12a
1031
(K12a
1031
)
A knot diagram
1
Linearized knot diagam
4 7 8 10 11 3 2 12 1 5 6 9
Solving Sequence
3,6 7,11
12 2 8 4 9 1 5 10
c
6
c
11
c
2
c
7
c
3
c
8
c
1
c
5
c
10
c
4
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−2.19647 × 10
18
u
56
− 5.09545 × 10
18
u
55
+ ··· + 7.20554 × 10
18
b + 1.28415 × 10
19
,
− 7.29984 × 10
18
u
56
− 1.44902 × 10
19
u
55
+ ··· + 7.20554 × 10
18
a + 7.38518 × 10
19
, u
57
+ 2u
56
+ ··· − 8u + 1i
I
u
2
= h−u
2
a − u
2
+ b − a + u − 2, 2u
2
a + a
2
+ u
2
+ 2a − 3u + 2, u
3
− u
2
+ 2u − 1i
I
u
3
= hb, −u
2
+ a − 1, u
3
+ u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 66 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−2.20×10
18
u
56
−5.10×10
18
u
55
+· · ·+7.21×10
18
b+1.28×10
19
, −7.30×
10
18
u
56
−1.45×10
19
u
55
+· · ·+7.21×10
18
a+7.39×10
19
, u
57
+2u
56
+· · ·−8u+1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
11
=
1.01309u
56
+ 2.01098u
55
+ ··· + 4.60352u − 10.2493
0.304830u
56
+ 0.707157u
55
+ ··· + 1.30428u − 1.78217
a
12
=
1.31792u
56
+ 2.71814u
55
+ ··· + 5.90780u − 12.0315
0.304830u
56
+ 0.707157u
55
+ ··· + 1.30428u − 1.78217
a
2
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
4
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
9
=
1.63554u
56
+ 3.51891u
55
+ ··· + 6.76669u − 12.1033
0.230341u
56
+ 0.550285u
55
+ ··· − 0.632533u − 1.56754
a
1
=
u
9
+ 4u
7
+ 5u
5
+ 2u
3
+ u
u
11
+ 5u
9
+ 8u
7
+ 3u
5
− u
3
+ u
a
5
=
−2.34155u
56
− 5.04434u
55
+ ··· − 6.58540u + 15.6235
−0.327980u
56
− 0.616638u
55
+ ··· + 0.0250809u + 2.21226
a
10
=
−1.69640u
56
− 3.90320u
55
+ ··· − 8.13650u + 12.0154
−0.122593u
56
− 0.0875092u
55
+ ··· + 1.34475u + 1.30123
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
6051548760743078920
3602767560289372397
u
56
−
8079139122240889311
3602767560289372397
u
55
+ ··· +
59050771530991937522
3602767560289372397
u +
45876737479885210010
3602767560289372397
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
57
− 10u
56
+ ··· + 1976u + 97
c
2
, c
6
, c
7
u
57
+ 2u
56
+ ··· − 8u + 1
c
3
u
57
− 2u
56
+ ··· − 7940u + 797
c
4
, c
5
, c
10
c
11
u
57
− u
56
+ ··· + 8u − 8
c
8
, c
9
, c
12
u
57
− 4u
56
+ ··· + 53u + 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
57
+ 38y
56
+ ··· + 7632286y −9409
c
2
, c
6
, c
7
y
57
+ 54y
56
+ ··· + 70y −1
c
3
y
57
+ 14y
56
+ ··· + 33141754y −635209
c
4
, c
5
, c
10
c
11
y
57
− 71y
56
+ ··· + 960y −64
c
8
, c
9
, c
12
y
57
− 60y
56
+ ··· + 737y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.587296 + 0.676990I
a = −0.992081 + 0.406961I
b = 1.68131 − 0.13190I
16.3335 − 4.5705I 10.56013 + 0.39432I
u = −0.587296 − 0.676990I
a = −0.992081 − 0.406961I
b = 1.68131 + 0.13190I
16.3335 + 4.5705I 10.56013 − 0.39432I
u = −0.774867 + 0.380193I
a = 0.56831 − 1.72511I
b = −1.67101 − 0.16539I
15.3379 + 9.2274I 8.88615 − 5.62844I
u = −0.774867 − 0.380193I
a = 0.56831 + 1.72511I
b = −1.67101 + 0.16539I
15.3379 − 9.2274I 8.88615 + 5.62844I
u = 0.831356
a = −0.649521
b = −1.63354
9.95066 7.73730
u = 0.699421 + 0.384713I
a = −0.52691 − 1.35735I
b = 0.861493 − 0.565331I
6.65268 − 6.38038I 7.53136 + 6.91744I
u = 0.699421 − 0.384713I
a = −0.52691 + 1.35735I
b = 0.861493 + 0.565331I
6.65268 + 6.38038I 7.53136 − 6.91744I
u = −0.077518 + 1.200830I
a = −0.441853 + 0.527455I
b = 0.400809 + 0.440176I
1.84442 + 1.53939I 0
u = −0.077518 − 1.200830I
a = −0.441853 − 0.527455I
b = 0.400809 − 0.440176I
1.84442 − 1.53939I 0
u = −0.668942 + 0.396082I
a = −1.19802 + 1.74930I
b = 1.62450 + 0.07831I
8.50640 + 4.83173I 6.80279 − 5.63364I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.668942 − 0.396082I
a = −1.19802 − 1.74930I
b = 1.62450 − 0.07831I
8.50640 − 4.83173I 6.80279 + 5.63364I
u = 0.540422 + 0.553317I
a = 0.0227086 + 0.0584366I
b = −0.926422 − 0.476199I
7.32379 + 2.19374I 9.37328 − 0.81466I
u = 0.540422 − 0.553317I
a = 0.0227086 − 0.0584366I
b = −0.926422 + 0.476199I
7.32379 − 2.19374I 9.37328 + 0.81466I
u = −0.558250 + 0.496865I
a = 1.63555 − 0.91390I
b = −1.61760 + 0.02546I
8.95263 − 0.76794I 8.17007 − 0.54035I
u = −0.558250 − 0.496865I
a = 1.63555 + 0.91390I
b = −1.61760 − 0.02546I
8.95263 + 0.76794I 8.17007 + 0.54035I
u = −0.271718 + 1.223630I
a = −0.099179 − 0.634045I
b = −0.666764 − 0.175562I
5.53795 + 3.61211I 0
u = −0.271718 − 1.223630I
a = −0.099179 + 0.634045I
b = −0.666764 + 0.175562I
5.53795 − 3.61211I 0
u = 0.206793 + 1.247570I
a = 1.18787 + 1.19469I
b = −1.46087 + 0.04757I
7.77360 − 3.06996I 0
u = 0.206793 − 1.247570I
a = 1.18787 − 1.19469I
b = −1.46087 − 0.04757I
7.77360 + 3.06996I 0
u = −0.601751 + 0.417912I
a = 0.568911 − 0.774796I
b = 0.064485 − 0.762242I
4.25189 + 1.93714I 5.45403 − 3.24449I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.601751 − 0.417912I
a = 0.568911 + 0.774796I
b = 0.064485 + 0.762242I
4.25189 − 1.93714I 5.45403 + 3.24449I
u = 0.382994 + 1.209880I
a = −0.90161 − 1.11860I
b = 1.63564 − 0.03917I
13.6861 − 4.3558I 0
u = 0.382994 − 1.209880I
a = −0.90161 + 1.11860I
b = 1.63564 + 0.03917I
13.6861 + 4.3558I 0
u = −0.719732
a = 1.06954
b = 0.676169
1.78784 6.76490
u = 0.089195 + 1.288090I
a = 1.043920 − 0.621762I
b = −0.408562 − 0.408532I
4.90426 − 1.61227I 0
u = 0.089195 − 1.288090I
a = 1.043920 + 0.621762I
b = −0.408562 + 0.408532I
4.90426 + 1.61227I 0
u = 0.597316 + 0.299927I
a = 0.75088 + 1.25022I
b = −0.698631 + 0.329985I
0.45756 − 3.37876I 3.99050 + 8.66171I
u = 0.597316 − 0.299927I
a = 0.75088 − 1.25022I
b = −0.698631 − 0.329985I
0.45756 + 3.37876I 3.99050 − 8.66171I
u = 0.027116 + 1.350160I
a = −2.00735 − 1.30623I
b = 1.43950 − 0.16113I
10.95220 − 0.52770I 0
u = 0.027116 − 1.350160I
a = −2.00735 + 1.30623I
b = 1.43950 + 0.16113I
10.95220 + 0.52770I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.636143
a = 0.899971
b = 1.45279
3.96768 −0.741900
u = −0.206696 + 1.359410I
a = 0.316733 + 0.033558I
b = 0.059845 − 0.466800I
3.73122 + 3.46475I 0
u = −0.206696 − 1.359410I
a = 0.316733 − 0.033558I
b = 0.059845 + 0.466800I
3.73122 − 3.46475I 0
u = 0.186663 + 1.395260I
a = 1.59178 + 0.51826I
b = −0.833776 − 0.025880I
6.49247 − 1.98805I 0
u = 0.186663 − 1.395260I
a = 1.59178 − 0.51826I
b = −0.833776 + 0.025880I
6.49247 + 1.98805I 0
u = 0.22801 + 1.41445I
a = −1.60611 − 0.79064I
b = 0.802162 − 0.369008I
5.94609 − 6.40773I 0
u = 0.22801 − 1.41445I
a = −1.60611 + 0.79064I
b = 0.802162 + 0.369008I
5.94609 + 6.40773I 0
u = −0.545024 + 0.153075I
a = −0.433814 + 0.605711I
b = −0.165016 + 0.402005I
−1.091470 + 0.719453I −4.41448 − 2.05784I
u = −0.545024 − 0.153075I
a = −0.433814 − 0.605711I
b = −0.165016 − 0.402005I
−1.091470 − 0.719453I −4.41448 + 2.05784I
u = −0.22391 + 1.45684I
a = −0.429922 − 0.136736I
b = −0.103875 + 0.846114I
10.28030 + 4.97302I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22391 − 1.45684I
a = −0.429922 + 0.136736I
b = −0.103875 − 0.846114I
10.28030 − 4.97302I 0
u = −0.24951 + 1.45945I
a = 2.70655 − 1.58438I
b = −1.65433 − 0.10023I
14.4830 + 8.1903I 0
u = −0.24951 − 1.45945I
a = 2.70655 + 1.58438I
b = −1.65433 + 0.10023I
14.4830 − 8.1903I 0
u = 0.26230 + 1.45913I
a = 1.44999 + 0.74312I
b = −0.870143 + 0.634792I
12.5896 − 9.8895I 0
u = 0.26230 − 1.45913I
a = 1.44999 − 0.74312I
b = −0.870143 − 0.634792I
12.5896 + 9.8895I 0
u = −0.19701 + 1.46962I
a = −3.02622 + 0.91279I
b = 1.66182 + 0.00364I
15.2659 + 1.9819I 0
u = −0.19701 − 1.46962I
a = −3.02622 − 0.91279I
b = 1.66182 − 0.00364I
15.2659 − 1.9819I 0
u = 0.17782 + 1.47551I
a = −1.061720 − 0.450691I
b = 1.047880 + 0.514208I
13.83430 − 0.35900I 0
u = 0.17782 − 1.47551I
a = −1.061720 + 0.450691I
b = 1.047880 − 0.514208I
13.83430 + 0.35900I 0
u = 0.381646 + 0.332840I
a = −0.862209 − 0.317769I
b = 0.612157 + 0.106984I
1.094310 + 0.315524I 8.13149 − 0.54452I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.381646 − 0.332840I
a = −0.862209 + 0.317769I
b = 0.612157 − 0.106984I
1.094310 − 0.315524I 8.13149 + 0.54452I
u = −0.29545 + 1.46817I
a = −2.14517 + 1.69345I
b = 1.67802 + 0.19089I
−18.1941 + 13.1223I 0
u = −0.29545 − 1.46817I
a = −2.14517 − 1.69345I
b = 1.67802 − 0.19089I
−18.1941 − 13.1223I 0
u = −0.14052 + 1.52399I
a = 2.53882 − 0.31091I
b = −1.72718 + 0.11615I
−15.8713 − 2.1155I 0
u = −0.14052 − 1.52399I
a = 2.53882 + 0.31091I
b = −1.72718 − 0.11615I
−15.8713 + 2.1155I 0
u = 0.357281
a = −1.94262
b = 0.364334
1.04543 13.7800
u = 0.132476
a = −8.67705
b = −1.39067
6.53451 14.2000
10
II.
I
u
2
= h−u
2
a − u
2
+b −a + u−2, 2u
2
a + a
2
+u
2
+ 2a −3u + 2, u
3
−u
2
+ 2u −1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
11
=
a
u
2
a + u
2
+ a − u + 2
a
12
=
u
2
a + u
2
+ 2a − u + 2
u
2
a + u
2
+ a − u + 2
a
2
=
u
u
2
− u + 1
a
8
=
u
2
+ 1
u
2
− u + 1
a
4
=
−1
0
a
9
=
−u
2
a − 2a + u − 1
−u
2
a − a − 1
a
1
=
u
2
+ 1
u
2
− u + 1
a
5
=
−u
2
a + au + u
2
− 2a − 2u + 1
2
a
10
=
−u
2
a − u
2
− 2a + u − 2
−u
2
a − u
2
− a + u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 4u + 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
− 1)
2
c
2
(u
3
+ u
2
+ 2u + 1)
2
c
3
(u
3
− u
2
+ 1)
2
c
4
, c
5
, c
10
c
11
(u
2
− 2)
3
c
6
, c
7
(u
3
− u
2
+ 2u − 1)
2
c
8
, c
9
(u − 1)
6
c
12
(u + 1)
6
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
(y
3
− y
2
+ 2y −1)
2
c
2
, c
6
, c
7
(y
3
+ 3y
2
+ 2y −1)
2
c
4
, c
5
, c
10
c
11
(y −2)
6
c
8
, c
9
, c
12
(y −1)
6
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.57853 − 1.61567I
b = 1.41421
9.60386 − 2.82812I 11.50976 + 2.97945I
u = 0.215080 + 1.307140I
a = 1.90324 + 0.49111I
b = −1.41421
9.60386 − 2.82812I 11.50976 + 2.97945I
u = 0.215080 − 1.307140I
a = −0.57853 + 1.61567I
b = 1.41421
9.60386 + 2.82812I 11.50976 − 2.97945I
u = 0.215080 − 1.307140I
a = 1.90324 − 0.49111I
b = −1.41421
9.60386 + 2.82812I 11.50976 − 2.97945I
u = 0.569840
a = −0.257160
b = 1.41421
5.46628 4.98050
u = 0.569840
a = −2.39228
b = −1.41421
5.46628 4.98050
14
III. I
u
3
= hb, −u
2
+ a − 1, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
11
=
u
2
+ 1
0
a
12
=
u
2
+ 1
0
a
2
=
u
−u
2
− u − 1
a
8
=
u
2
+ 1
u
2
+ u + 1
a
4
=
1
0
a
9
=
2u
2
+ 2
u
2
+ u + 1
a
1
=
−u
2
− 1
−u
2
− u − 1
a
5
=
1
0
a
10
=
u
2
+ 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6u
2
− 4u − 4
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
3
+ u
2
− 1
c
2
u
3
− u
2
+ 2u − 1
c
4
, c
5
, c
10
c
11
u
3
c
6
, c
7
u
3
+ u
2
+ 2u + 1
c
8
, c
9
(u + 1)
3
c
12
(u − 1)
3
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
3
− y
2
+ 2y −1
c
2
, c
6
, c
7
y
3
+ 3y
2
+ 2y −1
c
4
, c
5
, c
10
c
11
y
3
c
8
, c
9
, c
12
(y −1)
3
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.662359 − 0.562280I
b = 0
4.66906 + 2.82812I 6.83447 − 1.85489I
u = −0.215080 − 1.307140I
a = −0.662359 + 0.562280I
b = 0
4.66906 − 2.82812I 6.83447 + 1.85489I
u = −0.569840
a = 1.32472
b = 0
0.531480 −3.66890
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
+ u
2
− 1)
3
)(u
57
− 10u
56
+ ··· + 1976u + 97)
c
2
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
(u
57
+ 2u
56
+ ··· − 8u + 1)
c
3
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
57
− 2u
56
+ ··· − 7940u + 797)
c
4
, c
5
, c
10
c
11
u
3
(u
2
− 2)
3
(u
57
− u
56
+ ··· + 8u − 8)
c
6
, c
7
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
57
+ 2u
56
+ ··· − 8u + 1)
c
8
, c
9
((u − 1)
6
)(u + 1)
3
(u
57
− 4u
56
+ ··· + 53u + 7)
c
12
((u − 1)
3
)(u + 1)
6
(u
57
− 4u
56
+ ··· + 53u + 7)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
− y
2
+ 2y −1)
3
)(y
57
+ 38y
56
+ ··· + 7632286y −9409)
c
2
, c
6
, c
7
((y
3
+ 3y
2
+ 2y −1)
3
)(y
57
+ 54y
56
+ ··· + 70y −1)
c
3
((y
3
− y
2
+ 2y −1)
3
)(y
57
+ 14y
56
+ ··· + 3.31418 × 10
7
y −635209)
c
4
, c
5
, c
10
c
11
y
3
(y −2)
6
(y
57
− 71y
56
+ ··· + 960y −64)
c
8
, c
9
, c
12
((y −1)
9
)(y
57
− 60y
56
+ ··· + 737y −49)
20
