12n
0639
(K12n
0639
)
A knot diagram
1
Linearized knot diagam
3 8 12 8 11 10 2 4 3 5 6 9
Solving Sequence
3,12 4,8
5 9 10 2 1 7 6 11
c
3
c
4
c
8
c
9
c
2
c
1
c
7
c
6
c
11
c
5
, c
10
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h11u
19
− 95u
18
+ ··· + 2b − 86, −49u
19
+ 435u
18
+ ··· + 4a + 508, u
20
− 9u
19
+ ··· − 22u + 4i
I
u
2
= hu
14
+ 4u
13
+ 10u
12
+ 15u
11
+ 15u
10
+ 6u
9
− 6u
8
− 16u
7
− 16u
6
− 13u
5
− 6u
4
− 3u
3
+ b + u + 2,
− 4u
14
− 18u
13
+ ··· + a + 5,
u
15
+ 6u
14
+ 19u
13
+ 38u
12
+ 50u
11
+ 37u
10
− 4u
9
− 52u
8
− 74u
7
− 57u
6
− 18u
5
+ 12u
4
+ 18u
3
+ 8u
2
− 1i
I
u
3
= h−a
3
− 2a
2
u + 3u
2
a − a
2
+ 4au + u
2
+ b + 3a + u + 3,
a
3
u
2
+ a
4
+ 2a
3
u − 5a
2
u
2
+ a
3
− 4a
2
u − 3u
2
a − a
2
− 5au − 2u
2
− 4a − 1, u
3
+ u
2
− 1i
* 3 irreducible components of dim
C
= 0, with total 47 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
= h11u
19
− 95u
18
+ · · · + 2b − 86, −49u
19
+ 435u
18
+ · · · + 4a +
508, u
20
− 9u
19
+ · · · − 22u + 4i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
u
2
a
8
=
49
4
u
19
−
435
4
u
18
+ ··· +
1743
4
u − 127
−
11
2
u
19
+
95
2
u
18
+ ··· −
317
2
u + 43
a
5
=
−
1
2
u
19
+ 4u
18
+ ··· −
27
2
u +
7
2
−
1
2
u
19
+
7
2
u
18
+ ··· + 6u
2
−
1
2
u
a
9
=
43
4
u
19
−
365
4
u
18
+ ··· +
1173
4
u − 78
1
2
u
19
−
3
2
u
18
+ ··· −
129
2
u + 27
a
10
=
41
4
u
19
−
359
4
u
18
+ ··· +
1431
4
u − 105
1
2
u
19
−
3
2
u
18
+ ··· −
129
2
u + 27
a
2
=
1
2
u
18
−
7
2
u
17
+ ··· − 5u +
3
2
1
2
u
19
−
9
2
u
18
+ ··· +
21
2
u − 2
a
1
=
1
2
u
19
− 4u
18
+ ··· +
11
2
u −
1
2
1
2
u
19
−
9
2
u
18
+ ··· +
21
2
u − 2
a
7
=
12u
19
−
209
2
u
18
+ ··· + 404u −
237
2
−
13
2
u
19
+
107
2
u
18
+ ··· −
345
2
u + 48
a
6
=
169
4
u
19
−
1407
4
u
18
+ ··· +
4055
4
u − 251
−
63
2
u
19
+
521
2
u
18
+ ··· −
1441
2
u + 173
a
11
=
−
55
2
u
19
+ 227u
18
+ ··· −
1215
2
u +
285
2
57
2
u
19
−
469
2
u
18
+ ··· +
1243
2
u − 146
(ii) Obstruction class = −1
(iii) Cusp Shapes = 90u
19
− 751u
18
+ 3199u
17
− 8714u
16
+ 17011u
15
− 25494u
14
+
31435u
13
−32365u
12
+ 24669u
11
−4672u
10
−23026u
9
+ 47119u
8
−58278u
7
+ 55306u
6
−
40524u
5
+ 20676u
4
− 5078u
3
− 2030u
2
+ 2210u − 566
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 35u
19
+ ··· − 6u + 1
c
2
, c
7
, c
12
u
20
+ u
19
+ ··· − 2u − 1
c
3
u
20
− 9u
19
+ ··· − 22u + 4
c
4
, c
8
u
20
+ 15u
18
+ ··· − 3u − 1
c
5
, c
10
, c
11
u
20
− 7u
19
+ ··· − 16u − 8
c
6
u
20
+ 21u
19
+ ··· + 23824u + 2664
c
9
u
20
− 24u
18
+ ··· − 197u − 57
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 123y
19
+ ··· − 118y + 1
c
2
, c
7
, c
12
y
20
− 35y
19
+ ··· + 6y + 1
c
3
y
20
+ y
19
+ ··· − 172y + 16
c
4
, c
8
y
20
+ 30y
19
+ ··· − 41y + 1
c
5
, c
10
, c
11
y
20
− 19y
19
+ ··· − 352y + 64
c
6
y
20
− 7y
19
+ ··· − 72537184y + 7096896
c
9
y
20
− 48y
19
+ ··· + 66527y + 3249
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.263775 + 0.933613I
a = 0.063867 + 0.469394I
b = 0.635043 − 0.343749I
1.90324 + 1.62770I −2.35623 − 4.39456I
u = −0.263775 − 0.933613I
a = 0.063867 − 0.469394I
b = 0.635043 + 0.343749I
1.90324 − 1.62770I −2.35623 + 4.39456I
u = −1.08046
a = 0.288950
b = 0.649514
−5.71917 −17.5810
u = 0.288783 + 0.801332I
a = 0.234474 − 0.843337I
b = −0.484200 + 0.635373I
−2.01141 − 1.38113I −7.57284 − 0.08377I
u = 0.288783 − 0.801332I
a = 0.234474 + 0.843337I
b = −0.484200 − 0.635373I
−2.01141 + 1.38113I −7.57284 + 0.08377I
u = −0.626429 + 1.085280I
a = −0.082861 − 0.296469I
b = −0.711687 + 0.249730I
−2.18377 + 5.21287I −7.17987 − 5.41855I
u = −0.626429 − 1.085280I
a = −0.082861 + 0.296469I
b = −0.711687 − 0.249730I
−2.18377 − 5.21287I −7.17987 + 5.41855I
u = 0.727091 + 0.098566I
a = 0.27452 − 2.08093I
b = 0.036016 + 0.445425I
−6.93044 − 4.62901I −12.95845 − 0.90844I
u = 0.727091 − 0.098566I
a = 0.27452 + 2.08093I
b = 0.036016 − 0.445425I
−6.93044 + 4.62901I −12.95845 + 0.90844I
u = 0.622736 + 0.087717I
a = −0.13935 + 1.89543I
b = −0.007001 − 0.462890I
−1.13986 − 1.75347I −8.00254 + 2.62344I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.622736 − 0.087717I
a = −0.13935 − 1.89543I
b = −0.007001 + 0.462890I
−1.13986 + 1.75347I −8.00254 − 2.62344I
u = 1.15558 + 1.02594I
a = 0.454735 + 1.318780I
b = −2.17085 − 0.53930I
17.3877 − 12.0858I −13.7130 + 4.9422I
u = 1.15558 − 1.02594I
a = 0.454735 − 1.318780I
b = −2.17085 + 0.53930I
17.3877 + 12.0858I −13.7130 − 4.9422I
u = 1.04701 + 1.18287I
a = −0.989116 − 0.564620I
b = 2.06593 − 0.51777I
17.9143 + 3.9437I −13.79737 − 1.18244I
u = 1.04701 − 1.18287I
a = −0.989116 + 0.564620I
b = 2.06593 + 0.51777I
17.9143 − 3.9437I −13.79737 + 1.18244I
u = −0.408607
a = −0.763027
b = −0.439173
−0.693185 −14.4620
u = 1.16573 + 1.09145I
a = −0.587075 − 1.041030I
b = 2.09877 + 0.18587I
−14.6724 − 7.3263I −11.70204 + 4.29722I
u = 1.16573 − 1.09145I
a = −0.587075 + 1.041030I
b = 2.09877 − 0.18587I
−14.6724 + 7.3263I −11.70204 − 4.29722I
u = 1.12782 + 1.15737I
a = 0.757846 + 0.799117I
b = −2.06719 + 0.14611I
−14.4635 − 1.0817I −11.69610 + 0.I
u = 1.12782 − 1.15737I
a = 0.757846 − 0.799117I
b = −2.06719 − 0.14611I
−14.4635 + 1.0817I −11.69610 + 0.I
6
II.
I
u
2
= hu
14
+4u
13
+· · ·+b+2, −4u
14
−18u
13
+· · ·+a+5, u
15
+6u
14
+· · ·+8u
2
−1i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
u
2
a
8
=
4u
14
+ 18u
13
+ ··· − 11u − 5
−u
14
− 4u
13
+ ··· − u − 2
a
5
=
−u
12
− 5u
11
+ ··· − 10u − 7
−u
14
− 6u
13
+ ··· − 17u
2
− 8u
a
9
=
−2u
14
− 13u
13
+ ··· − 16u − 1
−4u
14
− 22u
13
+ ··· − 7u + 3
a
10
=
2u
14
+ 9u
13
+ ··· − 9u − 4
−4u
14
− 22u
13
+ ··· − 7u + 3
a
2
=
−u
13
− 6u
12
+ ··· − 18u − 7
−u
14
− 6u
13
+ ··· − 8u − 1
a
1
=
−u
14
− 7u
13
+ ··· − 26u − 8
−u
14
− 6u
13
+ ··· − 8u − 1
a
7
=
4u
14
+ 22u
13
+ ··· + 29u + 9
2u
14
+ 13u
13
+ ··· + 16u + 1
a
6
=
−2u
14
− 11u
13
+ ··· + 4u + 7
4u
14
+ 24u
13
+ ··· + 12u − 6
a
11
=
5u
14
+ 29u
13
+ ··· + 22u − 2
−4u
14
− 23u
13
+ ··· − 4u + 9
(ii) Obstruction class = 1
(iii) Cusp Shapes = 15u
14
+ 84u
13
+ 246u
12
+ 446u
11
+ 504u
10
+ 245u
9
− 262u
8
−
701u
7
− 742u
6
− 398u
5
+ 24u
4
+ 221u
3
+ 159u
2
+ 17u − 32
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
− 15u
14
+ ··· + 9u − 1
c
2
u
15
+ u
14
+ ··· − u − 1
c
3
u
15
+ 6u
14
+ ··· + 8u
2
− 1
c
4
u
15
+ 3u
13
+ ··· − 4u
2
− 1
c
5
u
15
− 8u
13
+ ··· − 12u
3
− 1
c
6
u
15
+ 8u
12
+ ··· + 3u
2
− 1
c
7
, c
12
u
15
− u
14
+ ··· − u + 1
c
8
u
15
+ 3u
13
+ ··· + 4u
2
+ 1
c
9
u
15
− 8u
13
+ ··· + 2u
2
+ 1
c
10
, c
11
u
15
− 8u
13
+ ··· − 12u
3
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 31y
14
+ ··· − 11y −1
c
2
, c
7
, c
12
y
15
− 15y
14
+ ··· + 9y −1
c
3
y
15
+ 2y
14
+ ··· + 16y −1
c
4
, c
8
y
15
+ 6y
14
+ ··· − 8y −1
c
5
, c
10
, c
11
y
15
− 16y
14
+ ··· + 12y
2
− 1
c
6
y
15
− 22y
13
+ ··· + 6y −1
c
9
y
15
− 16y
14
+ ··· − 4y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.02838
a = −0.673279
b = −1.40442
−8.97607 −19.1120
u = −0.865049 + 0.608135I
a = −0.280256 + 1.131020I
b = 0.638534 − 0.425883I
−1.66725 + 3.23030I −7.54630 − 5.89815I
u = −0.865049 − 0.608135I
a = −0.280256 − 1.131020I
b = 0.638534 + 0.425883I
−1.66725 − 3.23030I −7.54630 + 5.89815I
u = −0.034033 + 1.074520I
a = −0.708776 − 0.233805I
b = 1.075770 − 0.432119I
−3.77019 − 2.56256I −12.71337 + 2.06324I
u = −0.034033 − 1.074520I
a = −0.708776 + 0.233805I
b = 1.075770 + 0.432119I
−3.77019 + 2.56256I −12.71337 − 2.06324I
u = −0.764295 + 0.414957I
a = 0.38774 − 1.74222I
b = −0.518757 + 0.528802I
−6.94753 + 5.39923I −13.2804 − 8.9655I
u = −0.764295 − 0.414957I
a = 0.38774 + 1.74222I
b = −0.518757 − 0.528802I
−6.94753 − 5.39923I −13.2804 + 8.9655I
u = −0.410448 + 1.166870I
a = 0.612112 − 0.171673I
b = −0.945688 + 0.403215I
0.60597 + 1.58539I −9.91315 − 2.10632I
u = −0.410448 − 1.166870I
a = 0.612112 + 0.171673I
b = −0.945688 − 0.403215I
0.60597 − 1.58539I −9.91315 + 2.10632I
u = −1.122950 + 0.658583I
a = −0.067419 − 0.819750I
b = −0.652697 + 0.297689I
−5.15231 + 1.54512I −15.9184 − 2.4740I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.122950 − 0.658583I
a = −0.067419 + 0.819750I
b = −0.652697 − 0.297689I
−5.15231 − 1.54512I −15.9184 + 2.4740I
u = 0.625489
a = 1.58555
b = 1.61206
−6.43860 −4.48120
u = −0.76860 + 1.27462I
a = −0.362430 + 0.320539I
b = 0.872760 − 0.329613I
−2.95721 + 5.51163I −16.1590 − 8.3531I
u = −0.76860 − 1.27462I
a = −0.362430 − 0.320539I
b = 0.872760 + 0.329613I
−2.95721 − 5.51163I −16.1590 + 8.3531I
u = 0.276879
a = −6.07420
b = −2.14748
−13.8955 −11.3460
11
III. I
u
3
= h3u
2
a + u
2
+ · · · + 3a + 3, a
3
u
2
− 5a
2
u
2
+ · · · − 4a − 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
u
2
a
8
=
a
a
3
+ 2a
2
u − 3u
2
a + a
2
− 4au − u
2
− 3a − u − 3
a
5
=
a
3
u
2
− a
2
u
2
− 2u
2
a + 2a
2
− 4au − 2u
2
− a
−2a
3
u
2
+ 3a
2
u
2
+ 4u
2
a − 4a
2
+ 8au + 4u
2
+ 2a − 2u
a
9
=
a
3
+ 2a
2
u − 4u
2
a + a
2
− 4au − u
2
− 2a − u − 3
a
3
u
2
− a
2
u
2
+ a
3
+ 2a
2
u − 6u
2
a + 3a
2
− 8au − 4u
2
− 3a − 2u − 3
a
10
=
−a
3
u
2
+ a
2
u
2
+ 2u
2
a − 2a
2
+ 4au + 3u
2
+ a + u
a
3
u
2
− a
2
u
2
+ a
3
+ 2a
2
u − 6u
2
a + 3a
2
− 8au − 4u
2
− 3a − 2u − 3
a
2
=
a
3
u
2
− 2a
2
u
2
− 2u
2
a + 2a
2
− 4au − 2u
2
− a
−a
3
u
2
+ a
3
u + 3a
2
u
2
− a
3
− 2a
2
u + 5u
2
a − 2a
2
+ 6au − 4u + 2
a
1
=
a
3
u + a
2
u
2
− a
3
− 2a
2
u + 3u
2
a + 2au − 2u
2
− a − 4u + 2
−a
3
u
2
+ a
3
u + 3a
2
u
2
− a
3
− 2a
2
u + 5u
2
a − 2a
2
+ 6au − 4u + 2
a
7
=
3a
3
u
2
− 3a
2
u
2
− 6u
2
a + 6a
2
− 12au − 8u
2
− 3a − 2u
−5a
3
u
2
+ 4a
2
u
2
+ ··· + 10a + 4
a
6
=
a
3
u
2
− a
2
u
2
− 2u
2
a + 2a
2
− 4au − 3u
2
− a − u
−2a
3
u
2
− a
3
u − a
2
u + 5u
2
a − 4a
2
+ 10au + 6u
2
+ 5a + 5u + 1
a
11
=
a
3
u
2
− a
2
u
2
− 2u
2
a + 2a
2
− 4au − 2u
2
− a
−3a
3
u
2
+ 2a
2
u
2
+ ··· + 4a − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u − 18
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 21u
11
+ ··· + 54160u + 10201
c
2
, c
7
, c
12
u
12
− u
11
+ ··· − 78u + 101
c
3
(u
3
+ u
2
− 1)
4
c
4
, c
8
u
12
− 3u
11
+ ··· + 110u − 19
c
5
, c
10
, c
11
(u
2
+ u − 1)
6
c
6
(u
2
− 3u + 1)
6
c
9
u
12
− u
11
+ ··· + 170u + 211
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 37y
11
+ ··· − 490778160y + 104060401
c
2
, c
7
, c
12
y
12
− 21y
11
+ ··· − 54160y + 10201
c
3
(y
3
− y
2
+ 2y −1)
4
c
4
, c
8
y
12
+ 7y
11
+ ··· − 7160y + 361
c
5
, c
10
, c
11
(y
2
− 3y + 1)
6
c
6
(y
2
− 7y + 1)
6
c
9
y
12
− 29y
11
+ ··· − 124272y + 44521
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.290927 − 0.889122I
b = −0.901307 + 0.772844I
−2.89763 + 2.82812I −14.4902 − 2.9794I
u = −0.877439 + 0.744862I
a = −0.624540 + 1.001960I
b = 0.768380 + 0.035014I
−2.89763 + 2.82812I −14.4902 − 2.9794I
u = −0.877439 + 0.744862I
a = −0.550642 + 1.189890I
b = 1.58009 − 1.38831I
−10.79330 + 2.82812I −14.4902 − 2.9794I
u = −0.877439 + 0.744862I
a = 1.42405 − 1.48532I
b = −1.23208 − 0.72669I
−10.79330 + 2.82812I −14.4902 − 2.9794I
u = −0.877439 − 0.744862I
a = 0.290927 + 0.889122I
b = −0.901307 − 0.772844I
−2.89763 − 2.82812I −14.4902 + 2.9794I
u = −0.877439 − 0.744862I
a = −0.624540 − 1.001960I
b = 0.768380 − 0.035014I
−2.89763 − 2.82812I −14.4902 + 2.9794I
u = −0.877439 − 0.744862I
a = −0.550642 − 1.189890I
b = 1.58009 + 1.38831I
−10.79330 − 2.82812I −14.4902 + 2.9794I
u = −0.877439 − 0.744862I
a = 1.42405 + 1.48532I
b = −1.23208 + 0.72669I
−10.79330 − 2.82812I −14.4902 + 2.9794I
u = 0.754878
a = −0.777477
b = 2.73154
−14.9309 −21.0200
u = 0.754878
a = −0.297371
b = −1.83069
−7.03522 −21.0200
15
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.754878
a = 2.20067
b = 1.47851
−7.03522 −21.0200
u = 0.754878
a = −4.20541
b = −1.80951
−14.9309 −21.0200
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
+ 21u
11
+ ··· + 54160u + 10201)(u
15
− 15u
14
+ ··· + 9u − 1)
· (u
20
+ 35u
19
+ ··· − 6u + 1)
c
2
(u
12
− u
11
+ ··· − 78u + 101)(u
15
+ u
14
+ ··· − u − 1)
· (u
20
+ u
19
+ ··· − 2u − 1)
c
3
((u
3
+ u
2
− 1)
4
)(u
15
+ 6u
14
+ ··· + 8u
2
− 1)(u
20
− 9u
19
+ ··· − 22u + 4)
c
4
(u
12
− 3u
11
+ ··· + 110u − 19)(u
15
+ 3u
13
+ ··· − 4u
2
− 1)
· (u
20
+ 15u
18
+ ··· − 3u − 1)
c
5
((u
2
+ u − 1)
6
)(u
15
− 8u
13
+ ··· − 12u
3
− 1)(u
20
− 7u
19
+ ··· − 16u − 8)
c
6
((u
2
− 3u + 1)
6
)(u
15
+ 8u
12
+ ··· + 3u
2
− 1)
· (u
20
+ 21u
19
+ ··· + 23824u + 2664)
c
7
, c
12
(u
12
− u
11
+ ··· − 78u + 101)(u
15
− u
14
+ ··· − u + 1)
· (u
20
+ u
19
+ ··· − 2u − 1)
c
8
(u
12
− 3u
11
+ ··· + 110u − 19)(u
15
+ 3u
13
+ ··· + 4u
2
+ 1)
· (u
20
+ 15u
18
+ ··· − 3u − 1)
c
9
(u
12
− u
11
+ ··· + 170u + 211)(u
15
− 8u
13
+ ··· + 2u
2
+ 1)
· (u
20
− 24u
18
+ ··· − 197u − 57)
c
10
, c
11
((u
2
+ u − 1)
6
)(u
15
− 8u
13
+ ··· − 12u
3
+ 1)(u
20
− 7u
19
+ ··· − 16u − 8)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
12
− 37y
11
+ ··· − 490778160y + 104060401)
· (y
15
− 31y
14
+ ··· − 11y −1)(y
20
− 123y
19
+ ··· − 118y + 1)
c
2
, c
7
, c
12
(y
12
− 21y
11
+ ··· − 54160y + 10201)(y
15
− 15y
14
+ ··· + 9y −1)
· (y
20
− 35y
19
+ ··· + 6y + 1)
c
3
((y
3
− y
2
+ 2y −1)
4
)(y
15
+ 2y
14
+ ··· + 16y −1)
· (y
20
+ y
19
+ ··· − 172y + 16)
c
4
, c
8
(y
12
+ 7y
11
+ ··· − 7160y + 361)(y
15
+ 6y
14
+ ··· − 8y −1)
· (y
20
+ 30y
19
+ ··· − 41y + 1)
c
5
, c
10
, c
11
((y
2
− 3y + 1)
6
)(y
15
− 16y
14
+ ··· + 12y
2
− 1)
· (y
20
− 19y
19
+ ··· − 352y + 64)
c
6
((y
2
− 7y + 1)
6
)(y
15
− 22y
13
+ ··· + 6y −1)
· (y
20
− 7y
19
+ ··· − 72537184y + 7096896)
c
9
(y
12
− 29y
11
+ ··· − 124272y + 44521)(y
15
− 16y
14
+ ··· − 4y −1)
· (y
20
− 48y
19
+ ··· + 66527y + 3249)
18
