12n
0008
(K12n
0008
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 9 11 5 12 1 7 10
Solving Sequence
3,5
2 6
1,10
11 12 9 4 8 7
c
2
c
5
c
1
c
10
c
12
c
9
c
4
c
8
c
7
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−40170837860333u
44
+ 273898398784184u
43
+ ··· + 57194726799376b − 67982681260688,
19829193212635u
44
− 141090112593855u
43
+ ··· + 57194726799376a + 390935602889337,
u
45
− 7u
44
+ ··· + 13u − 1i
I
u
2
= h−4a
4
u − 2a
3
u − 2a
3
+ 15a
2
+ 15au + 5b − 3u − 3, a
5
+ 4a
3
u + 4a
3
− 5a
2
− 2au + u + 1, u
2
+ u + 1i
I
u
3
= hu
2
+ b − u + 1, −u
4
+ u
3
− u
2
+ a + 1, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 60 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−4.02 × 10
13
u
44
+ 2.74 × 10
14
u
43
+ · · · + 5.72 × 10
13
b − 6.80 ×
10
13
, 1.98 × 10
13
u
44
− 1.41 × 10
14
u
43
+ · · · + 5.72 × 10
13
a + 3.91 ×
10
14
, u
45
− 7u
44
+ · · · + 13u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
10
=
−0.346696u
44
+ 2.46684u
43
+ ··· + 26.5461u − 6.83517
0.702352u
44
− 4.78888u
43
+ ··· − 4.12647u + 1.18862
a
11
=
−0.557267u
44
+ 3.84559u
43
+ ··· + 30.7805u − 8.13179
0.738824u
44
− 5.11904u
43
+ ··· − 4.10862u + 1.16824
a
12
=
0.948786u
44
− 6.63640u
43
+ ··· − 38.0499u + 9.08094
−0.352063u
44
+ 2.54967u
43
+ ··· + 10.9763u − 1.85015
a
9
=
1.14775u
44
− 8.03211u
43
+ ··· − 29.6955u + 4.33683
−0.0391055u
44
+ 0.648251u
43
+ ··· + 12.0682u − 1.19731
a
4
=
u
4
+ u
2
+ 1
u
6
+ 2u
4
+ u
2
a
8
=
1.14775u
44
− 8.03211u
43
+ ··· − 29.6955u + 4.33683
0.256025u
44
− 1.48232u
43
+ ··· + 10.9482u − 1.19944
a
7
=
−u
2
− 1
1
16
u
43
−
3
8
u
42
+ ··· +
11
4
u −
1
16
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
60972380652257
57194726799376
u
44
−
218749636382387
28597363399688
u
43
+ ··· +
1862967060953113
57194726799376
u −
208231471984261
28597363399688
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
45
+ 29u
44
+ ··· + 23u − 1
c
2
, c
5
u
45
+ 7u
44
+ ··· + 13u + 1
c
3
u
45
− 7u
44
+ ··· + 3u + 1
c
4
, c
8
u
45
+ 2u
44
+ ··· + 3072u
2
− 1024
c
6
u
45
− 4u
44
+ ··· + 2u − 1
c
7
, c
11
u
45
− 3u
44
+ ··· + 32u − 32
c
9
, c
10
, c
12
u
45
− 8u
44
+ ··· − 8u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
45
− 19y
44
+ ··· + 3799y −1
c
2
, c
5
y
45
+ 29y
44
+ ··· + 23y −1
c
3
y
45
− 67y
44
+ ··· + 23y −1
c
4
, c
8
y
45
− 60y
44
+ ··· + 6291456y −1048576
c
6
y
45
− 62y
44
+ ··· + 14y −1
c
7
, c
11
y
45
+ 39y
44
+ ··· − 4608y −1024
c
9
, c
10
, c
12
y
45
− 50y
44
+ ··· + 70y
2
− 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.448428 + 0.886782I
a = 3.29054 − 2.26057I
b = −0.04313 − 3.34261I
−1.93664 − 1.84719I 16.6318 + 21.5784I
u = −0.448428 − 0.886782I
a = 3.29054 + 2.26057I
b = −0.04313 + 3.34261I
−1.93664 + 1.84719I 16.6318 − 21.5784I
u = 1.028510 + 0.052980I
a = 0.875035 − 0.619833I
b = 0.792997 + 0.140103I
−8.08739 − 3.08320I −6.91743 + 2.52914I
u = 1.028510 − 0.052980I
a = 0.875035 + 0.619833I
b = 0.792997 − 0.140103I
−8.08739 + 3.08320I −6.91743 − 2.52914I
u = 0.141103 + 1.020830I
a = 0.123326 − 0.575048I
b = −0.602098 − 0.954633I
−1.88299 + 2.68710I −8.64703 − 3.04236I
u = 0.141103 − 1.020830I
a = 0.123326 + 0.575048I
b = −0.602098 + 0.954633I
−1.88299 − 2.68710I −8.64703 + 3.04236I
u = 1.04389
a = 3.48098
b = 2.56398
−10.3675 −7.74840
u = −0.814101 + 0.473442I
a = −1.85931 − 0.05637I
b = −1.69396 + 0.05742I
−6.36217 − 1.58930I −8.13248 + 2.19731I
u = −0.814101 − 0.473442I
a = −1.85931 + 0.05637I
b = −1.69396 − 0.05742I
−6.36217 + 1.58930I −8.13248 − 2.19731I
u = 0.321638 + 1.027370I
a = −0.357029 − 0.516807I
b = 0.721264 + 1.082480I
−6.92269 + 6.44252I −13.9889 − 5.4327I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.321638 − 1.027370I
a = −0.357029 + 0.516807I
b = 0.721264 − 1.082480I
−6.92269 − 6.44252I −13.9889 + 5.4327I
u = 1.074090 + 0.154235I
a = −2.72392 + 0.96068I
b = −2.30814 + 0.29423I
−15.4421 − 7.7641I −8.55026 + 3.19844I
u = 1.074090 − 0.154235I
a = −2.72392 − 0.96068I
b = −2.30814 − 0.29423I
−15.4421 + 7.7641I −8.55026 − 3.19844I
u = −0.558842 + 0.933277I
a = −1.240750 − 0.138767I
b = −0.96715 + 1.42473I
−0.77833 − 2.85163I −9.27214 + 0.I
u = −0.558842 − 0.933277I
a = −1.240750 + 0.138767I
b = −0.96715 − 1.42473I
−0.77833 + 2.85163I −9.27214 + 0.I
u = 0.038638 + 1.093140I
a = 0.764813 − 0.243323I
b = −1.73664 − 1.54360I
−4.80137 + 0.66249I −10.84717 + 0.I
u = 0.038638 − 1.093140I
a = 0.764813 + 0.243323I
b = −1.73664 + 1.54360I
−4.80137 − 0.66249I −10.84717 + 0.I
u = −0.095599 + 1.102010I
a = 0.691464 + 0.377467I
b = −0.104979 + 0.147687I
−3.59727 − 1.96898I −11.38503 + 2.89411I
u = −0.095599 − 1.102010I
a = 0.691464 − 0.377467I
b = −0.104979 − 0.147687I
−3.59727 + 1.96898I −11.38503 − 2.89411I
u = −0.464257 + 0.743525I
a = −0.128607 + 0.894701I
b = 0.879012 + 0.408581I
−0.10245 − 1.42024I −3.50008 + 5.75375I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.464257 − 0.743525I
a = −0.128607 − 0.894701I
b = 0.879012 − 0.408581I
−0.10245 + 1.42024I −3.50008 − 5.75375I
u = 0.840032
a = −1.01212
b = −0.718511
−3.18298 0.134470
u = −0.209858 + 0.761572I
a = −0.588066 + 0.563430I
b = 0.557407 + 0.666651I
−0.270200 − 1.317790I −2.14262 + 3.99951I
u = −0.209858 − 0.761572I
a = −0.588066 − 0.563430I
b = 0.557407 − 0.666651I
−0.270200 + 1.317790I −2.14262 − 3.99951I
u = −0.715937 + 1.055430I
a = 0.47884 + 1.81232I
b = 1.310670 + 0.236185I
−7.98005 − 4.06553I 0
u = −0.715937 − 1.055430I
a = 0.47884 − 1.81232I
b = 1.310670 − 0.236185I
−7.98005 + 4.06553I 0
u = 0.414289 + 0.552563I
a = −0.545658 − 0.831909I
b = −0.997622 + 0.175155I
−5.50892 − 3.28114I −8.73853 + 6.07709I
u = 0.414289 − 0.552563I
a = −0.545658 + 0.831909I
b = −0.997622 − 0.175155I
−5.50892 + 3.28114I −8.73853 − 6.07709I
u = 0.456920 + 1.233790I
a = 0.146367 − 0.696106I
b = 0.998812 − 0.254559I
−6.87697 + 4.62897I 0
u = 0.456920 − 1.233790I
a = 0.146367 + 0.696106I
b = 0.998812 + 0.254559I
−6.87697 − 4.62897I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.236201 + 1.315500I
a = 0.223845 + 0.997497I
b = 2.56560 + 0.53987I
−11.96940 − 4.68670I 0
u = −0.236201 − 1.315500I
a = 0.223845 − 0.997497I
b = 2.56560 − 0.53987I
−11.96940 + 4.68670I 0
u = 0.53792 + 1.32330I
a = −0.520780 + 0.416398I
b = −1.64420 + 0.02216I
−12.0273 + 8.6721I 0
u = 0.53792 − 1.32330I
a = −0.520780 − 0.416398I
b = −1.64420 − 0.02216I
−12.0273 − 8.6721I 0
u = 0.47555 + 1.35442I
a = 0.116010 + 0.860979I
b = −0.012680 + 0.444909I
−12.52320 + 2.24438I 0
u = 0.47555 − 1.35442I
a = 0.116010 − 0.860979I
b = −0.012680 − 0.444909I
−12.52320 − 2.24438I 0
u = 0.60027 + 1.30875I
a = 1.41525 − 1.67542I
b = 3.06355 − 0.12659I
−19.0180 + 13.7371I 0
u = 0.60027 − 1.30875I
a = 1.41525 + 1.67542I
b = 3.06355 + 0.12659I
−19.0180 − 13.7371I 0
u = 0.51317 + 1.34856I
a = −0.98613 + 2.21993I
b = −3.05558 + 1.18362I
−14.5875 + 5.5390I 0
u = 0.51317 − 1.34856I
a = −0.98613 − 2.21993I
b = −3.05558 − 1.18362I
−14.5875 − 5.5390I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.40664 + 1.42221I
a = 0.22163 − 1.91263I
b = 1.91377 − 1.51963I
18.8970 − 2.5007I 0
u = 0.40664 − 1.42221I
a = 0.22163 + 1.91263I
b = 1.91377 + 1.51963I
18.8970 + 2.5007I 0
u = 0.027627 + 0.285904I
a = −1.54121 + 0.47713I
b = 0.419324 + 0.349696I
−0.299097 − 1.132870I −4.25483 + 6.05161I
u = 0.027627 − 0.285904I
a = −1.54121 − 0.47713I
b = 0.419324 − 0.349696I
−0.299097 + 1.132870I −4.25483 − 6.05161I
u = 0.129774
a = −5.18021
b = 1.04208
−2.19508 −3.54080
9
II. I
u
2
=
h−4a
4
u −2a
3
u +· · ·+ 15a
2
−3, a
5
+4a
3
u +4a
3
−5a
2
−2au + u+1, u
2
+u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−u − 1
a
6
=
u
u + 1
a
1
=
−u
−u − 1
a
10
=
a
4
5
a
4
u +
2
5
a
3
u + ··· − 3a
2
+
3
5
a
11
=
−
4
5
a
4
+
2
5
a
3
u − 3a
2
u − 3a
2
+ 3a +
3
5
u
8
5
a
4
u +
4
5
a
3
u + ··· − 6a
2
+
6
5
a
12
=
−
2
5
a
4
+
1
5
a
3
u − 2a
2
u − 2a
2
+ a −
1
5
u
−
1
5
a
4
u −
3
5
a
3
u + ··· −
3
5
a
3
−
2
5
a
9
=
0
−
1
5
a
4
u −
3
5
a
3
u + ··· −
3
5
a
3
−
2
5
a
4
=
0
u
a
8
=
0
−
1
5
a
4
u −
3
5
a
3
u + ··· −
3
5
a
3
−
2
5
a
7
=
u
−
2
5
a
4
u −
6
5
a
3
u + ··· −
6
5
a
3
+
6
5
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −
14
5
a
4
u − 3a
4
−
7
5
a
3
u −
2
5
a
3
− 15a
2
u − 3a
2
+ 11au + 16a +
27
5
u −
33
5
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
5
c
2
(u
2
+ u + 1)
5
c
4
, c
8
u
10
c
6
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
7
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
9
, c
10
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
11
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
12
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
5
c
4
, c
8
y
10
c
6
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
7
, c
11
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
9
, c
10
, c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 1.012010 − 0.734701I
b = 0.768927 − 0.124653I
−5.87256 + 2.37095I −11.57979 + 0.88917I
u = −0.500000 + 0.866025I
a = 0.130268 − 1.243770I
b = −0.492416 + 0.603584I
−5.87256 − 6.43072I −6.27578 + 5.55522I
u = −0.500000 + 0.866025I
a = −0.364485 − 0.347423I
b = −1.114310 + 0.148503I
−0.329100 − 0.499304I −6.44749 − 1.44665I
u = −0.500000 + 0.866025I
a = 0.483119 + 0.141942I
b = 0.685764 − 0.890773I
−0.32910 − 3.56046I −2.59686 + 8.38554I
u = −0.500000 + 0.866025I
a = −1.26091 + 2.18395I
b = 0.652039 + 1.129360I
−2.40108 − 2.02988I −7.10008 + 5.66929I
u = −0.500000 − 0.866025I
a = 1.012010 + 0.734701I
b = 0.768927 + 0.124653I
−5.87256 − 2.37095I −11.57979 − 0.88917I
u = −0.500000 − 0.866025I
a = 0.130268 + 1.243770I
b = −0.492416 − 0.603584I
−5.87256 + 6.43072I −6.27578 − 5.55522I
u = −0.500000 − 0.866025I
a = −0.364485 + 0.347423I
b = −1.114310 − 0.148503I
−0.329100 + 0.499304I −6.44749 + 1.44665I
u = −0.500000 − 0.866025I
a = 0.483119 − 0.141942I
b = 0.685764 + 0.890773I
−0.32910 + 3.56046I −2.59686 − 8.38554I
u = −0.500000 − 0.866025I
a = −1.26091 − 2.18395I
b = 0.652039 − 1.129360I
−2.40108 + 2.02988I −7.10008 − 5.66929I
13
III. I
u
3
= hu
2
+ b − u + 1, −u
4
+ u
3
− u
2
+ a + 1, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
10
=
u
4
− u
3
+ u
2
− 1
−u
2
+ u − 1
a
11
=
u
4
− u
3
+ 2u
2
u − 1
a
12
=
u
4
− u
3
+ 2u
2
u − 1
a
9
=
−u
2
− 1
−u
2
a
4
=
u
4
+ u
2
+ 1
u
4
− u
3
+ u
2
+ 1
a
8
=
−u
2
− 1
u
4
a
7
=
−u
2
− 1
u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
4
+ 5u
3
− 4u
2
− 9
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
2
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
3
, c
4
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
5
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
6
u
5
− 5u
4
+ 8u
3
− 3u
2
− u − 1
c
7
, c
11
u
5
c
8
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
9
, c
10
(u − 1)
5
c
12
(u + 1)
5
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
2
, c
5
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
3
, c
4
, c
8
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y −1
c
7
, c
11
y
5
c
9
, c
10
, c
12
(y −1)
5
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339110 + 0.822375I
a = −2.20635 + 0.34085I
b = −0.77780 + 1.38013I
−1.97403 − 1.53058I −3.52158 − 1.00973I
u = −0.339110 − 0.822375I
a = −2.20635 − 0.34085I
b = −0.77780 − 1.38013I
−1.97403 + 1.53058I −3.52158 + 1.00973I
u = 0.766826
a = −0.517119
b = −0.821196
−4.04602 −10.1350
u = 0.455697 + 1.200150I
a = −0.035087 − 0.621896I
b = 0.688402 + 0.106340I
−7.51750 + 4.40083I −14.4110 − 1.1901I
u = 0.455697 − 1.200150I
a = −0.035087 + 0.621896I
b = 0.688402 − 0.106340I
−7.51750 − 4.40083I −14.4110 + 1.1901I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
5
− 3u
4
+ ··· − u + 1)(u
45
+ 29u
44
+ ··· + 23u − 1)
c
2
((u
2
+ u + 1)
5
)(u
5
− u
4
+ ··· + u − 1)(u
45
+ 7u
44
+ ··· + 13u + 1)
c
3
((u
2
− u + 1)
5
)(u
5
+ u
4
+ ··· + u − 1)(u
45
− 7u
44
+ ··· + 3u + 1)
c
4
u
10
(u
5
+ u
4
+ ··· + u − 1)(u
45
+ 2u
44
+ ··· + 3072u
2
− 1024)
c
5
((u
2
− u + 1)
5
)(u
5
+ u
4
+ ··· + u + 1)(u
45
+ 7u
44
+ ··· + 13u + 1)
c
6
(u
5
− 5u
4
+ 8u
3
− 3u
2
− u − 1)(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
· (u
45
− 4u
44
+ ··· + 2u − 1)
c
7
u
5
(u
5
− u
4
+ ··· + u − 1)
2
(u
45
− 3u
44
+ ··· + 32u − 32)
c
8
u
10
(u
5
− u
4
+ ··· + u + 1)(u
45
+ 2u
44
+ ··· + 3072u
2
− 1024)
c
9
, c
10
((u − 1)
5
)(u
5
+ u
4
+ ··· + u − 1)
2
(u
45
− 8u
44
+ ··· − 8u − 1)
c
11
u
5
(u
5
+ u
4
+ ··· + u + 1)
2
(u
45
− 3u
44
+ ··· + 32u − 32)
c
12
((u + 1)
5
)(u
5
− u
4
+ ··· + u + 1)
2
(u
45
− 8u
44
+ ··· − 8u − 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)
5
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
· (y
45
− 19y
44
+ ··· + 3799y −1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
5
+ 3y
4
+ ··· − y −1)(y
45
+ 29y
44
+ ··· + 23y −1)
c
3
(y
2
+ y + 1)
5
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
· (y
45
− 67y
44
+ ··· + 23y −1)
c
4
, c
8
y
10
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
· (y
45
− 60y
44
+ ··· + 6291456y −1048576)
c
6
(y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y −1)(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
· (y
45
− 62y
44
+ ··· + 14y −1)
c
7
, c
11
y
5
(y
5
+ 3y
4
+ ··· − y −1)
2
(y
45
+ 39y
44
+ ··· − 4608y −1024)
c
9
, c
10
, c
12
((y −1)
5
)(y
5
− 5y
4
+ ··· − y −1)
2
(y
45
− 50y
44
+ ··· + 70y
2
− 1)
19
