12a
0591
(K12a
0591
)
A knot diagram
1
Linearized knot diagam
3 7 9 11 8 2 6 5 12 1 4 10
Solving Sequence
2,6
7 3 8 1 5
9,11
4 10 12
c
6
c
2
c
7
c
1
c
5
c
8
c
4
c
10
c
12
c
3
, c
9
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
58
− 4u
57
+ ··· + b + 2, −u
58
+ 3u
57
+ ··· + a − 3, u
59
− 2u
58
+ ··· + 6u
2
− 1i
I
u
2
= h−u
2
+ b, u
4
+ a − u + 1, u
5
+ u
4
− u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 64 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
= h2u
58
−4u
57
+· · ·+b+2, −u
58
+3u
57
+· · ·+a−3, u
59
−2u
58
+· · ·+6u
2
−1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
u
2
a
1
=
u
3
u
5
− u
3
+ u
a
5
=
u
4
− u
2
+ 1
−u
4
a
9
=
−u
6
+ u
4
− 2u
2
+ 1
u
6
+ u
2
a
11
=
u
58
− 3u
57
+ ··· + 8u + 3
−2u
58
+ 4u
57
+ ··· − 2u − 2
a
4
=
u
15
− 2u
13
+ 6u
11
− 8u
9
+ 10u
7
− 8u
5
+ 4u
3
− 2u
−u
15
+ u
13
− 4u
11
+ 3u
9
− 4u
7
+ 2u
5
− 2u
3
+ u
a
10
=
u
58
− 2u
57
+ ··· + 7u + 2
−u
58
+ 2u
57
+ ··· − 2u − 1
a
12
=
u
58
− u
57
+ ··· + 6u + 2
−u
56
+ 6u
54
+ ··· − 5u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6u
58
− 9u
57
+ ··· + 11u − 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
c
8
u
59
+ 12u
58
+ ··· + 12u + 1
c
2
, c
6
u
59
− 2u
58
+ ··· + 6u
2
− 1
c
3
u
59
− 2u
58
+ ··· + 770u − 769
c
4
, c
11
u
59
− u
58
+ ··· − 160u − 32
c
9
, c
10
, c
12
u
59
− 6u
58
+ ··· + 6u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
8
y
59
+ 72y
58
+ ··· + 524y
2
− 1
c
2
, c
6
y
59
− 12y
58
+ ··· + 12y −1
c
3
y
59
− 12y
58
+ ··· + 8725844y −591361
c
4
, c
11
y
59
+ 33y
58
+ ··· − 512y −1024
c
9
, c
10
, c
12
y
59
− 56y
58
+ ··· + 38y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.927503 + 0.362332I
a = 1.043690 + 0.616352I
b = −0.037342 − 1.128440I
−8.68686 − 0.29443I −15.0428 − 2.5911I
u = −0.927503 − 0.362332I
a = 1.043690 − 0.616352I
b = −0.037342 + 1.128440I
−8.68686 + 0.29443I −15.0428 + 2.5911I
u = 0.888453 + 0.481598I
a = −0.283319 − 1.162560I
b = −0.195891 + 1.201930I
−3.19795 − 4.45742I −10.97215 + 6.34811I
u = 0.888453 − 0.481598I
a = −0.283319 + 1.162560I
b = −0.195891 − 1.201930I
−3.19795 + 4.45742I −10.97215 − 6.34811I
u = 0.809645 + 0.552812I
a = 0.191552 + 0.505222I
b = 0.275906 − 0.423947I
1.77330 − 2.83842I −2.03651 + 4.98214I
u = 0.809645 − 0.552812I
a = 0.191552 − 0.505222I
b = 0.275906 + 0.423947I
1.77330 + 2.83842I −2.03651 − 4.98214I
u = −0.851357 + 0.451422I
a = −0.989465 − 0.251556I
b = −0.627413 + 0.684041I
−1.68118 + 2.03005I −12.04374 − 3.74651I
u = −0.851357 − 0.451422I
a = −0.989465 + 0.251556I
b = −0.627413 − 0.684041I
−1.68118 − 2.03005I −12.04374 + 3.74651I
u = −0.901916 + 0.511776I
a = 0.912140 + 0.147817I
b = 0.752079 + 0.229651I
−0.66335 + 6.70936I −8.90418 − 9.33390I
u = −0.901916 − 0.511776I
a = 0.912140 − 0.147817I
b = 0.752079 − 0.229651I
−0.66335 − 6.70936I −8.90418 + 9.33390I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.949474 + 0.086160I
a = −0.813004 − 0.661879I
b = 0.916859 − 0.826254I
−10.19780 − 5.55174I −17.2725 + 4.5392I
u = 0.949474 − 0.086160I
a = −0.813004 + 0.661879I
b = 0.916859 + 0.826254I
−10.19780 + 5.55174I −17.2725 − 4.5392I
u = 0.774062 + 0.735471I
a = −0.213712 − 0.409713I
b = −0.406362 − 0.573958I
−1.80101 − 2.68324I −9.89712 + 3.37596I
u = 0.774062 − 0.735471I
a = −0.213712 + 0.409713I
b = −0.406362 + 0.573958I
−1.80101 + 2.68324I −9.89712 − 3.37596I
u = −0.947787 + 0.525245I
a = −0.916835 + 0.152219I
b = −0.313188 − 0.887130I
−6.72517 + 10.58920I −11.8410 − 8.8623I
u = −0.947787 − 0.525245I
a = −0.916835 − 0.152219I
b = −0.313188 + 0.887130I
−6.72517 − 10.58920I −11.8410 + 8.8623I
u = 0.666174 + 0.595869I
a = −0.674977 + 0.451724I
b = 0.190354 + 0.211599I
2.23740 − 1.50357I −0.61420 + 3.68395I
u = 0.666174 − 0.595869I
a = −0.674977 − 0.451724I
b = 0.190354 − 0.211599I
2.23740 + 1.50357I −0.61420 − 3.68395I
u = −0.885590
a = −0.363107
b = 1.58758
−5.76666 −16.7450
u = 0.880462 + 0.048539I
a = 0.431969 + 1.195720I
b = −0.367625 + 0.381368I
−3.70562 − 2.28031I −15.9939 + 4.5234I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.880462 − 0.048539I
a = 0.431969 − 1.195720I
b = −0.367625 − 0.381368I
−3.70562 + 2.28031I −15.9939 − 4.5234I
u = −0.475499 + 0.709477I
a = 1.064150 + 0.054307I
b = 0.424369 + 0.494166I
−5.19700 − 6.02928I −8.35124 + 3.27137I
u = −0.475499 − 0.709477I
a = 1.064150 − 0.054307I
b = 0.424369 − 0.494166I
−5.19700 + 6.02928I −8.35124 − 3.27137I
u = −0.506438 + 0.625369I
a = −0.304837 − 0.259828I
b = −0.535812 − 0.823997I
0.58647 − 2.42870I −4.78642 + 3.26862I
u = −0.506438 − 0.625369I
a = −0.304837 + 0.259828I
b = −0.535812 + 0.823997I
0.58647 + 2.42870I −4.78642 − 3.26862I
u = 0.911469 + 0.817194I
a = −0.136623 − 0.058652I
b = −0.338056 − 0.728570I
−1.78315 − 3.05813I 0
u = 0.911469 − 0.817194I
a = −0.136623 + 0.058652I
b = −0.338056 + 0.728570I
−1.78315 + 3.05813I 0
u = −0.629873 + 0.451932I
a = −0.591616 + 0.250732I
b = −0.012775 + 1.272100I
−0.85526 + 1.48403I −10.53137 − 2.98098I
u = −0.629873 − 0.451932I
a = −0.591616 − 0.250732I
b = −0.012775 − 1.272100I
−0.85526 − 1.48403I −10.53137 + 2.98098I
u = −0.891610 + 0.896465I
a = −2.53033 + 1.42185I
b = 3.38449 + 1.43468I
5.57557 − 0.69509I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.891610 − 0.896465I
a = −2.53033 − 1.42185I
b = 3.38449 − 1.43468I
5.57557 + 0.69509I 0
u = 0.901021 + 0.888004I
a = −2.01271 − 0.18754I
b = 2.23237 − 1.26339I
6.70724 − 2.05355I 0
u = 0.901021 − 0.888004I
a = −2.01271 + 0.18754I
b = 2.23237 + 1.26339I
6.70724 + 2.05355I 0
u = 0.891289 + 0.906965I
a = 2.27708 + 1.72286I
b = −3.68686 + 0.28774I
8.48613 + 2.97023I 0
u = 0.891289 − 0.906965I
a = 2.27708 − 1.72286I
b = −3.68686 − 0.28774I
8.48613 − 2.97023I 0
u = 0.881355 + 0.919934I
a = −1.79426 − 2.72825I
b = 4.07557 + 0.84147I
2.80078 + 7.27092I 0
u = 0.881355 − 0.919934I
a = −1.79426 + 2.72825I
b = 4.07557 − 0.84147I
2.80078 − 7.27092I 0
u = 0.475104 + 0.546567I
a = 1.88094 − 0.67202I
b = −0.427394 − 0.066566I
−1.93785 + 0.48814I −6.57596 + 0.51918I
u = 0.475104 − 0.546567I
a = 1.88094 + 0.67202I
b = −0.427394 + 0.066566I
−1.93785 − 0.48814I −6.57596 − 0.51918I
u = 0.944066 + 0.868730I
a = 0.67891 + 2.04086I
b = −2.18407 − 0.79948I
6.56960 − 4.44376I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.944066 − 0.868730I
a = 0.67891 − 2.04086I
b = −2.18407 + 0.79948I
6.56960 + 4.44376I 0
u = −0.917197 + 0.901529I
a = 1.95344 − 0.68617I
b = −2.33755 − 1.31220I
10.83880 + 1.97378I 0
u = −0.917197 − 0.901529I
a = 1.95344 + 0.68617I
b = −2.33755 + 1.31220I
10.83880 − 1.97378I 0
u = −0.954974 + 0.867777I
a = 1.60650 − 2.23842I
b = −3.94977 + 0.62849I
5.37300 + 7.21470I 0
u = −0.954974 − 0.867777I
a = 1.60650 + 2.23842I
b = −3.94977 − 0.62849I
5.37300 − 7.21470I 0
u = −0.943653 + 0.888657I
a = −0.91907 + 1.76170I
b = 2.60787 − 0.77120I
10.75340 + 4.63087I 0
u = −0.943653 − 0.888657I
a = −0.91907 − 1.76170I
b = 2.60787 + 0.77120I
10.75340 − 4.63087I 0
u = 0.961984 + 0.873239I
a = −2.16430 − 2.05346I
b = 3.70194 − 0.40138I
8.25862 − 9.54022I 0
u = 0.961984 − 0.873239I
a = −2.16430 + 2.05346I
b = 3.70194 + 0.40138I
8.25862 + 9.54022I 0
u = −0.700512
a = −0.0483338
b = −0.474525
−1.03527 −9.24070
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.975652 + 0.873157I
a = 2.99487 + 1.42749I
b = −4.09520 + 1.68332I
2.49582 − 13.87990I 0
u = 0.975652 − 0.873157I
a = 2.99487 − 1.42749I
b = −4.09520 − 1.68332I
2.49582 + 13.87990I 0
u = −0.941988 + 0.917429I
a = −1.46830 − 1.66648I
b = −0.16686 + 2.84797I
8.35499 + 3.37505I 0
u = −0.941988 − 0.917429I
a = −1.46830 + 1.66648I
b = −0.16686 − 2.84797I
8.35499 − 3.37505I 0
u = −0.199230 + 0.628329I
a = 1.03443 + 1.42099I
b = −0.437533 − 0.953757I
−6.42147 + 3.73715I −8.91804 − 3.23014I
u = −0.199230 − 0.628329I
a = 1.03443 − 1.42099I
b = −0.437533 + 0.953757I
−6.42147 − 3.73715I −8.91804 + 3.23014I
u = −0.227842 + 0.411818I
a = −1.082040 − 0.120959I
b = 0.324797 + 0.691474I
−0.462772 + 1.214880I −5.21396 − 5.22189I
u = −0.227842 − 0.411818I
a = −1.082040 + 0.120959I
b = 0.324797 − 0.691474I
−0.462772 − 1.214880I −5.21396 + 5.22189I
u = 0.399421
a = 3.06288
b = −0.646854
−2.12944 −0.328760
10
II. I
u
2
= h−u
2
+ b, u
4
+ a − u + 1, u
5
+ u
4
− u
2
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
u
2
a
1
=
u
3
−u
4
− u
3
+ u
2
− 1
a
5
=
u
4
− u
2
+ 1
−u
4
a
9
=
−u
3
u
4
+ u
3
− u
2
+ 1
a
11
=
−u
4
+ u − 1
u
2
a
4
=
u
4
− u
2
+ 1
−u
4
a
10
=
−u
4
− u
3
+ u − 1
u
4
+ u
3
+ 1
a
12
=
−u
4
+ u − 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
4
− u
3
+ 5u
2
+ 7u − 18
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
2
u
5
− u
4
+ u
2
+ u − 1
c
4
, c
11
u
5
c
6
u
5
+ u
4
− u
2
+ u + 1
c
7
, c
8
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
9
, c
10
(u − 1)
5
c
12
(u + 1)
5
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
8
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
2
, c
6
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
4
, c
11
y
5
c
9
, c
10
, c
12
(y −1)
5
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.758138 + 0.584034I
a = 0.487744 + 0.170166I
b = 0.233677 + 0.885557I
0.17487 − 2.21397I −5.34777 + 4.39723I
u = 0.758138 − 0.584034I
a = 0.487744 − 0.170166I
b = 0.233677 − 0.885557I
0.17487 + 2.21397I −5.34777 − 4.39723I
u = −0.935538 + 0.903908I
a = 0.92150 + 1.10071I
b = 0.05818 − 1.69128I
9.31336 + 3.33174I −2.87586 − 2.18947I
u = −0.935538 − 0.903908I
a = 0.92150 − 1.10071I
b = 0.05818 + 1.69128I
9.31336 − 3.33174I −2.87586 + 2.18947I
u = −0.645200
a = −1.81849
b = 0.416284
−2.52712 −21.5530
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
59
+ 12u
58
+ ··· + 12u + 1)
c
2
(u
5
− u
4
+ u
2
+ u − 1)(u
59
− 2u
58
+ ··· + 6u
2
− 1)
c
3
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
59
− 2u
58
+ ··· + 770u − 769)
c
4
, c
11
u
5
(u
59
− u
58
+ ··· − 160u − 32)
c
6
(u
5
+ u
4
− u
2
+ u + 1)(u
59
− 2u
58
+ ··· + 6u
2
− 1)
c
7
, c
8
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
59
+ 12u
58
+ ··· + 12u + 1)
c
9
, c
10
((u − 1)
5
)(u
59
− 6u
58
+ ··· + 6u − 1)
c
12
((u + 1)
5
)(u
59
− 6u
58
+ ··· + 6u − 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
8
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
59
+ 72y
58
+ ··· + 524y
2
− 1)
c
2
, c
6
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
59
− 12y
58
+ ··· + 12y −1)
c
3
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
· (y
59
− 12y
58
+ ··· + 8725844y −591361)
c
4
, c
11
y
5
(y
59
+ 33y
58
+ ··· − 512y −1024)
c
9
, c
10
, c
12
((y −1)
5
)(y
59
− 56y
58
+ ··· + 38y −1)
16
