12n
0083
(K12n
0083
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 4 10 12 5 1 8 11
Solving Sequence
2,5 3,10
6 1 11 4 7 9 12 8
c
2
c
5
c
1
c
10
c
4
c
6
c
9
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−7.39937 × 10
54
u
61
− 7.95828 × 10
55
u
60
+ ··· + 9.36658 × 10
54
b + 4.21903 × 10
55
,
− 4.20292 × 10
54
u
61
− 2.34926 × 10
55
u
60
+ ··· + 4.68329 × 10
54
a + 1.31817 × 10
55
, u
62
+ 5u
61
+ ··· − 7u + 1i
I
u
2
= hu
2
+ b + u, a, u
3
+ u
2
− 1i
I
u
3
= hb
2
+ 3u
2
+ b + 5u + 4, a, u
3
+ u
2
− 1i
I
u
4
= hb, a + 1, u − 1i
* 4 irreducible components of dim
C
= 0, with total 72 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−7.40×10
54
u
61
−7.96×10
55
u
60
+· · ·+9.37×10
54
b+4.22×10
55
, −4.20×
10
54
u
61
−2.35×10
55
u
60
+· · ·+4.68×10
54
a+1.32×10
55
, u
62
+5u
61
+· · ·−7u+1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
10
=
0.897429u
61
+ 5.01625u
60
+ ··· − 31.2394u − 2.81463
0.789976u
61
+ 8.49647u
60
+ ··· + 33.8636u − 4.50435
a
6
=
0.284554u
61
+ 1.83768u
60
+ ··· − 2.18948u + 3.46404
−0.353321u
61
− 0.129700u
60
+ ··· − 1.33250u − 0.387951
a
1
=
−u
2
+ 1
−u
4
a
11
=
0.873580u
61
+ 3.94211u
60
+ ··· − 25.2656u − 4.12853
1.29842u
61
+ 10.9238u
60
+ ··· + 41.7471u − 5.56745
a
4
=
u
u
a
7
=
0.834045u
61
+ 3.97405u
60
+ ··· − 11.3813u + 4.68603
0.196170u
61
+ 2.00667u
60
+ ··· − 10.5243u + 0.834045
a
9
=
0.897429u
61
+ 5.01625u
60
+ ··· − 31.2394u − 2.81463
0.271747u
61
+ 6.65168u
60
+ ··· + 36.6699u − 5.03346
a
12
=
−0.835301u
61
− 3.12639u
60
+ ··· + 14.5834u + 4.37470
−5.11158u
61
− 24.5806u
60
+ ··· − 4.66960u + 0.596938
a
8
=
u
2
− 1
2.88576u
61
+ 15.7183u
60
+ ··· + 14.2886u − 2.18510
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6.52357u
61
+ 48.3926u
60
+ ··· + 104.147u − 23.8150
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
62
+ 35u
61
+ ··· + 141u + 1
c
2
, c
4
u
62
− 5u
61
+ ··· + 7u + 1
c
3
, c
6
u
62
− 4u
61
+ ··· + 10u − 2
c
5
, c
9
u
62
+ 4u
61
+ ··· + 512u + 512
c
7
u
62
− 3u
61
+ ··· + 26312u − 2116
c
8
, c
11
u
62
+ 5u
61
+ ··· + 11u − 1
c
10
, c
12
u
62
+ 23u
61
+ ··· + 261u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
62
− 11y
61
+ ··· − 18829y + 1
c
2
, c
4
y
62
− 35y
61
+ ··· − 141y + 1
c
3
, c
6
y
62
+ 18y
61
+ ··· − 459y
2
+ 4
c
5
, c
9
y
62
+ 48y
61
+ ··· + 1703936y + 262144
c
7
y
62
− 47y
61
+ ··· − 1209188200y + 4477456
c
8
, c
11
y
62
− 23y
61
+ ··· − 261y + 1
c
10
, c
12
y
62
+ 37y
61
+ ··· − 59949y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.174819 + 1.015820I
a = 1.39308 + 0.32509I
b = −0.169145 + 0.123365I
−0.70247 − 10.05410I −8.00000 + 0.I
u = −0.174819 − 1.015820I
a = 1.39308 − 0.32509I
b = −0.169145 − 0.123365I
−0.70247 + 10.05410I −8.00000 + 0.I
u = −0.180060 + 0.947177I
a = −1.40332 − 0.28966I
b = 0.218716 − 0.044373I
0.58990 − 4.34695I −4.42907 + 2.23582I
u = −0.180060 − 0.947177I
a = −1.40332 + 0.28966I
b = 0.218716 + 0.044373I
0.58990 + 4.34695I −4.42907 − 2.23582I
u = 0.008820 + 0.960888I
a = 1.46292 + 0.33775I
b = −0.0127547 − 0.0398138I
−5.70261 − 3.52968I −11.39023 + 3.16583I
u = 0.008820 − 0.960888I
a = 1.46292 − 0.33775I
b = −0.0127547 + 0.0398138I
−5.70261 + 3.52968I −11.39023 − 3.16583I
u = −0.925664 + 0.154080I
a = 1.55623 + 0.95238I
b = 1.16403 + 0.90014I
−3.09925 + 0.70693I −5.36300 − 9.97003I
u = −0.925664 − 0.154080I
a = 1.55623 − 0.95238I
b = 1.16403 − 0.90014I
−3.09925 − 0.70693I −5.36300 + 9.97003I
u = −0.961954 + 0.456687I
a = −0.996485 − 0.482853I
b = −0.773429 − 0.351449I
1.77868 + 2.87670I 0
u = −0.961954 − 0.456687I
a = −0.996485 + 0.482853I
b = −0.773429 + 0.351449I
1.77868 − 2.87670I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.072360 + 0.027388I
a = 0.302091 + 0.387550I
b = −0.44401 + 2.52455I
−2.88487 − 0.32439I 0
u = 1.072360 − 0.027388I
a = 0.302091 − 0.387550I
b = −0.44401 − 2.52455I
−2.88487 + 0.32439I 0
u = 1.026130 + 0.340287I
a = −0.816921 + 0.074019I
b = 0.010122 + 1.005580I
−0.058930 + 1.340430I 0
u = 1.026130 − 0.340287I
a = −0.816921 − 0.074019I
b = 0.010122 − 1.005580I
−0.058930 − 1.340430I 0
u = 0.263341 + 0.858285I
a = 1.45496 + 0.37428I
b = 0.129842 − 0.231061I
−2.48962 + 3.11257I −8.69664 − 2.35415I
u = 0.263341 − 0.858285I
a = 1.45496 − 0.37428I
b = 0.129842 + 0.231061I
−2.48962 − 3.11257I −8.69664 + 2.35415I
u = −0.799508 + 0.362955I
a = 0.895892 − 0.982033I
b = −0.36497 − 1.79872I
3.99898 + 4.77294I −4.23166 − 6.20197I
u = −0.799508 − 0.362955I
a = 0.895892 + 0.982033I
b = −0.36497 + 1.79872I
3.99898 − 4.77294I −4.23166 + 6.20197I
u = 0.871772 + 0.018873I
a = −0.012598 + 0.371703I
b = 0.29638 + 4.67077I
1.75714 − 2.86066I −51.0320 + 5.8085I
u = 0.871772 − 0.018873I
a = −0.012598 − 0.371703I
b = 0.29638 − 4.67077I
1.75714 + 2.86066I −51.0320 − 5.8085I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.896158 + 0.689044I
a = −0.409815 − 0.214201I
b = −0.330845 − 0.132174I
2.20517 + 2.65821I 0
u = −0.896158 − 0.689044I
a = −0.409815 + 0.214201I
b = −0.330845 + 0.132174I
2.20517 − 2.65821I 0
u = 1.129020 + 0.261697I
a = 0.757928 − 0.225431I
b = −0.024907 − 1.263230I
−0.33332 − 3.49089I 0
u = 1.129020 − 0.261697I
a = 0.757928 + 0.225431I
b = −0.024907 + 1.263230I
−0.33332 + 3.49089I 0
u = −1.097400 + 0.397924I
a = 1.239460 + 0.321056I
b = 0.994156 + 0.243940I
0.08735 + 7.90185I 0
u = −1.097400 − 0.397924I
a = 1.239460 − 0.321056I
b = 0.994156 − 0.243940I
0.08735 − 7.90185I 0
u = 0.819617
a = −0.307172
b = 0.434770
−1.19404 −8.40790
u = −0.847597 + 0.872677I
a = 0.347460 − 0.563753I
b = 0.206725 − 0.482515I
5.15087 + 0.64514I 0
u = −0.847597 − 0.872677I
a = 0.347460 + 0.563753I
b = 0.206725 + 0.482515I
5.15087 − 0.64514I 0
u = −0.680841 + 0.367855I
a = −1.112590 + 0.853351I
b = 0.40199 + 1.42776I
4.31390 − 1.39878I −2.91240 + 0.35592I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.680841 − 0.367855I
a = −1.112590 − 0.853351I
b = 0.40199 − 1.42776I
4.31390 + 1.39878I −2.91240 − 0.35592I
u = −0.915257 + 0.866442I
a = −0.468319 + 0.452475I
b = −0.323390 + 0.408101I
4.96004 + 5.72433I 0
u = −0.915257 − 0.866442I
a = −0.468319 − 0.452475I
b = −0.323390 − 0.408101I
4.96004 − 5.72433I 0
u = −1.208710 + 0.423576I
a = 0.149684 − 1.293320I
b = −0.07945 − 2.58390I
−4.51527 + 5.72958I 0
u = −1.208710 − 0.423576I
a = 0.149684 + 1.293320I
b = −0.07945 + 2.58390I
−4.51527 − 5.72958I 0
u = −1.240470 + 0.336251I
a = −0.16206 + 1.41432I
b = 0.04078 + 2.56231I
−7.10231 + 0.65623I 0
u = −1.240470 − 0.336251I
a = −0.16206 − 1.41432I
b = 0.04078 − 2.56231I
−7.10231 − 0.65623I 0
u = 1.176840 + 0.522272I
a = 0.189318 + 1.049630I
b = 0.70577 + 2.17746I
−3.81660 − 2.96163I 0
u = 1.176840 − 0.522272I
a = 0.189318 − 1.049630I
b = 0.70577 − 2.17746I
−3.81660 + 2.96163I 0
u = 0.067472 + 0.687187I
a = −1.57662 − 0.37750I
b = 0.048649 + 0.283986I
−0.89271 − 1.62585I −5.09936 + 3.39490I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.067472 − 0.687187I
a = −1.57662 + 0.37750I
b = 0.048649 − 0.283986I
−0.89271 + 1.62585I −5.09936 − 3.39490I
u = 1.289650 + 0.343544I
a = 0.359603 + 0.917247I
b = 0.58072 + 2.17986I
−4.20310 + 0.00438I 0
u = 1.289650 − 0.343544I
a = 0.359603 − 0.917247I
b = 0.58072 − 2.17986I
−4.20310 − 0.00438I 0
u = 1.204210 + 0.581904I
a = −0.199785 − 1.111010I
b = −0.70126 − 2.14987I
−5.30672 − 8.46798I 0
u = 1.204210 − 0.581904I
a = −0.199785 + 1.111010I
b = −0.70126 + 2.14987I
−5.30672 + 8.46798I 0
u = −1.247410 + 0.558996I
a = −0.008282 − 1.191920I
b = −0.03995 − 2.65434I
−2.68389 + 9.79746I 0
u = −1.247410 − 0.558996I
a = −0.008282 + 1.191920I
b = −0.03995 + 2.65434I
−2.68389 − 9.79746I 0
u = 0.433777 + 0.453147I
a = −1.191930 − 0.656817I
b = −0.050682 + 0.450591I
−1.02754 − 1.38144I −8.05413 + 4.71760I
u = 0.433777 − 0.453147I
a = −1.191930 + 0.656817I
b = −0.050682 − 0.450591I
−1.02754 + 1.38144I −8.05413 − 4.71760I
u = −1.293820 + 0.477820I
a = −0.023615 + 1.302060I
b = 0.04451 + 2.61060I
−9.73955 + 8.61640I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.293820 − 0.477820I
a = −0.023615 − 1.302060I
b = 0.04451 − 2.61060I
−9.73955 − 8.61640I 0
u = −0.315761 + 0.523302I
a = −0.18061 − 1.62642I
b = −0.419756 − 0.854977I
3.48000 + 1.09960I −1.12389 − 2.21841I
u = −0.315761 − 0.523302I
a = −0.18061 + 1.62642I
b = −0.419756 + 0.854977I
3.48000 − 1.09960I −1.12389 + 2.21841I
u = 1.303730 + 0.481258I
a = −0.318337 − 1.050310I
b = −0.65397 − 2.15550I
−9.72761 − 1.62399I 0
u = 1.303730 − 0.481258I
a = −0.318337 + 1.050310I
b = −0.65397 + 2.15550I
−9.72761 + 1.62399I 0
u = −1.274470 + 0.579615I
a = 0.050317 + 1.202790I
b = 0.01989 + 2.64791I
−4.1048 + 15.7750I 0
u = −1.274470 − 0.579615I
a = 0.050317 − 1.202790I
b = 0.01989 − 2.64791I
−4.1048 − 15.7750I 0
u = 1.369400 + 0.350028I
a = −0.430304 − 0.959677I
b = −0.58989 − 2.12605I
−5.80592 + 5.27823I 0
u = 1.369400 − 0.350028I
a = −0.430304 + 0.959677I
b = −0.58989 + 2.12605I
−5.80592 − 5.27823I 0
u = −0.109124 + 0.515994I
a = −0.17301 + 1.83308I
b = 0.531901 + 0.923027I
2.76743 − 4.27360I −2.53028 + 4.40261I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.109124 − 0.515994I
a = −0.17301 − 1.83308I
b = 0.531901 − 0.923027I
2.76743 + 4.27360I −2.53028 − 4.40261I
u = 0.0853866
a = −6.04151
b = 0.733691
−1.41710 −6.18580
11
II. I
u
2
= hu
2
+ b + u, a, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
10
=
0
−u
2
− u
a
6
=
0
u
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
11
=
−u + 1
−2u
a
4
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
9
=
0
−u
2
− u
a
12
=
u
2
u
2
− u − 1
a
8
=
u
2
− 1
u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
2
− 11u − 10
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
10
u
3
− u
2
+ 2u − 1
c
2
, c
8
u
3
+ u
2
− 1
c
4
, c
11
u
3
− u
2
+ 1
c
5
, c
9
u
3
c
6
, c
12
u
3
+ u
2
+ 2u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
10
, c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
4
, c
8
c
11
y
3
− y
2
+ 2y −1
c
5
, c
9
y
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0
b = 0.662359 + 0.562280I
6.04826 + 5.65624I −0.77833 − 5.57920I
u = −0.877439 − 0.744862I
a = 0
b = 0.662359 − 0.562280I
6.04826 − 5.65624I −0.77833 + 5.57920I
u = 0.754878
a = 0
b = −1.32472
−2.22691 −19.4430
15
III. I
u
3
= hb
2
+ 3u
2
+ b + 5u + 4, a, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
10
=
0
b
a
6
=
0
u
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
11
=
u
2
b − bu
2u
2
b
a
4
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
9
=
0
b
a
12
=
−u
2
b − 2bu − u
2
+ 2b − u + 1
−2bu + u
2
+ 2b + u + 3
a
8
=
u
2
− 1
−u
2
b + 2u
2
+ b + 3u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
b + 12bu + 9u
2
− 10b + 14u − 9
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
10
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
8
(u
3
+ u
2
− 1)
2
c
4
, c
11
(u
3
− u
2
+ 1)
2
c
5
, c
9
u
6
c
6
, c
12
(u
3
+ u
2
+ 2u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
, c
8
c
11
(y
3
− y
2
+ 2y −1)
2
c
5
, c
9
y
6
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0
b = −0.807599 − 0.320410I
6.04826 −1.68265 + 0.98317I
u = −0.877439 + 0.744862I
a = 0
b = −0.192401 + 0.320410I
1.91067 + 2.82812I −17.1302 − 8.6725I
u = −0.877439 − 0.744862I
a = 0
b = −0.807599 + 0.320410I
6.04826 −1.68265 − 0.98317I
u = −0.877439 − 0.744862I
a = 0
b = −0.192401 − 0.320410I
1.91067 − 2.82812I −17.1302 + 8.6725I
u = 0.754878
a = 0
b = −0.50000 + 3.03873I
1.91067 + 2.82812I 6.31282 + 2.33391I
u = 0.754878
a = 0
b = −0.50000 − 3.03873I
1.91067 − 2.82812I 6.31282 − 2.33391I
19
IV. I
u
4
= hb, a + 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
10
=
−1
0
a
6
=
1
1
a
1
=
0
−1
a
11
=
−1
−1
a
4
=
1
1
a
7
=
1
1
a
9
=
−1
−1
a
12
=
−1
−2
a
8
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
8
, c
10
u − 1
c
3
, c
6
u
c
4
, c
9
, c
11
c
12
u + 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y −1
c
3
, c
6
y
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
23
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
3
− u
2
+ 2u − 1)
3
(u
62
+ 35u
61
+ ··· + 141u + 1)
c
2
(u − 1)(u
3
+ u
2
− 1)
3
(u
62
− 5u
61
+ ··· + 7u + 1)
c
3
u(u
3
− u
2
+ 2u − 1)
3
(u
62
− 4u
61
+ ··· + 10u − 2)
c
4
(u + 1)(u
3
− u
2
+ 1)
3
(u
62
− 5u
61
+ ··· + 7u + 1)
c
5
u
9
(u − 1)(u
62
+ 4u
61
+ ··· + 512u + 512)
c
6
u(u
3
+ u
2
+ 2u + 1)
3
(u
62
− 4u
61
+ ··· + 10u − 2)
c
7
(u − 1)(u
3
− u
2
+ 2u − 1)
3
(u
62
− 3u
61
+ ··· + 26312u − 2116)
c
8
(u − 1)(u
3
+ u
2
− 1)
3
(u
62
+ 5u
61
+ ··· + 11u − 1)
c
9
u
9
(u + 1)(u
62
+ 4u
61
+ ··· + 512u + 512)
c
10
(u − 1)(u
3
− u
2
+ 2u − 1)
3
(u
62
+ 23u
61
+ ··· + 261u + 1)
c
11
(u + 1)(u
3
− u
2
+ 1)
3
(u
62
+ 5u
61
+ ··· + 11u − 1)
c
12
(u + 1)(u
3
+ u
2
+ 2u + 1)
3
(u
62
+ 23u
61
+ ··· + 261u + 1)
24
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
3
+ 3y
2
+ 2y −1)
3
(y
62
− 11y
61
+ ··· − 18829y + 1)
c
2
, c
4
(y −1)(y
3
− y
2
+ 2y −1)
3
(y
62
− 35y
61
+ ··· − 141y + 1)
c
3
, c
6
y(y
3
+ 3y
2
+ 2y −1)
3
(y
62
+ 18y
61
+ ··· − 459y
2
+ 4)
c
5
, c
9
y
9
(y −1)(y
62
+ 48y
61
+ ··· + 1703936y + 262144)
c
7
(y −1)(y
3
+ 3y
2
+ 2y −1)
3
· (y
62
− 47y
61
+ ··· − 1209188200y + 4477456)
c
8
, c
11
(y −1)(y
3
− y
2
+ 2y −1)
3
(y
62
− 23y
61
+ ··· − 261y + 1)
c
10
, c
12
(y −1)(y
3
+ 3y
2
+ 2y −1)
3
(y
62
+ 37y
61
+ ··· − 59949y + 1)
25
