12a
0837
(K12a
0837
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 10 11 12 3 1 8 7 6
Solving Sequence
7,12 3,8
9 4 11 6 1 10 5 2
c
7
c
8
c
3
c
11
c
6
c
12
c
10
c
5
c
2
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
78
− 30u
76
+ ··· + b − 2u, −u
78
− u
77
+ ··· + a + 2, u
80
+ 2u
79
+ ··· − 4u + 1i
I
u
2
= h−u
7
+ 2u
5
+ u
4
− u
3
− u
2
+ b − u, u
7
− u
6
− 2u
5
+ 2u
4
+ u
3
− u
2
+ a + u − 1,
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 88 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
= hu
78
−30u
76
+· · ·+b−2u, −u
78
−u
77
+· · ·+a+2, u
80
+2u
79
+· · ·−4u+1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
3
=
u
78
+ u
77
+ ··· − 5u − 2
−u
78
+ 30u
76
+ ··· + 2u
2
+ 2u
a
8
=
1
−u
2
a
9
=
u
15
− 6u
13
+ 14u
11
− 14u
9
+ 2u
7
+ 6u
5
− 2u
3
− 2u
−u
15
+ 5u
13
− 8u
11
+ u
9
+ 8u
7
− 4u
5
− 2u
3
+ u
a
4
=
u
78
+ u
77
+ ··· − u − 3
2u
79
+ u
78
+ ··· − 7u + 2
a
11
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
1
=
u
5
− 2u
3
+ u
−u
5
+ u
3
+ u
a
10
=
u
3
− 2u
−u
5
+ u
3
+ u
a
5
=
u
10
− 5u
8
+ 8u
6
− 3u
4
− 3u
2
+ 1
−u
12
+ 4u
10
− 4u
8
− 2u
6
+ 3u
4
+ 2u
2
a
2
=
u
78
+ u
77
+ ··· − 2u − 2
u
79
− 32u
77
+ ··· − 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −3u
79
+ 98u
77
+ ··· − 2u − 11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
80
− 9u
79
+ ··· + 6u − 1
c
3
, c
8
u
80
+ u
79
+ ··· − 384u − 256
c
5
u
80
+ 2u
79
+ ··· + 1220u + 757
c
6
, c
7
, c
11
u
80
− 2u
79
+ ··· + 4u + 1
c
9
u
80
− 6u
79
+ ··· + 123136u − 10736
c
10
, c
12
u
80
+ 6u
79
+ ··· − 292u − 53
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
80
− 79y
79
+ ··· + 20y + 1
c
3
, c
8
y
80
− 51y
79
+ ··· − 638976y + 65536
c
5
y
80
+ 6y
79
+ ··· + 12140628y + 573049
c
6
, c
7
, c
11
y
80
− 66y
79
+ ··· − 24y + 1
c
9
y
80
+ 42y
79
+ ··· − 4371656480y + 115261696
c
10
, c
12
y
80
+ 54y
79
+ ··· − 91836y + 2809
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.876291 + 0.160225I
a = 0.126511 + 0.194287I
b = 0.991591 − 0.142647I
−3.22938 + 0.02843I −1.84884 + 0.48915I
u = −0.876291 − 0.160225I
a = 0.126511 − 0.194287I
b = 0.991591 + 0.142647I
−3.22938 − 0.02843I −1.84884 − 0.48915I
u = 0.061548 + 0.829572I
a = −2.75467 − 0.96522I
b = −0.479375 − 0.000835I
−14.1634 − 1.3518I −7.92930 + 0.28677I
u = 0.061548 − 0.829572I
a = −2.75467 + 0.96522I
b = −0.479375 + 0.000835I
−14.1634 + 1.3518I −7.92930 − 0.28677I
u = 0.124055 + 0.821527I
a = 2.86514 + 0.84565I
b = 0.663517 + 0.615007I
−12.1701 + 11.2092I −5.92334 − 6.70693I
u = 0.124055 − 0.821527I
a = 2.86514 − 0.84565I
b = 0.663517 − 0.615007I
−12.1701 − 11.2092I −5.92334 + 6.70693I
u = 0.110007 + 0.808995I
a = −2.92348 − 0.92359I
b = −0.827690 − 0.452767I
−5.77427 + 7.00502I −3.79403 − 6.58682I
u = 0.110007 − 0.808995I
a = −2.92348 + 0.92359I
b = −0.827690 + 0.452767I
−5.77427 − 7.00502I −3.79403 + 6.58682I
u = 1.123910 + 0.371621I
a = −1.247920 + 0.018594I
b = 2.82867 + 1.63771I
−9.11869 − 6.88395I 0
u = 1.123910 − 0.371621I
a = −1.247920 − 0.018594I
b = 2.82867 − 1.63771I
−9.11869 + 6.88395I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.098406 + 0.810214I
a = −0.63923 − 1.38592I
b = −0.389110 + 0.252548I
−8.29890 − 4.46765I −5.51069 + 3.55720I
u = −0.098406 − 0.810214I
a = −0.63923 + 1.38592I
b = −0.389110 − 0.252548I
−8.29890 + 4.46765I −5.51069 − 3.55720I
u = 0.085461 + 0.805086I
a = 2.93657 + 0.81283I
b = 0.809055 + 0.115028I
−6.57809 + 1.74161I −5.90436 − 0.72823I
u = 0.085461 − 0.805086I
a = 2.93657 − 0.81283I
b = 0.809055 − 0.115028I
−6.57809 − 1.74161I −5.90436 + 0.72823I
u = 1.145300 + 0.351627I
a = 1.67209 − 0.06130I
b = −3.24427 − 1.61589I
−2.62101 − 2.78978I 0
u = 1.145300 − 0.351627I
a = 1.67209 + 0.06130I
b = −3.24427 + 1.61589I
−2.62101 + 2.78978I 0
u = −1.168410 + 0.294889I
a = 0.1148200 + 0.0781573I
b = −0.327938 − 0.588526I
0.663853 − 0.774238I 0
u = −1.168410 − 0.294889I
a = 0.1148200 − 0.0781573I
b = −0.327938 + 0.588526I
0.663853 + 0.774238I 0
u = −1.161670 + 0.354754I
a = −0.349914 − 0.227723I
b = 0.76660 + 1.23782I
−5.05506 + 0.24657I 0
u = −1.161670 − 0.354754I
a = −0.349914 + 0.227723I
b = 0.76660 − 1.23782I
−5.05506 − 0.24657I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.104056 + 0.769226I
a = 0.343299 + 0.727264I
b = 0.247120 − 0.134794I
−2.54055 − 3.11403I 2.14109 + 3.96126I
u = −0.104056 − 0.769226I
a = 0.343299 − 0.727264I
b = 0.247120 + 0.134794I
−2.54055 + 3.11403I 2.14109 − 3.96126I
u = 1.178980 + 0.350632I
a = −1.98357 − 0.34974I
b = 3.41068 + 2.02898I
−3.23753 + 2.44113I 0
u = 1.178980 − 0.350632I
a = −1.98357 + 0.34974I
b = 3.41068 − 2.02898I
−3.23753 − 2.44113I 0
u = −0.173756 + 0.720531I
a = 0.909003 − 0.853662I
b = −0.420842 − 0.340767I
−5.51527 − 3.58891I −5.34819 + 4.59326I
u = −0.173756 − 0.720531I
a = 0.909003 + 0.853662I
b = −0.420842 + 0.340767I
−5.51527 + 3.58891I −5.34819 − 4.59326I
u = 1.202850 + 0.378423I
a = 1.81156 + 0.67727I
b = −3.24144 − 2.11727I
−10.65550 + 5.69954I 0
u = 1.202850 − 0.378423I
a = 1.81156 − 0.67727I
b = −3.24144 + 2.11727I
−10.65550 − 5.69954I 0
u = 0.026848 + 0.719407I
a = 1.81521 − 0.24784I
b = 0.321846 − 0.756545I
−4.06965 + 1.05961I −4.75524 + 0.35267I
u = 0.026848 − 0.719407I
a = 1.81521 + 0.24784I
b = 0.321846 + 0.756545I
−4.06965 − 1.05961I −4.75524 − 0.35267I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.268360 + 0.284560I
a = −1.38083 − 1.56784I
b = 1.50576 + 2.71525I
−0.22850 + 2.54847I 0
u = 1.268360 − 0.284560I
a = −1.38083 + 1.56784I
b = 1.50576 − 2.71525I
−0.22850 − 2.54847I 0
u = −1.282660 + 0.242315I
a = −0.063197 + 0.146379I
b = −0.784403 + 0.183442I
2.57911 − 1.28933I 0
u = −1.282660 − 0.242315I
a = −0.063197 − 0.146379I
b = −0.784403 − 0.183442I
2.57911 + 1.28933I 0
u = −0.079891 + 0.677812I
a = −1.109980 + 0.265114I
b = 0.080705 + 0.500690I
−1.11785 − 1.88218I 1.71266 + 4.77492I
u = −0.079891 − 0.677812I
a = −1.109980 − 0.265114I
b = 0.080705 − 0.500690I
−1.11785 + 1.88218I 1.71266 − 4.77492I
u = −1.319450 + 0.050809I
a = 0.659804 − 0.757470I
b = −1.95095 + 1.27020I
3.13099 − 0.46029I 0
u = −1.319450 − 0.050809I
a = 0.659804 + 0.757470I
b = −1.95095 − 1.27020I
3.13099 + 0.46029I 0
u = −1.293290 + 0.302897I
a = −0.712362 − 0.212083I
b = 2.00229 − 0.73491I
0.05794 − 4.76170I 0
u = −1.293290 − 0.302897I
a = −0.712362 + 0.212083I
b = 2.00229 + 0.73491I
0.05794 + 4.76170I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.525555 + 0.406126I
a = −1.43656 − 1.57364I
b = 0.849649 + 0.237312I
−7.60137 + 7.31394I −2.35633 − 7.01924I
u = 0.525555 − 0.406126I
a = −1.43656 + 1.57364I
b = 0.849649 − 0.237312I
−7.60137 − 7.31394I −2.35633 + 7.01924I
u = 1.346940 + 0.076038I
a = −0.76105 − 1.54121I
b = 0.80314 + 1.89379I
1.81267 + 2.78487I 0
u = 1.346940 − 0.076038I
a = −0.76105 + 1.54121I
b = 0.80314 − 1.89379I
1.81267 − 2.78487I 0
u = 1.319870 + 0.292706I
a = 0.590163 + 1.204040I
b = −0.62449 − 1.86179I
3.28613 + 5.43808I 0
u = 1.319870 − 0.292706I
a = 0.590163 − 1.204040I
b = −0.62449 + 1.86179I
3.28613 − 5.43808I 0
u = −1.343590 + 0.190900I
a = 0.716855 + 0.205344I
b = −0.104581 − 0.432832I
−2.97413 + 1.50171I 0
u = −1.343590 − 0.190900I
a = 0.716855 − 0.205344I
b = −0.104581 + 0.432832I
−2.97413 − 1.50171I 0
u = −1.308570 + 0.369166I
a = 0.93076 − 1.51894I
b = −1.50790 + 3.63906I
−9.88263 − 2.95791I 0
u = −1.308570 − 0.369166I
a = 0.93076 + 1.51894I
b = −1.50790 − 3.63906I
−9.88263 + 2.95791I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.364130 + 0.025641I
a = 0.587914 + 0.874232I
b = −0.683473 − 1.056130I
6.59459 + 0.84569I 0
u = 1.364130 − 0.025641I
a = 0.587914 − 0.874232I
b = −0.683473 + 1.056130I
6.59459 − 0.84569I 0
u = −1.323710 + 0.350892I
a = −1.60329 + 1.52554I
b = 2.37847 − 3.93632I
−2.16036 − 5.90707I 0
u = −1.323710 − 0.350892I
a = −1.60329 − 1.52554I
b = 2.37847 + 3.93632I
−2.16036 + 5.90707I 0
u = −1.368010 + 0.072608I
a = −0.16776 + 1.58493I
b = 1.07759 − 2.56824I
4.39110 − 4.94022I 0
u = −1.368010 − 0.072608I
a = −0.16776 − 1.58493I
b = 1.07759 + 2.56824I
4.39110 + 4.94022I 0
u = 1.332520 + 0.330599I
a = −0.880089 + 0.059703I
b = 1.338240 + 0.031915I
1.97422 + 7.09169I 0
u = 1.332520 − 0.330599I
a = −0.880089 − 0.059703I
b = 1.338240 − 0.031915I
1.97422 − 7.09169I 0
u = 1.331770 + 0.353406I
a = 1.51891 − 0.09536I
b = −2.36445 − 0.02150I
−3.81014 + 8.66087I 0
u = 1.331770 − 0.353406I
a = 1.51891 + 0.09536I
b = −2.36445 + 0.02150I
−3.81014 − 8.66087I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.344834 + 0.516443I
a = 1.018030 + 0.976489I
b = −0.859848 − 0.693807I
−8.17586 − 3.95463I −4.10155 − 0.16840I
u = 0.344834 − 0.516443I
a = 1.018030 − 0.976489I
b = −0.859848 + 0.693807I
−8.17586 + 3.95463I −4.10155 + 0.16840I
u = −1.338410 + 0.351784I
a = 1.73352 − 1.98362I
b = −2.23896 + 4.53468I
−1.22329 − 11.18980I 0
u = −1.338410 − 0.351784I
a = 1.73352 + 1.98362I
b = −2.23896 − 4.53468I
−1.22329 + 11.18980I 0
u = 1.356590 + 0.303136I
a = 0.332649 − 1.319690I
b = −0.64556 + 1.86817I
−0.69617 + 7.31030I 0
u = 1.356590 − 0.303136I
a = 0.332649 + 1.319690I
b = −0.64556 − 1.86817I
−0.69617 − 7.31030I 0
u = −1.347690 + 0.357270I
a = −1.63097 + 2.21403I
b = 2.00997 − 4.65273I
−7.5407 − 15.4564I 0
u = −1.347690 − 0.357270I
a = −1.63097 − 2.21403I
b = 2.00997 + 4.65273I
−7.5407 + 15.4564I 0
u = −1.391820 + 0.084980I
a = −0.29047 − 1.82633I
b = −0.32123 + 2.86162I
−1.58820 − 8.79008I 0
u = −1.391820 − 0.084980I
a = −0.29047 + 1.82633I
b = −0.32123 − 2.86162I
−1.58820 + 8.79008I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.40474
a = −1.32314
b = 1.61553
3.28642 0
u = 0.463371 + 0.351769I
a = 1.64704 + 1.48525I
b = −0.749958 − 0.354643I
−1.28196 + 3.66343I 0.29546 − 7.86407I
u = 0.463371 − 0.351769I
a = 1.64704 − 1.48525I
b = −0.749958 + 0.354643I
−1.28196 − 3.66343I 0.29546 + 7.86407I
u = −0.403667 + 0.370681I
a = 1.087240 − 0.181150I
b = 0.117334 − 0.754012I
−3.56503 − 1.41634I −1.57318 + 4.55275I
u = −0.403667 − 0.370681I
a = 1.087240 + 0.181150I
b = 0.117334 + 0.754012I
−3.56503 + 1.41634I −1.57318 − 4.55275I
u = 0.319076 + 0.393637I
a = −1.59691 − 0.85238I
b = 0.699834 + 0.532015I
−1.69342 − 0.82758I −2.07495 − 0.65962I
u = 0.319076 − 0.393637I
a = −1.59691 + 0.85238I
b = 0.699834 − 0.532015I
−1.69342 + 0.82758I −2.07495 + 0.65962I
u = −0.464901 + 0.134379I
a = −0.471147 + 0.220662I
b = −0.244788 + 0.348012I
0.993986 − 0.381631I 9.03609 + 2.08695I
u = −0.464901 − 0.134379I
a = −0.471147 − 0.220662I
b = −0.244788 − 0.348012I
0.993986 + 0.381631I 9.03609 − 2.08695I
u = 0.227761
a = −3.50429
b = 0.602821
−1.28207 −11.0840
12
II. I
u
2
= h−u
7
+ 2u
5
+ u
4
− u
3
− u
2
+ b − u, u
7
− u
6
− 2u
5
+ 2u
4
+ u
3
− u
2
+
a + u − 1, u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
3
=
−u
7
+ u
6
+ 2u
5
− 2u
4
− u
3
+ u
2
− u + 1
u
7
− 2u
5
− u
4
+ u
3
+ u
2
+ u
a
8
=
1
−u
2
a
9
=
1
−u
2
a
4
=
−u
7
+ u
6
+ 2u
5
− 2u
4
− u
3
+ u
2
− u + 1
u
7
− 2u
5
− u
4
+ u
3
+ u
2
+ u
a
11
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
1
=
u
5
− 2u
3
+ u
−u
5
+ u
3
+ u
a
10
=
u
3
− 2u
−u
5
+ u
3
+ u
a
5
=
−u
5
+ 2u
3
− u
u
5
− u
3
− u
a
2
=
−u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ u
2
+ 1
u
7
− 3u
5
− u
4
+ 2u
3
+ u
2
+ 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
7
+ 6u
6
+ 2u
5
− 16u
4
− 5u
3
+ 9u
2
+ 8u + 9
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
8
u
8
c
4
(u + 1)
8
c
5
, c
9
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
6
, c
7
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
10
, c
12
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
11
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
8
y
8
c
5
, c
9
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
6
, c
7
, c
11
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
10
, c
12
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.180120 + 0.268597I
a = 0.663977 − 0.849844I
b = −0.33804 + 1.54318I
−0.604279 − 1.131230I −1.074136 + 0.216470I
u = −1.180120 − 0.268597I
a = 0.663977 + 0.849844I
b = −0.33804 − 1.54318I
−0.604279 + 1.131230I −1.074136 − 0.216470I
u = −0.108090 + 0.747508I
a = −0.727959 − 0.566792I
b = −0.306664 − 0.427719I
−3.80435 − 2.57849I −3.22623 + 3.25417I
u = −0.108090 − 0.747508I
a = −0.727959 + 0.566792I
b = −0.306664 + 0.427719I
−3.80435 + 2.57849I −3.22623 − 3.25417I
u = 1.37100
a = −0.910598
b = 1.71160
4.85780 7.89920
u = 1.334530 + 0.318930I
a = 0.690511 − 0.438656I
b = −1.53294 + 0.14882I
0.73474 + 6.44354I 2.34782 − 4.54733I
u = 1.334530 − 0.318930I
a = 0.690511 + 0.438656I
b = −1.53294 − 0.14882I
0.73474 − 6.44354I 2.34782 + 4.54733I
u = −0.463640
a = 1.65754
b = −0.356309
−0.799899 7.00590
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
8
)(u
80
− 9u
79
+ ··· + 6u − 1)
c
3
, c
8
u
8
(u
80
+ u
79
+ ··· − 384u − 256)
c
4
((u + 1)
8
)(u
80
− 9u
79
+ ··· + 6u − 1)
c
5
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
· (u
80
+ 2u
79
+ ··· + 1220u + 757)
c
6
, c
7
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
80
− 2u
79
+ ··· + 4u + 1)
c
9
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
· (u
80
− 6u
79
+ ··· + 123136u − 10736)
c
10
, c
12
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
80
+ 6u
79
+ ··· − 292u − 53)
c
11
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
80
− 2u
79
+ ··· + 4u + 1)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
8
)(y
80
− 79y
79
+ ··· + 20y + 1)
c
3
, c
8
y
8
(y
80
− 51y
79
+ ··· − 638976y + 65536)
c
5
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
80
+ 6y
79
+ ··· + 12140628y + 573049)
c
6
, c
7
, c
11
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
80
− 66y
79
+ ··· − 24y + 1)
c
9
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
80
+ 42y
79
+ ··· − 4371656480y + 115261696)
c
10
, c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
80
+ 54y
79
+ ··· − 91836y + 2809)
18
