12n
0444
(K12n
0444
)
A knot diagram
1
Linearized knot diagam
3 6 12 7 2 10 12 11 3 5 8 10
Solving Sequence
6,10 3,7
2 1 5 11 4 9 8 12
c
6
c
2
c
1
c
5
c
10
c
4
c
9
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h5.32469 × 10
28
u
19
− 1.94113 × 10
29
u
18
+ ··· + 2.78354 × 10
30
b − 8.37086 × 10
31
,
− 3.59288 × 10
32
u
19
+ 1.29989 × 10
33
u
18
+ ··· + 6.03193 × 10
33
a + 5.69917 × 10
35
,
u
20
− 5u
19
+ ··· − 5504u + 2167i
I
u
2
= h−39167u
9
+ 24055u
8
+ ··· + 90803b + 150893, 39167u
9
− 24055u
8
+ ··· + 90803a − 241696,
u
10
− 6u
8
+ 27u
6
+ 30u
5
+ 30u
4
+ 14u
3
+ 10u
2
+ 2u + 1i
I
u
3
= hb + 1, a − u + 1, u
3
− 2u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 33 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
= h5.32 × 10
28
u
19
− 1.94 × 10
29
u
18
+ · · · + 2.78 × 10
30
b − 8.37 ×
10
31
, −3.59 × 10
32
u
19
+ 1.30 × 10
33
u
18
+ · · · + 6.03 × 10
33
a + 5.70 ×
10
35
, u
20
− 5u
19
+ · · · − 5504u + 2167i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
0.0595644u
19
− 0.215502u
18
+ ··· + 169.749u − 94.4834
−0.0191292u
19
+ 0.0697361u
18
+ ··· − 55.2312u + 30.0727
a
7
=
1
u
2
a
2
=
0.0404352u
19
− 0.145766u
18
+ ··· + 114.518u − 64.4106
−0.0191292u
19
+ 0.0697361u
18
+ ··· − 55.2312u + 30.0727
a
1
=
−0.0614601u
19
+ 0.224116u
18
+ ··· − 176.369u + 97.1863
0.0732431u
19
− 0.266224u
18
+ ··· + 209.407u − 116.188
a
5
=
−0.0367341u
19
+ 0.133131u
18
+ ··· − 104.597u + 56.9934
0.0484428u
19
− 0.177657u
18
+ ··· + 141.186u − 79.6977
a
11
=
−0.109966u
19
+ 0.400168u
18
+ ··· − 316.982u + 175.815
−0.00769082u
19
+ 0.0267093u
18
+ ··· − 20.2609u + 10.6953
a
4
=
−0.153606u
19
+ 0.560090u
18
+ ··· − 444.350u + 246.211
−0.164874u
19
+ 0.598489u
18
+ ··· − 471.899u + 261.395
a
9
=
0.0266987u
19
− 0.0960595u
18
+ ··· + 75.9779u − 40.6092
−0.115096u
19
+ 0.419259u
18
+ ··· − 332.122u + 184.578
a
8
=
0.0291891u
19
− 0.106960u
18
+ ··· + 85.6547u − 46.9664
0.0785887u
19
− 0.287166u
18
+ ··· + 225.983u − 125.342
a
12
=
−0.0614601u
19
+ 0.224116u
18
+ ··· − 176.369u + 97.1863
−0.0400440u
19
+ 0.146686u
18
+ ··· − 115.256u + 64.0732
(ii) Obstruction class = −1
(iii) Cusp Shapes = −0.0313654u
19
+ 0.114631u
18
+ ··· − 89.3865u + 39.1345
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 12u
19
+ ··· − 19u + 4
c
2
, c
5
u
20
− 2u
19
+ ··· − 5u + 2
c
3
u
20
− 5u
19
+ ··· − 4312u + 581
c
4
u
20
− u
19
+ ··· − 622u + 97
c
6
u
20
+ 5u
19
+ ··· + 5504u + 2167
c
7
, c
8
, c
11
u
20
+ 3u
19
+ ··· − 71u + 62
c
9
u
20
− u
19
+ ··· − 78u + 17
c
10
u
20
− u
19
+ ··· − 52u + 17
c
12
u
20
+ u
19
+ ··· − 294u + 151
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 8y
19
+ ··· − 417y + 16
c
2
, c
5
y
20
− 12y
19
+ ··· + 19y + 4
c
3
y
20
− 75y
19
+ ··· + 153632486y + 337561
c
4
y
20
+ 63y
19
+ ··· − 66784y + 9409
c
6
y
20
− 35y
19
+ ··· − 16056826y + 4695889
c
7
, c
8
, c
11
y
20
+ 37y
19
+ ··· + 27075y + 3844
c
9
y
20
+ 39y
19
+ ··· + 6088y + 289
c
10
y
20
− 5y
19
+ ··· − 120y + 289
c
12
y
20
+ 47y
19
+ ··· + 412468y + 22801
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.703317 + 0.600603I
a = −0.437463 − 0.884492I
b = −0.924571 + 0.362873I
−2.03672 + 1.24145I −6.74073 − 2.13278I
u = 0.703317 − 0.600603I
a = −0.437463 + 0.884492I
b = −0.924571 − 0.362873I
−2.03672 − 1.24145I −6.74073 + 2.13278I
u = 0.459645 + 0.743049I
a = 0.260880 + 0.613052I
b = 0.118127 − 0.454177I
0.254769 − 1.344490I 2.10960 + 5.34511I
u = 0.459645 − 0.743049I
a = 0.260880 − 0.613052I
b = 0.118127 + 0.454177I
0.254769 + 1.344490I 2.10960 − 5.34511I
u = −0.971034 + 0.814910I
a = 0.177512 − 0.054619I
b = 0.759811 + 0.367137I
1.28418 − 1.69463I 6.51624 + 5.12886I
u = −0.971034 − 0.814910I
a = 0.177512 + 0.054619I
b = 0.759811 − 0.367137I
1.28418 + 1.69463I 6.51624 − 5.12886I
u = 1.358840 + 0.009534I
a = −0.18107 + 1.59414I
b = 1.34261 − 0.46139I
16.5713 + 0.1764I −5.92523 − 0.80143I
u = 1.358840 − 0.009534I
a = −0.18107 − 1.59414I
b = 1.34261 + 0.46139I
16.5713 − 0.1764I −5.92523 + 0.80143I
u = −1.39927 + 0.25928I
a = −0.153915 − 1.075640I
b = −0.150857 + 0.864815I
−5.96211 − 0.50208I −3.53348 + 0.08568I
u = −1.39927 − 0.25928I
a = −0.153915 + 1.075640I
b = −0.150857 − 0.864815I
−5.96211 + 0.50208I −3.53348 − 0.08568I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.35298 + 0.59131I
a = 0.282108 − 0.480951I
b = 1.105280 + 0.366189I
−2.42744 − 4.72916I −3.24036 + 7.15543I
u = 1.35298 − 0.59131I
a = 0.282108 + 0.480951I
b = 1.105280 − 0.366189I
−2.42744 + 4.72916I −3.24036 − 7.15543I
u = −1.33962 + 0.86373I
a = −0.145047 + 1.067730I
b = 1.296470 − 0.378610I
−10.43980 + 3.82524I −6.93583 − 2.76644I
u = −1.33962 − 0.86373I
a = −0.145047 − 1.067730I
b = 1.296470 + 0.378610I
−10.43980 − 3.82524I −6.93583 + 2.76644I
u = 1.73280 + 0.79935I
a = −0.300874 − 1.047990I
b = −0.057341 + 0.997519I
−18.5073 − 4.9976I −2.68482 + 1.85210I
u = 1.73280 − 0.79935I
a = −0.300874 + 1.047990I
b = −0.057341 − 0.997519I
−18.5073 + 4.9976I −2.68482 − 1.85210I
u = −1.90586 + 0.34611I
a = 0.024972 + 0.827392I
b = −1.189400 − 0.543552I
−9.03300 − 5.58421I −6.18314 + 4.04952I
u = −1.90586 − 0.34611I
a = 0.024972 − 0.827392I
b = −1.189400 + 0.543552I
−9.03300 + 5.58421I −6.18314 − 4.04952I
u = 2.50819 + 1.00282I
a = 0.136951 + 0.755014I
b = −1.300140 − 0.529897I
17.1367 − 10.4420I −5.38225 + 4.71177I
u = 2.50819 − 1.00282I
a = 0.136951 − 0.755014I
b = −1.300140 + 0.529897I
17.1367 + 10.4420I −5.38225 − 4.71177I
6
II. I
u
2
= h−39167u
9
+ 24055u
8
+ · · · + 90803b + 150893, 39167u
9
−
24055u
8
+ · · · + 90803a − 241696, u
10
− 6u
8
+ · · · + 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
−0.431340u
9
+ 0.264914u
8
+ ··· − 1.16678u + 2.66176
0.431340u
9
− 0.264914u
8
+ ··· + 1.16678u − 1.66176
a
7
=
1
u
2
a
2
=
1
0.431340u
9
− 0.264914u
8
+ ··· + 1.16678u − 1.66176
a
1
=
2.92164u
9
− 0.926676u
8
+ ··· + 10.7123u − 1.76757
−1.35298u
9
+ 0.191591u
8
+ ··· − 5.87905u − 0.570664
a
5
=
−0.431340u
9
+ 0.264914u
8
+ ··· − 1.16678u + 2.66176
1.13732u
9
− 0.470172u
8
+ ··· + 3.66644u − 1.67648
a
11
=
−u
9
+ 6u
7
− 27u
5
− 30u
4
− 30u
3
− 14u
2
− 10u − 2
0.926676u
9
+ 0.384426u
8
+ ··· + 9.61086u + 2.92164
a
4
=
−1.61557u
9
+ 0.780393u
8
+ ··· − 4.93171u + 4.07332
1.20899u
9
− 0.543143u
8
+ ··· + 3.81972u − 2.19195
a
9
=
−u
9
+ 6u
7
− 27u
5
− 30u
4
− 30u
3
− 14u
2
− 10u − 2
0.735086u
9
+ 0.0469147u
8
+ ··· + 7.47556u + 1.56866
a
8
=
−0.429336u
9
− 1.35298u
8
+ ··· − 12.2006u − 6.73772
−0.234717u
9
+ 0.913527u
8
+ ··· + 4.86712u + 4.70396
a
12
=
2.92164u
9
− 0.926676u
8
+ ··· + 10.7123u − 1.76757
−1.73741u
9
+ 0.411198u
8
+ ··· − 6.94734u + 0.356012
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
418432
90803
u
9
−
85648
90803
u
8
+
2619224
90803
u
7
+
532096
90803
u
6
−
11982728
90803
u
5
−
14963376
90803
u
4
−
12033384
90803
u
3
−
4718044
90803
u
2
−
2327168
90803
u −
672892
90803
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
2
, c
5
u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1
c
3
u
10
+ 4u
9
+ ··· + 10u + 1
c
4
u
10
− 2u
9
+ u
8
− 4u
7
+ 7u
6
+ 10u
5
− 8u
4
+ 14u
3
+ 53u
2
+ 25
c
6
u
10
− 6u
8
+ 27u
6
+ 30u
5
+ 30u
4
+ 14u
3
+ 10u
2
+ 2u + 1
c
7
, c
8
, c
11
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
c
9
, c
10
(u
2
+ 1)
5
c
12
u
10
+ u
8
− 10u
7
− u
6
+ 10u
5
+ 40u
4
+ 4u
3
− 3u
2
− 50u + 25
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
2
, c
5
(y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
c
3
y
10
+ 12y
9
+ ··· − 16y + 1
c
4
y
10
− 2y
9
+ ··· + 2650y + 625
c
6
y
10
− 12y
9
+ ··· + 16y + 1
c
7
, c
8
, c
11
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
c
9
, c
10
(y + 1)
10
c
12
y
10
+ 2y
9
+ ··· − 2650y + 625
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.482881 + 0.629714I
a = 1.000000 − 0.766826I
b = 0.766826I
−2.40108 −1.48114 + 0.I
u = −0.482881 − 0.629714I
a = 1.000000 + 0.766826I
b = − 0.766826I
−2.40108 −1.48114 + 0.I
u = 0.098692 + 0.530370I
a = 1.82238 + 0.33911I
b = −0.822375 − 0.339110I
−0.32910 − 1.53058I −0.51511 + 4.43065I
u = 0.098692 − 0.530370I
a = 1.82238 − 0.33911I
b = −0.822375 + 0.339110I
−0.32910 + 1.53058I −0.51511 − 4.43065I
u = −0.090267 + 0.435818I
a = 2.20015 − 0.45570I
b = −1.200150 + 0.455697I
−5.87256 + 4.40083I −4.74431 − 3.49859I
u = −0.090267 − 0.435818I
a = 2.20015 + 0.45570I
b = −1.200150 − 0.455697I
−5.87256 − 4.40083I −4.74431 + 3.49859I
u = −1.83956 + 0.80797I
a = −0.200152 + 0.455697I
b = 1.200150 − 0.455697I
−5.87256 + 4.40083I −4.74431 − 3.49859I
u = −1.83956 − 0.80797I
a = −0.200152 − 0.455697I
b = 1.200150 + 0.455697I
−5.87256 − 4.40083I −4.74431 + 3.49859I
u = 2.31402 + 1.21207I
a = 0.177625 + 0.339110I
b = 0.822375 − 0.339110I
−0.32910 + 1.53058I −0.51511 − 4.43065I
u = 2.31402 − 1.21207I
a = 0.177625 − 0.339110I
b = 0.822375 + 0.339110I
−0.32910 − 1.53058I −0.51511 + 4.43065I
10
III. I
u
3
= hb + 1, a − u + 1, u
3
− 2u
2
+ u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
u − 1
−1
a
7
=
1
u
2
a
2
=
u − 2
−1
a
1
=
−1
0
a
5
=
u − 1
−1
a
11
=
1
u
2
a
4
=
u
2
− 1
u
2
− 2u − 2
a
9
=
1
u
2
a
8
=
1
u
2
a
12
=
−1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u + 1)
3
c
3
, c
4
, c
9
c
10
u
3
+ u + 1
c
6
u
3
+ 2u
2
+ u − 1
c
7
, c
8
, c
11
u
3
c
12
u
3
− 2u
2
+ u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
9
c
10
y
3
+ 2y
2
+ y −1
c
6
, c
12
y
3
− 2y
2
+ 5y −1
c
7
, c
8
, c
11
y
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.23279 + 0.79255I
a = 0.232786 + 0.792552I
b = −1.00000
−1.64493 −6.00000
u = 1.23279 − 0.79255I
a = 0.232786 − 0.792552I
b = −1.00000
−1.64493 −6.00000
u = −0.465571
a = −1.46557
b = −1.00000
−1.64493 −6.00000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u + 1)
3
)(u
5
− 3u
4
+ ··· − u + 1)
2
(u
20
+ 12u
19
+ ··· − 19u + 4)
c
2
, c
5
((u + 1)
3
)(u
10
− 3u
8
+ ··· − u
2
+ 1)(u
20
− 2u
19
+ ··· − 5u + 2)
c
3
(u
3
+ u + 1)(u
10
+ 4u
9
+ ··· + 10u + 1)(u
20
− 5u
19
+ ··· − 4312u + 581)
c
4
(u
3
+ u + 1)(u
10
− 2u
9
+ ··· + 53u
2
+ 25)
· (u
20
− u
19
+ ··· − 622u + 97)
c
6
(u
3
+ 2u
2
+ u − 1)
· (u
10
− 6u
8
+ 27u
6
+ 30u
5
+ 30u
4
+ 14u
3
+ 10u
2
+ 2u + 1)
· (u
20
+ 5u
19
+ ··· + 5504u + 2167)
c
7
, c
8
, c
11
u
3
(u
10
+ 5u
8
+ ··· − u
2
+ 1)(u
20
+ 3u
19
+ ··· − 71u + 62)
c
9
((u
2
+ 1)
5
)(u
3
+ u + 1)(u
20
− u
19
+ ··· − 78u + 17)
c
10
((u
2
+ 1)
5
)(u
3
+ u + 1)(u
20
− u
19
+ ··· − 52u + 17)
c
12
(u
3
− 2u
2
+ u + 1)
· (u
10
+ u
8
− 10u
7
− u
6
+ 10u
5
+ 40u
4
+ 4u
3
− 3u
2
− 50u + 25)
· (u
20
+ u
19
+ ··· − 294u + 151)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
3
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
· (y
20
− 8y
19
+ ··· − 417y + 16)
c
2
, c
5
((y −1)
3
)(y
5
− 3y
4
+ ··· − y + 1)
2
(y
20
− 12y
19
+ ··· + 19y + 4)
c
3
(y
3
+ 2y
2
+ y −1)(y
10
+ 12y
9
+ ··· − 16y + 1)
· (y
20
− 75y
19
+ ··· + 153632486y + 337561)
c
4
(y
3
+ 2y
2
+ y −1)(y
10
− 2y
9
+ ··· + 2650y + 625)
· (y
20
+ 63y
19
+ ··· − 66784y + 9409)
c
6
(y
3
− 2y
2
+ 5y −1)(y
10
− 12y
9
+ ··· + 16y + 1)
· (y
20
− 35y
19
+ ··· − 16056826y + 4695889)
c
7
, c
8
, c
11
y
3
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
· (y
20
+ 37y
19
+ ··· + 27075y + 3844)
c
9
((y + 1)
10
)(y
3
+ 2y
2
+ y −1)(y
20
+ 39y
19
+ ··· + 6088y + 289)
c
10
((y + 1)
10
)(y
3
+ 2y
2
+ y −1)(y
20
− 5y
19
+ ··· − 120y + 289)
c
12
(y
3
− 2y
2
+ 5y −1)(y
10
+ 2y
9
+ ··· − 2650y + 625)
· (y
20
+ 47y
19
+ ··· + 412468y + 22801)
16
