12n
0082
(K12n
0082
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 9 4 11 5 7 12 8 10
Solving Sequence
7,11
8
4,12
3 6 10 1 9 5 2
c
7
c
11
c
3
c
6
c
10
c
12
c
9
c
5
c
2
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−1075292995u
33
− 4308557086u
32
+ ··· + 1783382596b − 2108461367,
− 16834407501u
33
− 77191989494u
32
+ ··· + 1783382596a − 50354623549,
u
34
+ 5u
33
+ ··· + 17u + 1i
I
u
2
= hb, 3u
7
+ u
6
− 4u
5
− 4u
4
+ 5u
3
+ 3u
2
+ a − u − 5, u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1i
I
u
3
= h−u
2
a − 2u
2
+ b + a + u + 2, a
2
+ 2au + 4u
2
+ a − 4u + 4, u
3
− u
2
+ 1i
I
u
4
= hu
2
+ b, a − u, u
3
− u
2
+ 1i
* 4 irreducible components of dim
C
= 0, with total 51 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−1.08 × 10
9
u
33
− 4.31 × 10
9
u
32
+ · · · + 1.78 × 10
9
b − 2.11 × 10
9
, −1.68 ×
10
10
u
33
−7.72×10
10
u
32
+· · ·+1.78×10
9
a−5.04×10
10
, u
34
+5u
33
+· · ·+17u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
4
=
9.43959u
33
+ 43.2840u
32
+ ··· + 295.933u + 28.2355
0.602951u
33
+ 2.41595u
32
+ ··· + 10.5760u + 1.18228
a
12
=
u
−u
3
+ u
a
3
=
10.0425u
33
+ 45.7000u
32
+ ··· + 306.509u + 29.4177
0.602951u
33
+ 2.41595u
32
+ ··· + 10.5760u + 1.18228
a
6
=
−8.32985u
33
− 34.9914u
32
+ ··· − 174.842u − 14.4784
1.50407u
33
+ 6.20192u
32
+ ··· + 28.1402u + 1.51861
a
10
=
−u
3
u
5
− u
3
+ u
a
1
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
−u
5
− u
u
5
− u
3
+ u
a
5
=
−3.94864u
33
− 16.6164u
32
+ ··· − 92.4717u − 9.25517
−0.647049u
33
− 2.83405u
32
+ ··· − 12.4240u − 1.06772
a
2
=
7.18982u
33
+ 33.9328u
32
+ ··· + 248.452u + 22.9281
0.647049u
33
+ 2.83405u
32
+ ··· + 12.4240u + 1.06772
(ii) Obstruction class = −1
(iii) Cusp Shap es = −
749238525
445845649
u
33
−
25224574109
1783382596
u
32
+ ··· −
515748589297
1783382596
u −
13447700611
445845649
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
34
+ 40u
32
+ ··· + 1011u + 1
c
2
, c
4
u
34
− 12u
33
+ ··· + 27u + 1
c
3
, c
6
u
34
− 4u
33
+ ··· + 896u − 256
c
5
, c
8
u
34
− 2u
33
+ ··· − 1536u − 512
c
7
, c
11
u
34
− 5u
33
+ ··· − 17u + 1
c
9
u
34
+ 5u
33
+ ··· − 34224u + 2116
c
10
, c
12
u
34
− 15u
33
+ ··· − 147u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
34
+ 80y
33
+ ··· − 1049111y + 1
c
2
, c
4
y
34
+ 40y
32
+ ··· − 1011y + 1
c
3
, c
6
y
34
+ 60y
33
+ ··· − 2146304y + 65536
c
5
, c
8
y
34
− 56y
33
+ ··· − 1441792y + 262144
c
7
, c
11
y
34
− 15y
33
+ ··· − 147y + 1
c
9
y
34
− 83y
33
+ ··· − 660094664y + 4477456
c
10
, c
12
y
34
+ 13y
33
+ ··· − 18915y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.675479 + 0.712354I
a = −0.270719 + 0.161827I
b = 0.097932 − 0.857151I
0.11560 + 2.14584I 4.84697 − 4.28283I
u = −0.675479 − 0.712354I
a = −0.270719 − 0.161827I
b = 0.097932 + 0.857151I
0.11560 − 2.14584I 4.84697 + 4.28283I
u = 0.787838 + 0.675769I
a = −0.413189 + 0.782228I
b = 0.360422 − 0.507830I
−2.05683 + 2.19416I 2.78133 − 4.16804I
u = 0.787838 − 0.675769I
a = −0.413189 − 0.782228I
b = 0.360422 + 0.507830I
−2.05683 − 2.19416I 2.78133 + 4.16804I
u = −0.994788 + 0.364572I
a = −1.43728 − 0.32761I
b = 1.097360 − 0.422286I
0.27995 − 2.43239I 4.42150 + 3.66524I
u = −0.994788 − 0.364572I
a = −1.43728 + 0.32761I
b = 1.097360 + 0.422286I
0.27995 + 2.43239I 4.42150 − 3.66524I
u = 0.883198 + 0.167536I
a = −0.82822 − 2.77937I
b = 0.390635 + 1.309810I
4.40661 − 2.39050I 8.72484 − 4.03068I
u = 0.883198 − 0.167536I
a = −0.82822 + 2.77937I
b = 0.390635 − 1.309810I
4.40661 + 2.39050I 8.72484 + 4.03068I
u = −0.439134 + 1.012860I
a = −0.14054 + 2.08289I
b = −0.21712 − 2.33804I
12.94380 − 1.52834I 2.89091 + 1.52951I
u = −0.439134 − 1.012860I
a = −0.14054 − 2.08289I
b = −0.21712 + 2.33804I
12.94380 + 1.52834I 2.89091 − 1.52951I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.554438 + 0.988490I
a = 0.37372 − 1.78363I
b = −0.69939 + 2.06047I
12.1480 + 7.3055I 2.34511 − 2.34014I
u = −0.554438 − 0.988490I
a = 0.37372 + 1.78363I
b = −0.69939 − 2.06047I
12.1480 − 7.3055I 2.34511 + 2.34014I
u = 0.877826 + 0.731952I
a = −2.89148 − 0.75774I
b = −0.433493 + 0.039285I
−4.30564 + 2.78916I −43.5792 − 1.9775I
u = 0.877826 − 0.731952I
a = −2.89148 + 0.75774I
b = −0.433493 − 0.039285I
−4.30564 − 2.78916I −43.5792 + 1.9775I
u = 0.960792 + 0.643718I
a = −1.56912 − 0.34499I
b = 0.705310 + 0.607713I
−1.51157 + 2.93275I 3.49597 − 2.09060I
u = 0.960792 − 0.643718I
a = −1.56912 + 0.34499I
b = 0.705310 − 0.607713I
−1.51157 − 2.93275I 3.49597 + 2.09060I
u = −0.988094 + 0.655947I
a = 0.798191 − 0.915301I
b = 0.242079 + 0.838628I
1.06007 − 7.41163I 6.37706 + 10.12845I
u = −0.988094 − 0.655947I
a = 0.798191 + 0.915301I
b = 0.242079 − 0.838628I
1.06007 + 7.41163I 6.37706 − 10.12845I
u = 1.134070 + 0.366301I
a = 1.64472 + 3.48212I
b = −0.39273 − 1.87590I
6.17981 + 3.91713I 6.50133 − 4.05550I
u = 1.134070 − 0.366301I
a = 1.64472 − 3.48212I
b = −0.39273 + 1.87590I
6.17981 − 3.91713I 6.50133 + 4.05550I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.759460 + 0.212841I
a = 2.07551 − 3.57649I
b = 0.167778 + 0.535642I
−0.839427 − 0.307998I 10.22671 + 6.42071I
u = −0.759460 − 0.212841I
a = 2.07551 + 3.57649I
b = 0.167778 − 0.535642I
−0.839427 + 0.307998I 10.22671 − 6.42071I
u = −0.199261 + 0.726308I
a = 0.009119 − 0.678462I
b = −0.695938 + 1.120220I
2.39265 − 0.51912I 3.37017 + 1.09835I
u = −0.199261 − 0.726308I
a = 0.009119 + 0.678462I
b = −0.695938 − 1.120220I
2.39265 + 0.51912I 3.37017 − 1.09835I
u = −1.155100 + 0.509124I
a = 0.68550 + 3.05854I
b = −1.31414 − 1.23344I
5.17110 − 4.13601I 6.26603 + 3.46699I
u = −1.155100 − 0.509124I
a = 0.68550 − 3.05854I
b = −1.31414 + 1.23344I
5.17110 + 4.13601I 6.26603 − 3.46699I
u = 1.331470 + 0.064618I
a = 0.73597 − 5.01515I
b = −0.62913 + 2.45336I
19.5904 + 4.8180I 6.85242 − 2.12746I
u = 1.331470 − 0.064618I
a = 0.73597 + 5.01515I
b = −0.62913 − 2.45336I
19.5904 − 4.8180I 6.85242 + 2.12746I
u = −0.647070
a = −0.602677
b = −0.176681
0.883121 11.7300
u = −1.141270 + 0.734626I
a = −1.07593 + 3.44611I
b = −0.83809 − 1.99730I
13.9751 − 13.5871I 3.82321 + 6.38940I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.141270 − 0.734626I
a = −1.07593 − 3.44611I
b = −0.83809 + 1.99730I
13.9751 + 13.5871I 3.82321 − 6.38940I
u = −1.202000 + 0.684331I
a = 1.86014 − 3.56344I
b = 0.00069 + 2.47834I
15.3278 − 4.6468I 5.01393 + 2.36080I
u = −1.202000 − 0.684331I
a = 1.86014 + 3.56344I
b = 0.00069 − 2.47834I
15.3278 + 4.6468I 5.01393 − 2.36080I
u = −0.0852780
a = 8.48988
b = 0.492352
−1.21008 −9.44660
8
II. I
u
2
= hb, 3u
7
+ u
6
− 4u
5
− 4u
4
+ 5u
3
+ 3u
2
+ a − u − 5, u
8
+ u
7
− u
6
−
2u
5
+ u
4
+ 2u
3
− 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
4
=
−3u
7
− u
6
+ 4u
5
+ 4u
4
− 5u
3
− 3u
2
+ u + 5
0
a
12
=
u
−u
3
+ u
a
3
=
−3u
7
− u
6
+ 4u
5
+ 4u
4
− 5u
3
− 3u
2
+ u + 5
0
a
6
=
1
0
a
10
=
−u
3
u
5
− u
3
+ u
a
1
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
−u
5
− u
u
5
− u
3
+ u
a
5
=
−u
5
− u
u
7
− u
5
+ 2u
3
− u
a
2
=
−3u
7
− u
6
+ 5u
5
+ 4u
4
− 5u
3
− 3u
2
+ 2u + 5
−u
7
+ u
5
− 2u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −17u
7
− 6u
6
+ 24u
5
+ 22u
4
− 31u
3
− 17u
2
+ 12u + 31
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
6
u
8
c
4
(u + 1)
8
c
5
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
7
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
8
, c
9
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
10
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
11
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
12
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
6
y
8
c
5
, c
8
, c
9
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
7
, c
11
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
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
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.570868 + 0.730671I
a = 0.615431 − 0.295452I
b = 0
−0.604279 + 1.131230I 1.048604 + 0.799861I
u = −0.570868 − 0.730671I
a = 0.615431 + 0.295452I
b = 0
−0.604279 − 1.131230I 1.048604 − 0.799861I
u = 0.855237 + 0.665892I
a = −1.68119 − 0.49658I
b = 0
−3.80435 + 2.57849I −0.86993 − 2.07507I
u = 0.855237 − 0.665892I
a = −1.68119 + 0.49658I
b = 0
−3.80435 − 2.57849I −0.86993 + 2.07507I
u = 1.09818
a = 0.532015
b = 0
4.85780 9.68010
u = −1.031810 + 0.655470I
a = 0.473764 + 0.240160I
b = 0
0.73474 − 6.44354I 3.69048 + 2.66284I
u = −1.031810 − 0.655470I
a = 0.473764 − 0.240160I
b = 0
0.73474 + 6.44354I 3.69048 − 2.66284I
u = −0.603304
a = 4.65198
b = 0
−0.799899 25.5820
12
III.
I
u
3
= h−u
2
a − 2u
2
+ b + a + u + 2, a
2
+ 2au + 4u
2
+ a − 4u + 4, u
3
− u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
4
=
a
u
2
a + 2u
2
− a − u − 2
a
12
=
u
−u
2
+ u + 1
a
3
=
u
2
a + 2u
2
− u − 2
u
2
a + 2u
2
− a − u − 2
a
6
=
u
2
a − au − a − 3
u
2
a + au + 3u
2
a
10
=
−u
2
+ 1
−u
2
a
1
=
−1
0
a
9
=
1
−u
2
a
5
=
u
2
a − au − a − 3
u
2
a + au + 3u
2
a
2
=
2u
2
a + 3u
2
− a − 5
u
2
a + au + 3u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
a − 7au − 12u
2
− 3a − 4u + 4
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
c
12
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
11
(u
3
+ u
2
− 1)
2
c
4
, c
7
(u
3
− u
2
+ 1)
2
c
5
, c
8
u
6
c
6
, c
10
(u
3
+ u
2
+ 2u + 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
9
, c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
, c
7
c
11
(y
3
− y
2
+ 2y −1)
2
c
5
, c
8
y
6
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −1.06984 − 1.06527I
b = −0.215080 + 1.307140I
5.65624I 3.29784 − 4.97572I
u = 0.877439 + 0.744862I
a = −1.68504 − 0.42445I
b = −0.569840
−4.13758 + 2.82812I 11.29331 − 8.29280I
u = 0.877439 − 0.744862I
a = −1.06984 + 1.06527I
b = −0.215080 − 1.307140I
− 5.65624I 3.29784 + 4.97572I
u = 0.877439 − 0.744862I
a = −1.68504 + 0.42445I
b = −0.569840
−4.13758 − 2.82812I 11.29331 + 8.29280I
u = −0.754878
a = 0.25488 + 3.03873I
b = −0.215080 − 1.307140I
4.13758 − 2.82812I 0.90884 + 8.67250I
u = −0.754878
a = 0.25488 − 3.03873I
b = −0.215080 + 1.307140I
4.13758 + 2.82812I 0.90884 − 8.67250I
16
IV. I
u
4
= hu
2
+ b, a − u, u
3
− u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
4
=
u
−u
2
a
12
=
u
−u
2
+ u + 1
a
3
=
−u
2
+ u
−u
2
a
6
=
u
2
−u
2
+ u + 1
a
10
=
−u
2
+ 1
−u
2
a
1
=
−1
0
a
9
=
1
−u
2
a
5
=
u
2
−u
2
+ u + 1
a
2
=
u − 1
−u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
2
+ 3u + 2
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
c
12
u
3
− u
2
+ 2u − 1
c
2
, c
11
u
3
+ u
2
− 1
c
4
, c
7
u
3
− u
2
+ 1
c
5
, c
8
u
3
c
6
, c
10
u
3
+ u
2
+ 2u + 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
9
, c
10
, c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
4
, c
7
c
11
y
3
− y
2
+ 2y −1
c
5
, c
8
y
3
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.877439 + 0.744862I
b = −0.215080 − 1.307140I
0 4.20216 − 0.37970I
u = 0.877439 − 0.744862I
a = 0.877439 − 0.744862I
b = −0.215080 + 1.307140I
0 4.20216 + 0.37970I
u = −0.754878
a = −0.754878
b = −0.569840
0 −1.40430
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
3
− u
2
+ 2u − 1)
3
(u
34
+ 40u
32
+ ··· + 1011u + 1)
c
2
((u − 1)
8
)(u
3
+ u
2
− 1)
3
(u
34
− 12u
33
+ ··· + 27u + 1)
c
3
u
8
(u
3
− u
2
+ 2u − 1)
3
(u
34
− 4u
33
+ ··· + 896u − 256)
c
4
((u + 1)
8
)(u
3
− u
2
+ 1)
3
(u
34
− 12u
33
+ ··· + 27u + 1)
c
5
u
9
(u
8
− u
7
+ ··· − 2u − 1)(u
34
− 2u
33
+ ··· − 1536u − 512)
c
6
u
8
(u
3
+ u
2
+ 2u + 1)
3
(u
34
− 4u
33
+ ··· + 896u − 256)
c
7
(u
3
− u
2
+ 1)
3
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
· (u
34
− 5u
33
+ ··· − 17u + 1)
c
8
u
9
(u
8
+ u
7
+ ··· + 2u − 1)(u
34
− 2u
33
+ ··· − 1536u − 512)
c
9
(u
3
− u
2
+ 2u − 1)
3
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
· (u
34
+ 5u
33
+ ··· − 34224u + 2116)
c
10
(u
3
+ u
2
+ 2u + 1)
3
· (u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
34
− 15u
33
+ ··· − 147u + 1)
c
11
(u
3
+ u
2
− 1)
3
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
· (u
34
− 5u
33
+ ··· − 17u + 1)
c
12
(u
3
− u
2
+ 2u − 1)
3
· (u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
34
− 15u
33
+ ··· − 147u + 1)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
3
+ 3y
2
+ 2y −1)
3
(y
34
+ 80y
33
+ ··· − 1049111y + 1)
c
2
, c
4
((y −1)
8
)(y
3
− y
2
+ 2y −1)
3
(y
34
+ 40y
32
+ ··· − 1011y + 1)
c
3
, c
6
y
8
(y
3
+ 3y
2
+ 2y −1)
3
(y
34
+ 60y
33
+ ··· − 2146304y + 65536)
c
5
, c
8
y
9
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
34
− 56y
33
+ ··· − 1441792y + 262144)
c
7
, c
11
(y
3
− y
2
+ 2y −1)
3
· (y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
34
− 15y
33
+ ··· − 147y + 1)
c
9
(y
3
+ 3y
2
+ 2y −1)
3
· (y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
34
− 83y
33
+ ··· − 660094664y + 4477456)
c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
3
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
34
+ 13y
33
+ ··· − 18915y + 1)
22
