12n
0241
(K12n
0241
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 12 11 10 4 5 6 7 9
Solving Sequence
3,9 4,5
10 2 1 8 7 12 6 11
c
3
c
9
c
2
c
1
c
8
c
7
c
12
c
5
c
11
c
4
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.03137 × 10
66
u
36
+ 6.98280 × 10
65
u
35
+ ··· + 2.95081 × 10
68
b − 1.06268 × 10
68
,
3.77823 × 10
67
u
36
− 2.87660 × 10
67
u
35
+ ··· + 2.36065 × 10
69
a + 2.08922 × 10
69
,
u
37
− u
36
+ ··· + 128u + 256i
I
v
1
= ha, b − 1, v
8
− v
7
− v
6
+ 2v
5
+ v
4
− 2v
3
+ 2v −1i
* 2 irreducible components of dim
C
= 0, with total 45 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−1.03 × 10
66
u
36
+ 6.98 × 10
65
u
35
+ · · · + 2.95 × 10
68
b − 1.06 ×
10
68
, 3.78 × 10
67
u
36
− 2.88 × 10
67
u
35
+ · · · + 2.36 × 10
69
a + 2.09 ×
10
69
, u
37
− u
36
+ · · · + 128u + 256i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
5
=
−0.0160050u
36
+ 0.0121856u
35
+ ··· − 8.06518u − 0.885020
0.00349520u
36
− 0.00236640u
35
+ ··· + 1.79889u + 0.360133
a
10
=
−0.0133199u
36
+ 0.0418348u
35
+ ··· + 0.507839u + 12.6265
0.00381941u
36
− 0.0132824u
35
+ ··· + 0.836374u − 4.09729
a
2
=
−0.0160050u
36
+ 0.0121856u
35
+ ··· − 8.06518u − 0.885020
0.00596782u
36
− 0.00520306u
35
+ ··· + 2.78728u + 0.617635
a
1
=
−0.0100372u
36
+ 0.00698258u
35
+ ··· − 5.27790u − 0.267385
0.00596782u
36
− 0.00520306u
35
+ ··· + 2.78728u + 0.617635
a
8
=
u
u
3
+ u
a
7
=
−0.0154489u
36
+ 0.0231599u
35
+ ··· − 4.79943u + 5.30651
−0.000302840u
36
+ 0.00840424u
35
+ ··· + 2.35144u + 2.54004
a
12
=
−0.0100372u
36
+ 0.00698258u
35
+ ··· − 5.27790u − 0.267385
0.0110751u
36
− 0.00839995u
35
+ ··· + 5.74781u + 1.39963
a
6
=
−0.0440610u
36
+ 0.0321408u
35
+ ··· − 23.2326u − 5.35412
0.0221974u
36
− 0.0147892u
35
+ ··· + 12.6696u + 3.48847
a
11
=
−0.0148066u
36
+ 0.00829427u
35
+ ··· − 11.4914u − 1.08615
−0.00186082u
36
+ 0.00671544u
35
+ ··· + 3.81289u + 3.67874
(ii) Obstruction class = −1
(iii) Cusp Shapes = −0.0751690u
36
+ 0.0952069u
35
+ ··· − 23.9143u + 7.79267
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
37
+ 5u
36
+ ··· − 5u + 1
c
2
, c
4
u
37
− 9u
36
+ ··· − 7u + 1
c
3
, c
8
u
37
− u
36
+ ··· + 128u + 256
c
5
, c
7
u
37
+ 6u
36
+ ··· + 35u + 5
c
6
, c
10
, c
11
u
37
− 2u
36
+ ··· + 3u + 1
c
9
u
37
+ 2u
36
+ ··· + 3u + 1
c
12
u
37
− 8u
36
+ ··· + 2082719u − 154033
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
37
+ 63y
36
+ ··· − y −1
c
2
, c
4
y
37
− 5y
36
+ ··· − 5y −1
c
3
, c
8
y
37
+ 51y
36
+ ··· − 475136y −65536
c
5
, c
7
y
37
+ 16y
36
+ ··· + 735y −25
c
6
, c
10
, c
11
y
37
− 32y
36
+ ··· + 27y −1
c
9
y
37
− 48y
36
+ ··· + 27y −1
c
12
y
37
− 84y
36
+ ··· + 1848165923715y −23726165089
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.878368 + 0.497461I
a = 0.626443 + 0.343621I
b = 0.227102 − 0.673098I
−0.66892 + 2.27408I 1.99793 − 4.76069I
u = −0.878368 − 0.497461I
a = 0.626443 − 0.343621I
b = 0.227102 + 0.673098I
−0.66892 − 2.27408I 1.99793 + 4.76069I
u = −0.756357 + 0.569885I
a = 0.466384 − 0.081501I
b = 1.080620 + 0.363590I
0.61705 − 4.41139I 2.87485 + 3.50022I
u = −0.756357 − 0.569885I
a = 0.466384 + 0.081501I
b = 1.080620 − 0.363590I
0.61705 + 4.41139I 2.87485 − 3.50022I
u = −0.491761 + 0.952107I
a = 0.695040 + 0.682945I
b = −0.267990 − 0.719272I
6.82750 − 1.30632I 10.37535 + 1.68426I
u = −0.491761 − 0.952107I
a = 0.695040 − 0.682945I
b = −0.267990 + 0.719272I
6.82750 + 1.30632I 10.37535 − 1.68426I
u = 0.886724 + 0.087993I
a = 0.586081 − 0.184026I
b = 0.553122 + 0.487673I
2.29551 + 0.52381I 5.03075 + 0.94889I
u = 0.886724 − 0.087993I
a = 0.586081 + 0.184026I
b = 0.553122 − 0.487673I
2.29551 − 0.52381I 5.03075 − 0.94889I
u = −0.366615 + 0.752594I
a = 0.451654 − 0.034286I
b = 1.201400 + 0.167112I
0.11703 + 2.14687I 4.21868 − 4.14096I
u = −0.366615 − 0.752594I
a = 0.451654 + 0.034286I
b = 1.201400 − 0.167112I
0.11703 − 2.14687I 4.21868 + 4.14096I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.040136 + 0.830344I
a = 1.19957 − 1.19424I
b = −0.581327 + 0.416814I
1.73939 + 7.30260I 6.40314 − 7.32057I
u = 0.040136 − 0.830344I
a = 1.19957 + 1.19424I
b = −0.581327 − 0.416814I
1.73939 − 7.30260I 6.40314 + 7.32057I
u = 0.563224 + 0.605095I
a = 0.466156 + 0.056547I
b = 1.114100 − 0.256450I
−3.61155 + 1.05730I −2.65114 + 0.19593I
u = 0.563224 − 0.605095I
a = 0.466156 − 0.056547I
b = 1.114100 + 0.256450I
−3.61155 − 1.05730I −2.65114 − 0.19593I
u = 0.981996 + 0.654437I
a = 0.572407 − 0.396675I
b = 0.180218 + 0.817886I
4.10036 − 5.71187I 7.23097 + 5.43034I
u = 0.981996 − 0.654437I
a = 0.572407 + 0.396675I
b = 0.180218 − 0.817886I
4.10036 + 5.71187I 7.23097 − 5.43034I
u = −0.066568 + 0.771395I
a = 1.33592 + 1.03307I
b = −0.531570 − 0.362239I
−2.71712 − 3.28265I 1.87410 + 4.96573I
u = −0.066568 − 0.771395I
a = 1.33592 − 1.03307I
b = −0.531570 + 0.362239I
−2.71712 + 3.28265I 1.87410 − 4.96573I
u = 0.419103 + 0.595983I
a = 0.910599 − 0.447643I
b = −0.115558 + 0.434784I
1.109940 + 0.478757I 7.85612 − 2.59030I
u = 0.419103 − 0.595983I
a = 0.910599 + 0.447643I
b = −0.115558 − 0.434784I
1.109940 − 0.478757I 7.85612 + 2.59030I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.077849 + 0.690240I
a = 1.50612 − 0.77029I
b = −0.473705 + 0.269170I
0.539059 − 0.667337I 5.56096 − 0.61157I
u = 0.077849 − 0.690240I
a = 1.50612 + 0.77029I
b = −0.473705 − 0.269170I
0.539059 + 0.667337I 5.56096 + 0.61157I
u = −0.407194
a = 0.525318
b = 0.903607
−1.31151 −10.2650
u = 0.39124 + 1.82665I
a = −0.068005 + 0.975912I
b = −1.07106 − 1.01973I
9.29659 − 4.13983I 0
u = 0.39124 − 1.82665I
a = −0.068005 − 0.975912I
b = −1.07106 + 1.01973I
9.29659 + 4.13983I 0
u = 0.24251 + 1.87487I
a = 0.000094 + 0.931629I
b = −0.99989 − 1.07339I
9.54816 − 3.51968I 0
u = 0.24251 − 1.87487I
a = 0.000094 − 0.931629I
b = −0.99989 + 1.07339I
9.54816 + 3.51968I 0
u = −0.48778 + 1.83564I
a = −0.120525 − 0.976988I
b = −1.12438 + 1.00821I
6.99297 + 8.38148I 0
u = −0.48778 − 1.83564I
a = −0.120525 + 0.976988I
b = −1.12438 − 1.00821I
6.99297 − 8.38148I 0
u = −0.13124 + 1.93107I
a = 0.037828 − 0.886275I
b = −0.95193 + 1.12627I
7.58072 − 0.60625I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.13124 − 1.93107I
a = 0.037828 + 0.886275I
b = −0.95193 − 1.12627I
7.58072 + 0.60625I 0
u = 0.53822 + 1.86696I
a = −0.147624 + 0.960917I
b = −1.15619 − 1.01668I
12.0430 − 12.4368I 0
u = 0.53822 − 1.86696I
a = −0.147624 − 0.960917I
b = −1.15619 + 1.01668I
12.0430 + 12.4368I 0
u = 0.09684 + 2.00647I
a = 0.036196 + 0.850721I
b = −0.95008 − 1.17335I
12.76300 + 4.48472I 0
u = 0.09684 − 2.00647I
a = 0.036196 − 0.850721I
b = −0.95008 + 1.17335I
12.76300 − 4.48472I 0
u = −0.35556 + 2.00636I
a = −0.066983 − 0.886854I
b = −1.08468 + 1.12119I
16.7972 + 4.0807I 0
u = −0.35556 − 2.00636I
a = −0.066983 + 0.886854I
b = −1.08468 − 1.12119I
16.7972 − 4.0807I 0
8
II. I
v
1
= ha, b − 1, v
8
− v
7
− v
6
+ 2v
5
+ v
4
− 2v
3
+ 2v − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
v
0
a
4
=
1
0
a
5
=
0
1
a
10
=
v
−v
a
2
=
1
−1
a
1
=
0
−1
a
8
=
v
0
a
7
=
−v
3
+ v
v
3
a
12
=
v
2
−1
a
6
=
v
4
−v
2
+ 1
a
11
=
−v
7
+ v
6
+ 2v
5
− v
4
− 2v
3
+ 2v
2
+ 2v − 1
v
7
− 2v
5
+ 2v
3
− 2v
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6v
7
− v
6
− 11v
5
+ 7v
4
+ 12v
3
− 6v
2
− 6v + 10
9
(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
7
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
6
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
9
, c
12
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
10
, c
11
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 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
8
y
8
c
5
, c
7
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
6
, c
10
, c
11
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
9
, c
12
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.570868 + 0.730671I
a = 0
b = 1.00000
−0.604279 − 1.131230I −1.074136 + 0.216470I
v = 0.570868 − 0.730671I
a = 0
b = 1.00000
−0.604279 + 1.131230I −1.074136 − 0.216470I
v = −0.855237 + 0.665892I
a = 0
b = 1.00000
−3.80435 − 2.57849I −3.22623 + 3.25417I
v = −0.855237 − 0.665892I
a = 0
b = 1.00000
−3.80435 + 2.57849I −3.22623 − 3.25417I
v = −1.09818
a = 0
b = 1.00000
4.85780 7.89920
v = 1.031810 + 0.655470I
a = 0
b = 1.00000
0.73474 + 6.44354I 2.34782 − 4.54733I
v = 1.031810 − 0.655470I
a = 0
b = 1.00000
0.73474 − 6.44354I 2.34782 + 4.54733I
v = 0.603304
a = 0
b = 1.00000
−0.799899 7.00590
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
37
+ 5u
36
+ ··· − 5u + 1)
c
2
((u − 1)
8
)(u
37
− 9u
36
+ ··· − 7u + 1)
c
3
, c
8
u
8
(u
37
− u
36
+ ··· + 128u + 256)
c
4
((u + 1)
8
)(u
37
− 9u
36
+ ··· − 7u + 1)
c
5
, c
7
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
37
+ 6u
36
+ ··· + 35u + 5)
c
6
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
37
− 2u
36
+ ··· + 3u + 1)
c
9
(u
8
− u
7
+ ··· + 2u − 1)(u
37
+ 2u
36
+ ··· + 3u + 1)
c
10
, c
11
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
37
− 2u
36
+ ··· + 3u + 1)
c
12
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
· (u
37
− 8u
36
+ ··· + 2082719u − 154033)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
8
)(y
37
+ 63y
36
+ ··· − y − 1)
c
2
, c
4
((y − 1)
8
)(y
37
− 5y
36
+ ··· − 5y − 1)
c
3
, c
8
y
8
(y
37
+ 51y
36
+ ··· − 475136y − 65536)
c
5
, c
7
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
37
+ 16y
36
+ ··· + 735y − 25)
c
6
, c
10
, c
11
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
37
− 32y
36
+ ··· + 27y − 1)
c
9
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
37
− 48y
36
+ ··· + 27y − 1)
c
12
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
37
− 84y
36
+ ··· + 1848165923715y − 23726165089)
14
