12n
0469
(K12n
0469
)
A knot diagram
1
Linearized knot diagam
3 6 9 8 2 12 11 3 4 6 7 10
Solving Sequence
6,12 3,7
2 1 5 11 8 9 4 10
c
6
c
2
c
1
c
5
c
11
c
7
c
8
c
4
c
10
c
3
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h271358647u
35
− 529043524u
34
+ ··· + 488920819b − 215606766,
334439774u
35
− 1269481739u
34
+ ··· + 977841638a − 5972341699, u
36
− 2u
35
+ ··· − 4u − 1i
I
u
2
= hb + 1, −u
2
+ a − u − 2, u
3
+ u
2
+ 2u + 1i
I
u
3
= hb − 1, 2u
2
a + a
2
− 2au + 4a + u − 1, u
3
− u
2
+ 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 45 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
= h2.71 × 10
8
u
35
− 5.29 × 10
8
u
34
+ · · · + 4.89 × 10
8
b − 2.16 × 10
8
, 3.34 ×
10
8
u
35
−1.27×10
9
u
34
+· · · +9.78×10
8
a−5.97×10
9
, u
36
−2u
35
+· · · − 4u −1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
3
=
−0.342018u
35
+ 1.29825u
34
+ ··· − 14.1375u + 6.10768
−0.555016u
35
+ 1.08206u
34
+ ··· − 2.96956u + 0.440985
a
7
=
1
−u
2
a
2
=
−0.897034u
35
+ 2.38031u
34
+ ··· − 17.1071u + 6.54866
−0.555016u
35
+ 1.08206u
34
+ ··· − 2.96956u + 0.440985
a
1
=
u
7
+ 4u
5
+ 4u
3
−u
7
− 3u
5
− 2u
3
+ u
a
5
=
1.04147u
35
− 2.64685u
34
+ ··· + 19.6688u − 5.62579
0.678551u
35
− 1.22102u
34
+ ··· + 3.80116u − 0.427325
a
11
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
1.07868u
35
− 2.14851u
34
+ ··· + 24.9712u − 6.81062
−0.00885463u
35
− 0.0464490u
34
+ ··· + 2.49588u − 1.07868
a
4
=
0.440985u
35
− 1.43699u
34
+ ··· + 15.6653u − 4.73350
0.614212u
35
− 1.25941u
34
+ ··· + 4.73960u − 0.342018
a
10
=
−u
3
− 2u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
259508424
488920819
u
35
−
190324579
488920819
u
34
+ ··· +
7239334870
488920819
u +
1820082994
488920819
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
36
+ 46u
35
+ ··· + 1827u + 49
c
2
, c
5
u
36
+ 4u
35
+ ··· + 91u − 7
c
3
, c
8
, c
9
u
36
+ u
35
+ ··· − 8u − 8
c
4
u
36
− 3u
35
+ ··· + 8u − 8
c
6
, c
7
, c
11
u
36
− 2u
35
+ ··· − 4u − 1
c
10
u
36
+ 2u
35
+ ··· − 2232u − 481
c
12
u
36
+ 4u
35
+ ··· − 16u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
36
− 102y
35
+ ··· − 1510327y + 2401
c
2
, c
5
y
36
− 46y
35
+ ··· − 1827y + 49
c
3
, c
8
, c
9
y
36
− 29y
35
+ ··· − 832y + 64
c
4
y
36
+ 55y
35
+ ··· − 1728y + 64
c
6
, c
7
, c
11
y
36
+ 36y
35
+ ··· − 52y + 1
c
10
y
36
+ 20y
35
+ ··· − 9532084y + 231361
c
12
y
36
+ 44y
35
+ ··· − 532y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.746590 + 0.534072I
a = −1.14025 − 1.14185I
b = 1.71941 + 0.06440I
−10.55280 − 2.49065I 2.28481 + 2.78156I
u = −0.746590 − 0.534072I
a = −1.14025 + 1.14185I
b = 1.71941 − 0.06440I
−10.55280 + 2.49065I 2.28481 − 2.78156I
u = 0.664083 + 0.631741I
a = 1.014470 − 0.540455I
b = −1.67188 − 0.14945I
−6.71821 − 3.14670I 4.48459 + 0.40539I
u = 0.664083 − 0.631741I
a = 1.014470 + 0.540455I
b = −1.67188 + 0.14945I
−6.71821 + 3.14670I 4.48459 − 0.40539I
u = 0.784223 + 0.433590I
a = 1.14935 − 1.67783I
b = −1.65415 + 0.25580I
−6.05301 + 8.02106I 5.69652 − 5.48227I
u = 0.784223 − 0.433590I
a = 1.14935 + 1.67783I
b = −1.65415 − 0.25580I
−6.05301 − 8.02106I 5.69652 + 5.48227I
u = −0.754684
a = 2.04702
b = −1.31973
0.746951 8.79890
u = −0.202995 + 1.255340I
a = −0.951854 − 0.726217I
b = 0.175613 + 0.296050I
2.08619 − 3.03413I 10.51311 + 3.79199I
u = −0.202995 − 1.255340I
a = −0.951854 + 0.726217I
b = 0.175613 − 0.296050I
2.08619 + 3.03413I 10.51311 − 3.79199I
u = 0.597158 + 0.410952I
a = −0.03383 + 1.92390I
b = 0.617286 − 0.749282I
1.66187 + 4.15562I 8.00287 − 6.91393I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.597158 − 0.410952I
a = −0.03383 − 1.92390I
b = 0.617286 + 0.749282I
1.66187 − 4.15562I 8.00287 + 6.91393I
u = −0.326127 + 1.248070I
a = 0.633554 + 0.938984I
b = −1.359350 + 0.009058I
−3.12452 − 3.89594I 3.97835 + 4.14640I
u = −0.326127 − 1.248070I
a = 0.633554 − 0.938984I
b = −1.359350 − 0.009058I
−3.12452 + 3.89594I 3.97835 − 4.14640I
u = 0.504679 + 0.396227I
a = −1.116890 − 0.669513I
b = 0.659666 + 0.583318I
1.51779 − 0.52826I 7.65473 − 0.24051I
u = 0.504679 − 0.396227I
a = −1.116890 + 0.669513I
b = 0.659666 − 0.583318I
1.51779 + 0.52826I 7.65473 + 0.24051I
u = −0.631586
a = −1.76477
b = 0.225322
5.91452 16.7470
u = 0.096853 + 1.373970I
a = 0.057014 − 0.382407I
b = 0.233991 + 0.646152I
−3.76081 + 1.82148I 4.18456 − 2.97690I
u = 0.096853 − 1.373970I
a = 0.057014 + 0.382407I
b = 0.233991 − 0.646152I
−3.76081 − 1.82148I 4.18456 + 2.97690I
u = −0.032206 + 1.383860I
a = 0.93309 + 1.09446I
b = 1.227970 − 0.334502I
−1.31636 − 0.59124I 2.44003 + 0.I
u = −0.032206 − 1.383860I
a = 0.93309 − 1.09446I
b = 1.227970 + 0.334502I
−1.31636 + 0.59124I 2.44003 + 0.I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.209916 + 1.383740I
a = −0.161570 + 0.134658I
b = 0.763567 + 0.425752I
−4.01585 + 2.05418I 4.74740 + 1.07135I
u = 0.209916 − 1.383740I
a = −0.161570 − 0.134658I
b = 0.763567 − 0.425752I
−4.01585 − 2.05418I 4.74740 − 1.07135I
u = −0.10642 + 1.44688I
a = −0.474212 + 0.967148I
b = −1.044990 − 0.635725I
−7.20243 − 2.62870I 0. + 1.56071I
u = −0.10642 − 1.44688I
a = −0.474212 − 0.967148I
b = −1.044990 + 0.635725I
−7.20243 + 2.62870I 0. − 1.56071I
u = 0.21082 + 1.46540I
a = 0.586263 + 1.079000I
b = 0.693691 − 0.900679I
−4.41494 + 7.10948I 6.00000 − 5.97206I
u = 0.21082 − 1.46540I
a = 0.586263 − 1.079000I
b = 0.693691 + 0.900679I
−4.41494 − 7.10948I 6.00000 + 5.97206I
u = 0.29360 + 1.49537I
a = −0.31496 − 1.59906I
b = −1.68265 + 0.33795I
−12.2864 + 11.9589I 0
u = 0.29360 − 1.49537I
a = −0.31496 + 1.59906I
b = −1.68265 − 0.33795I
−12.2864 − 11.9589I 0
u = −0.334693 + 0.338551I
a = 0.58755 + 1.75938I
b = −0.790374 − 0.317839I
−1.37785 − 1.03574I −0.30331 + 3.73142I
u = −0.334693 − 0.338551I
a = 0.58755 − 1.75938I
b = −0.790374 + 0.317839I
−1.37785 + 1.03574I −0.30331 − 3.73142I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.25500 + 1.53176I
a = 0.269560 − 1.152100I
b = 1.79484 + 0.16936I
−17.3048 − 6.1571I 0
u = −0.25500 − 1.53176I
a = 0.269560 + 1.152100I
b = 1.79484 − 0.16936I
−17.3048 + 6.1571I 0
u = 0.19273 + 1.54392I
a = −0.368636 − 0.625846I
b = −1.78522 − 0.06510I
−13.90280 − 0.08031I 0
u = 0.19273 − 1.54392I
a = −0.368636 + 0.625846I
b = −1.78522 + 0.06510I
−13.90280 + 0.08031I 0
u = 0.431723
a = 0.157248
b = 0.236175
0.680363 15.1480
u = −0.145505
a = 8.22319
b = 1.06341
3.33955 1.75020
8
II. I
u
2
= hb + 1, −u
2
+ a − u − 2, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
3
=
u
2
+ u + 2
−1
a
7
=
1
−u
2
a
2
=
u
2
+ u + 1
−1
a
1
=
−1
0
a
5
=
u
2
+ u + 2
−1
a
11
=
−u
−u
2
− u − 1
a
8
=
u
2
+ 1
−u
2
− u − 1
a
9
=
u
2
+ 1
−u
2
− u − 1
a
4
=
u
2
+ u + 2
−1
a
10
=
u
2
+ 1
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
2
+ 4u + 4
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
8
c
9
u
3
c
5
(u + 1)
3
c
6
, c
7
u
3
+ u
2
+ 2u + 1
c
10
, c
12
u
3
+ u
2
− 1
c
11
u
3
− u
2
+ 2u − 1
10
(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
8
c
9
y
3
c
6
, c
7
, c
11
y
3
+ 3y
2
+ 2y −1
c
10
, c
12
y
3
− y
2
+ 2y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0.122561 + 0.744862I
b = −1.00000
−4.66906 − 2.82812I −0.18504 + 4.10401I
u = −0.215080 − 1.307140I
a = 0.122561 − 0.744862I
b = −1.00000
−4.66906 + 2.82812I −0.18504 − 4.10401I
u = −0.569840
a = 1.75488
b = −1.00000
−0.531480 2.37010
12
III. I
u
3
= hb − 1, 2u
2
a + a
2
− 2au + 4a + u − 1, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
3
=
a
1
a
7
=
1
−u
2
a
2
=
a + 1
1
a
1
=
1
0
a
5
=
−a
−1
a
11
=
−u
u
2
− u + 1
a
8
=
u
2
+ 1
−u
2
+ u − 1
a
9
=
u
2
a + a + 1
−u
2
a + au − 2u
2
− a + u − 2
a
4
=
u
2
− u + 2
−au − 2
a
10
=
−u
2
− 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 4u + 12
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
8
c
9
(u
2
− 2)
3
c
6
, c
7
(u
3
− u
2
+ 2u − 1)
2
c
10
(u
3
− u
2
+ 1)
2
c
11
(u
3
+ u
2
+ 2u + 1)
2
c
12
(u
3
+ u
2
− 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
8
c
9
(y −2)
6
c
6
, c
7
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
10
, c
12
(y
3
− y
2
+ 2y −1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.814156 − 0.050322I
b = 1.00000
0.26574 + 2.82812I 4.49024 − 2.97945I
u = 0.215080 + 1.307140I
a = −1.05928 + 1.54005I
b = 1.00000
0.26574 + 2.82812I 4.49024 − 2.97945I
u = 0.215080 − 1.307140I
a = 0.814156 + 0.050322I
b = 1.00000
0.26574 − 2.82812I 4.49024 + 2.97945I
u = 0.215080 − 1.307140I
a = −1.05928 − 1.54005I
b = 1.00000
0.26574 − 2.82812I 4.49024 + 2.97945I
u = 0.569840
a = 0.118556
b = 1.00000
4.40332 11.0200
u = 0.569840
a = −3.62831
b = 1.00000
4.40332 11.0200
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
36
+ 46u
35
+ ··· + 1827u + 49)
c
2
((u − 1)
3
)(u + 1)
6
(u
36
+ 4u
35
+ ··· + 91u − 7)
c
3
, c
8
, c
9
u
3
(u
2
− 2)
3
(u
36
+ u
35
+ ··· − 8u − 8)
c
4
u
3
(u
2
− 2)
3
(u
36
− 3u
35
+ ··· + 8u − 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
36
+ 4u
35
+ ··· + 91u − 7)
c
6
, c
7
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
36
− 2u
35
+ ··· − 4u − 1)
c
10
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
36
+ 2u
35
+ ··· − 2232u − 481)
c
11
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
(u
36
− 2u
35
+ ··· − 4u − 1)
c
12
((u
3
+ u
2
− 1)
3
)(u
36
+ 4u
35
+ ··· − 16u + 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
36
− 102y
35
+ ··· − 1510327y + 2401)
c
2
, c
5
((y −1)
9
)(y
36
− 46y
35
+ ··· − 1827y + 49)
c
3
, c
8
, c
9
y
3
(y −2)
6
(y
36
− 29y
35
+ ··· − 832y + 64)
c
4
y
3
(y −2)
6
(y
36
+ 55y
35
+ ··· − 1728y + 64)
c
6
, c
7
, c
11
((y
3
+ 3y
2
+ 2y −1)
3
)(y
36
+ 36y
35
+ ··· − 52y + 1)
c
10
((y
3
− y
2
+ 2y −1)
3
)(y
36
+ 20y
35
+ ··· − 9532084y + 231361)
c
12
((y
3
− y
2
+ 2y −1)
3
)(y
36
+ 44y
35
+ ··· − 532y + 1)
18
