18848I
−10.08510 + 7.74293I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.962450 + 0.919824I
a = 0.713562 − 0.200841I
b = −1.20978 + 1.14699I
−14.5747 − 5.2989I 0
u = −0.962450 − 0.919824I
a = 0.713562 + 0.200841I
b = −1.20978 − 1.14699I
−14.5747 + 5.2989I 0
u = 0.961309 + 0.961016I
a = −0.125970 + 0.554955I
b = 1.019520 − 0.111695I
−9.11622 + 3.51279I 0
u = 0.961309 − 0.961016I
a = −0.125970 − 0.554955I
b = 1.019520 + 0.111695I
−9.11622 − 3.51279I 0
u = −1.039620 + 0.876652I
a = −0.63077 + 1.75489I
b = −1.15233 − 1.13531I
−14.5120 − 13.9039I 0
u = −1.039620 − 0.876652I
a = −0.63077 − 1.75489I
b = −1.15233 + 1.13531I
−14.5120 + 13.9039I 0
u = 0.629652 + 0.060391I
a = −0.09621 + 3.42389I
b = 0.189078 − 1.367880I
3.87413 + 2.95359I −12.9329 − 8.6631I
u = 0.629652 − 0.060391I
a = −0.09621 − 3.42389I
b = 0.189078 + 1.367880I
3.87413 − 2.95359I −12.9329 + 8.6631I
u = −0.512134 + 0.182325I
a = 2.61667 + 0.05062I
b = 0.497781 − 0.337919I
−1.008150 − 0.694652I −6.89267 − 0.99274I
u = −0.512134 − 0.182325I
a = 2.61667 − 0.05062I
b = 0.497781 + 0.337919I
−1.008150 + 0.694652I −6.89267 + 0.99274I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.113052
a = −3.84398
b = 0.612202
−1.00335 −10.2280
9
II.
I
u
2
= h−u
2
a − 2u
2
+ b + a + u + 2, a
2
+ 2au + 4u
2
+ a − 4u + 4, u
3
− u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
4
=
a
u
2
a + 2u
2
− a − u − 2
a
12
=
u
−u
2
+ u + 1
a
3
=
u
2
a + 2u
2
− u − 2
u
2
a + 2u
2
− a − u − 2
a
6
=
u
2
a − au − a − 3
u
2
a + au + 3u
2
a
5
=
u
2
a − au − a − 3
u
2
a + au + 3u
2
a
2
=
2u
2
a + 3u
2
− a − 5
u
2
a + au + 3u
2
a
10
=
−u
2
+ 1
−u
2
a
1
=
−1
0
a
9
=
1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
a + 15au + 20u
2
+ 11a + 4u + 4
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
c
12
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
11
(u
3
+ u
2
− 1)
2
c
4
, c
7
(u
3
− u
2
+ 1)
2
c
5
, c
8
u
6
c
6
, c
10
(u
3
+ u
2
+ 2u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
9
, c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
, c
7
c
11
(y
3
− y
2
+ 2y −1)
2
c
5
, c
8
y
6
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −1.06984 − 1.06527I
b = −0.215080 + 1.307140I
5.65624I −3.29784 − 6.94206I
u = 0.877439 + 0.744862I
a = −1.68504 − 0.42445I
b = −0.569840
−4.13758 + 2.82812I −24.3518 + 2.3339I
u = 0.877439 − 0.744862I
a = −1.06984 + 1.06527I
b = −0.215080 − 1.307140I
− 5.65624I −3.29784 + 6.94206I
u = 0.877439 − 0.744862I
a = −1.68504 + 0.42445I
b = −0.569840
−4.13758 − 2.82812I −24.3518 − 2.3339I
u = −0.754878
a = 0.25488 + 3.03873I
b = −0.215080 − 1.307140I
4.13758 − 2.82812I 12.14969 − 2.71361I
u = −0.754878
a = 0.25488 − 3.03873I
b = −0.215080 + 1.307140I
4.13758 + 2.82812I 12.14969 + 2.71361I
13
III. I
u
3
= hu
2
+ b, a − u, u
3
− u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
4
=
u
−u
2
a
12
=
u
−u
2
+ u + 1
a
3
=
−u
2
+ u
−u
2
a
6
=
u
2
−u
2
+ u + 1
a
5
=
u
2
−u
2
+ u + 1
a
2
=
u − 1
−u
2
+ u + 1
a
10
=
−u
2
+ 1
−u
2
a
1
=
−1
0
a
9
=
1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
2
− 3u − 2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
c
12
u
3
− u
2
+ 2u − 1
c
2
, c
11
u
3
+ u
2
− 1
c
4
, c
7
u
3
− u
2
+ 1
c
5
, c
8
u
3
c
6
, c
10
u
3
+ u
2
+ 2u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
9
, c
10
, c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
4
, c
7
c
11
y
3
− y
2
+ 2y −1
c
5
, c
8
y
3
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.877439 + 0.744862I
b = −0.215080 − 1.307140I
0 −4.20216 + 0.37970I
u = 0.877439 − 0.744862I
a = 0.877439 − 0.744862I
b = −0.215080 + 1.307140I
0 −4.20216 − 0.37970I
u = −0.754878
a = −0.754878
b = −0.569840
0 1.40430
17
IV. I
u
4
= hb, a + 1, u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
−1
a
8
=
1
−1
a
4
=
−1
0
a
12
=
−1
0
a
3
=
−1
0
a
6
=
1
0
a
5
=
2
−1
a
2
=
−3
1
a
10
=
1
−1
a
1
=
−2
1
a
9
=
2
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
8
c
9
, c
11
, c
12
u − 1
c
3
, c
6
u
c
4
, c
5
, c
7
c
10
u + 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y −1
c
3
, c
6
y
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
0 0
21
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
3
− u
2
+ 2u − 1)
3
(u
41
+ 29u
40
+ ··· + 71u + 1)
c
2
(u − 1)(u
3
+ u
2
− 1)
3
(u
41
− 5u
40
+ ··· + 7u − 1)
c
3
u(u
3
− u
2
+ 2u − 1)
3
(u
41
− 4u
40
+ ··· − 10u + 2)
c
4
(u + 1)(u
3
− u
2
+ 1)
3
(u
41
− 5u
40
+ ··· + 7u − 1)
c
5
u
9
(u + 1)(u
41
− 4u
40
+ ··· + 512u − 512)
c
6
u(u
3
+ u
2
+ 2u + 1)
3
(u
41
− 4u
40
+ ··· − 10u + 2)
c
7
(u + 1)(u
3
− u
2
+ 1)
3
(u
41
− 5u
40
+ ··· + 5u + 1)
c
8
u
9
(u − 1)(u
41
− 4u
40
+ ··· + 512u − 512)
c
9
(u − 1)(u
3
− u
2
+ 2u − 1)
3
(u
41
+ 3u
40
+ ··· − 19597u + 2017)
c
10
(u + 1)(u
3
+ u
2
+ 2u + 1)
3
(u
41
− 9u
40
+ ··· + 95u − 1)
c
11
(u − 1)(u
3
+ u
2
− 1)
3
(u
41
− 5u
40
+ ··· + 5u + 1)
c
12
(u − 1)(u
3
− u
2
+ 2u − 1)
3
(u
41
− 9u
40
+ ··· + 95u − 1)
22
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
3
+ 3y
2
+ 2y −1)
3
(y
41
− 29y
40
+ ··· + 1223y −1)
c
2
, c
4
(y −1)(y
3
− y
2
+ 2y −1)
3
(y
41
− 29y
40
+ ··· + 71y −1)
c
3
, c
6
y(y
3
+ 3y
2
+ 2y −1)
3
(y
41
+ 42y
39
+ ··· + 56y −4)
c
5
, c
8
y
9
(y −1)(y
41
− 50y
40
+ ··· + 6160384y −262144)
c
7
, c
11
(y −1)(y
3
− y
2
+ 2y −1)
3
(y
41
− 9y
40
+ ··· + 95y −1)
c
9
(y −1)(y
3
+ 3y
2
+ 2y −1)
3
· (y
41
+ 135y
40
+ ··· + 36517343y −4068289)
c
10
, c
12
(y −1)(y
3
+ 3y
2
+ 2y −1)
3
(y
41
+ 51y
40
+ ··· + 6127y −1)
23
