12a
0909
(K12a
0909
)
A knot diagram
1
Linearized knot diagam
4 6 9 2 10 11 12 1 5 3 7 8
Solving Sequence
8,12 1,4
2 5 9 3 7 11 6 10
c
12
c
1
c
4
c
8
c
3
c
7
c
11
c
6
c
10
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−2.13178 × 10
29
u
50
+ 5.14130 × 10
28
u
49
+ ··· + 1.65837 × 10
29
b − 2.15608 × 10
27
,
− 1.15795 × 10
29
u
50
+ 4.35679 × 10
28
u
49
+ ··· + 1.65837 × 10
29
a + 6.77432 × 10
28
, u
51
− 2u
50
+ ··· − u + 1i
I
u
2
= hb − 1, a − 1, u + 1i
* 2 irreducible components of dim
C
= 0, with total 52 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−2.13×10
29
u
50
+5.14×10
28
u
49
+· · ·+1.66×10
29
b−2.16×10
27
, −1.16×
10
29
u
50
+4.36×10
28
u
49
+· · ·+1.66×10
29
a+6.77×10
28
, u
51
−2u
50
+· · ·−u+1i
(i) Arc colorings
a
8
=
0
u
a
12
=
1
0
a
1
=
1
u
2
a
4
=
0.698249u
50
− 0.262716u
49
+ ··· − 11.7227u − 0.408493
1.28547u
50
− 0.310022u
49
+ ··· + 0.0538566u + 0.0130012
a
2
=
0.674876u
50
− 0.229531u
49
+ ··· − 12.1488u + 0.603479
1.22229u
50
− 0.317787u
49
+ ··· + 0.264395u + 0.0354543
a
5
=
0.137080u
50
+ 0.0189458u
49
+ ··· + 0.718302u − 1.06415
0.412183u
50
+ 0.00870488u
49
+ ··· − 0.371603u − 0.128201
a
9
=
−u
−u
3
+ u
a
3
=
−0.568625u
50
+ 0.203922u
49
+ ··· − 12.2253u + 0.626397
−1.07781u
50
+ 0.232253u
49
+ ··· − 0.243835u + 1.04522
a
7
=
u
u
a
11
=
−u
2
+ 1
−u
2
a
6
=
−u
3
+ 2u
−u
3
+ u
a
10
=
−0.275103u
50
+ 0.0102409u
49
+ ··· + 1.08990u − 0.935948
−0.412183u
50
− 0.00870488u
49
+ ··· + 0.371603u + 0.128201
(ii) Obstruction class = −1
(iii) Cusp Shapes = −1.08773u
50
+ 4.19995u
49
+ ··· − 16.8202u − 2.33236
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
51
− 3u
50
+ ··· − 87u + 9
c
2
17(17u
51
− 112u
50
+ ··· + 151u − 17)
c
3
17(17u
51
− 228u
50
+ ··· − 4473u + 2377)
c
5
, c
9
u
51
− 15u
49
+ ··· − 3u − 1
c
6
, c
7
, c
8
c
11
, c
12
u
51
− 2u
50
+ ··· − u + 1
c
10
u
51
+ 3u
50
+ ··· − 291u − 51
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
51
− 39y
50
+ ··· + 5175y −81
c
2
289(289y
51
− 39098y
50
+ ··· + 18449y −289)
c
3
289(289y
51
− 30598y
50
+ ··· + 2.10558 × 10
8
y −5650129)
c
5
, c
9
y
51
− 30y
50
+ ··· + 9y −1
c
6
, c
7
, c
8
c
11
, c
12
y
51
− 70y
50
+ ··· + 9y −1
c
10
y
51
+ 9y
50
+ ··· + 43575y −2601
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.047260 + 0.151588I
a = −0.338923 + 0.162440I
b = −0.065433 + 0.787075I
−4.15821 − 2.51727I 0
u = 1.047260 − 0.151588I
a = −0.338923 − 0.162440I
b = −0.065433 − 0.787075I
−4.15821 + 2.51727I 0
u = −1.077900 + 0.210349I
a = −0.414562 + 0.513194I
b = −0.211649 − 0.453660I
−1.01874 + 6.10618I 0
u = −1.077900 − 0.210349I
a = −0.414562 − 0.513194I
b = −0.211649 + 0.453660I
−1.01874 − 6.10618I 0
u = −0.890510
a = 0.780361
b = 0.512763
−1.69978 −4.88630
u = −1.120720 + 0.024853I
a = −2.15005 − 0.20348I
b = −1.208940 − 0.347028I
−6.84706 + 0.17814I 0
u = −1.120720 − 0.024853I
a = −2.15005 + 0.20348I
b = −1.208940 + 0.347028I
−6.84706 − 0.17814I 0
u = 1.126260 + 0.086381I
a = −1.61198 + 0.21718I
b = −0.738481 + 0.732354I
−5.85482 − 3.55367I 0
u = 1.126260 − 0.086381I
a = −1.61198 − 0.21718I
b = −0.738481 − 0.732354I
−5.85482 + 3.55367I 0
u = −0.438374 + 0.708518I
a = 0.366935 + 0.769343I
b = −0.844551 − 0.361467I
−4.09880 + 2.34673I −13.7642 − 6.0920I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.438374 − 0.708518I
a = 0.366935 − 0.769343I
b = −0.844551 + 0.361467I
−4.09880 − 2.34673I −13.7642 + 6.0920I
u = −1.170700 + 0.336077I
a = 1.80050 + 0.84662I
b = 1.55082 + 0.18031I
−5.46811 + 12.11320I 0
u = −1.170700 − 0.336077I
a = 1.80050 − 0.84662I
b = 1.55082 − 0.18031I
−5.46811 − 12.11320I 0
u = −1.140670 + 0.479406I
a = 1.072900 + 0.758069I
b = 0.825477 − 0.456373I
−4.33250 − 0.67320I 0
u = −1.140670 − 0.479406I
a = 1.072900 − 0.758069I
b = 0.825477 + 0.456373I
−4.33250 + 0.67320I 0
u = 0.436309 + 0.619055I
a = 0.571431 − 1.171610I
b = −0.742635 + 0.081884I
−0.40580 − 8.83491I −5.93498 + 8.65489I
u = 0.436309 − 0.619055I
a = 0.571431 + 1.171610I
b = −0.742635 − 0.081884I
−0.40580 + 8.83491I −5.93498 − 8.65489I
u = 1.190350 + 0.364165I
a = 1.55085 − 0.82569I
b = 1.381760 + 0.003819I
−9.25343 − 5.99551I 0
u = 1.190350 − 0.364165I
a = 1.55085 + 0.82569I
b = 1.381760 − 0.003819I
−9.25343 + 5.99551I 0
u = 0.299142 + 0.682434I
a = −0.097433 − 0.386283I
b = −0.974446 + 0.425322I
0.02412 + 4.64525I −6.01152 − 5.00660I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.299142 − 0.682434I
a = −0.097433 + 0.386283I
b = −0.974446 − 0.425322I
0.02412 − 4.64525I −6.01152 + 5.00660I
u = −1.34834
a = 1.42591
b = 1.94067
−2.22043 0
u = 0.380677 + 0.380330I
a = 1.68381 − 0.54757I
b = −0.0243995 − 0.1158830I
3.01854 + 1.08074I −0.68742 + 2.46829I
u = 0.380677 − 0.380330I
a = 1.68381 + 0.54757I
b = −0.0243995 + 0.1158830I
3.01854 − 1.08074I −0.68742 − 2.46829I
u = 0.295738 + 0.444672I
a = −0.187973 + 0.763692I
b = −0.269582 + 0.799798I
3.28922 − 3.89695I −0.59685 + 7.26004I
u = 0.295738 − 0.444672I
a = −0.187973 − 0.763692I
b = −0.269582 − 0.799798I
3.28922 + 3.89695I −0.59685 − 7.26004I
u = −0.390664 + 0.229819I
a = −0.02035 − 1.55893I
b = 0.805995 − 0.679439I
−1.04347 + 2.52944I −8.36979 − 8.40912I
u = −0.390664 − 0.229819I
a = −0.02035 + 1.55893I
b = 0.805995 + 0.679439I
−1.04347 − 2.52944I −8.36979 + 8.40912I
u = 0.400437
a = −0.797854
b = 1.13406
−2.10246 −9.62010
u = −0.216099 + 0.323578I
a = 0.636723 − 0.775106I
b = 0.035450 − 0.334693I
−0.210137 + 0.904024I −4.53571 − 7.39815I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.216099 − 0.323578I
a = 0.636723 + 0.775106I
b = 0.035450 + 0.334693I
−0.210137 − 0.904024I −4.53571 + 7.39815I
u = 0.330272
a = −1.79695
b = 0.951690
−2.13241 −5.70430
u = −0.111190 + 0.292412I
a = 4.56602 − 1.68963I
b = 0.566891 + 0.031398I
−0.202413 − 0.749454I 1.38825 − 9.49663I
u = −0.111190 − 0.292412I
a = 4.56602 + 1.68963I
b = 0.566891 − 0.031398I
−0.202413 + 0.749454I 1.38825 + 9.49663I
u = 1.72101 + 0.02434I
a = 0.523521 + 0.113299I
b = 1.54117 + 0.66770I
−11.27570 − 0.29543I 0
u = 1.72101 − 0.02434I
a = 0.523521 − 0.113299I
b = 1.54117 − 0.66770I
−11.27570 + 0.29543I 0
u = −1.73799
a = −7.71129
b = −22.2616
−13.2279 0
u = −1.74438 + 0.03663I
a = −0.436565 + 0.244633I
b = −0.910713 − 0.160686I
−14.2469 + 3.2818I 0
u = −1.74438 − 0.03663I
a = −0.436565 − 0.244633I
b = −0.910713 + 0.160686I
−14.2469 − 3.2818I 0
u = 1.74976 + 0.05001I
a = −0.269790 − 0.861011I
b = −0.63067 − 1.27223I
−11.21760 − 7.16998I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.74976 − 0.05001I
a = −0.269790 + 0.861011I
b = −0.63067 + 1.27223I
−11.21760 + 7.16998I 0
u = 1.76103 + 0.00664I
a = −2.28196 + 0.03399I
b = −5.41561 + 0.35170I
−17.3264 − 0.3153I 0
u = 1.76103 − 0.00664I
a = −2.28196 − 0.03399I
b = −5.41561 − 0.35170I
−17.3264 + 0.3153I 0
u = −1.76201 + 0.01966I
a = −1.82088 + 0.20804I
b = −4.16074 − 0.13010I
−16.3513 + 3.9914I 0
u = −1.76201 − 0.01966I
a = −1.82088 − 0.20804I
b = −4.16074 + 0.13010I
−16.3513 − 3.9914I 0
u = 1.77148 + 0.08896I
a = 2.05324 − 0.41872I
b = 5.09853 − 1.25029I
−16.0463 − 13.9639I 0
u = 1.77148 − 0.08896I
a = 2.05324 + 0.41872I
b = 5.09853 + 1.25029I
−16.0463 + 13.9639I 0
u = −1.77662 + 0.09436I
a = 1.91281 + 0.46826I
b = 4.74582 + 1.21071I
19.5628 + 7.9919I 0
u = −1.77662 − 0.09436I
a = 1.91281 − 0.46826I
b = 4.74582 − 1.21071I
19.5628 − 7.9919I 0
u = 1.79336 + 0.11215I
a = 1.64752 − 0.41305I
b = 4.21301 − 0.77958I
−14.9728 − 1.9325I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.79336 − 0.11215I
a = 1.64752 + 0.41305I
b = 4.21301 + 0.77958I
−14.9728 + 1.9325I 0
10
II. I
u
2
= hb − 1, a − 1, u + 1i
(i) Arc colorings
a
8
=
0
−1
a
12
=
1
0
a
1
=
1
1
a
4
=
1
1
a
2
=
1
1
a
5
=
1
1
a
9
=
1
0
a
3
=
0
1
a
7
=
−1
−1
a
11
=
0
−1
a
6
=
−1
0
a
10
=
0
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
10
u
c
2
u − 1
c
3
, c
5
, c
6
c
7
, c
8
, c
9
c
11
, c
12
u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
10
y
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
9
, c
11
, c
12
y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 1.00000
−1.64493 −6.00000
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
u(u
51
− 3u
50
+ ··· − 87u + 9)
c
2
17(u − 1)(17u
51
− 112u
50
+ ··· + 151u − 17)
c
3
17(u + 1)(17u
51
− 228u
50
+ ··· − 4473u + 2377)
c
5
, c
9
(u + 1)(u
51
− 15u
49
+ ··· − 3u − 1)
c
6
, c
7
, c
8
c
11
, c
12
(u + 1)(u
51
− 2u
50
+ ··· − u + 1)
c
10
u(u
51
+ 3u
50
+ ··· − 291u − 51)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
y(y
51
− 39y
50
+ ··· + 5175y −81)
c
2
289(y −1)(289y
51
− 39098y
50
+ ··· + 18449y −289)
c
3
289(y −1)(289y
51
− 30598y
50
+ ··· + 2.10558 × 10
8
y −5650129)
c
5
, c
9
(y −1)(y
51
− 30y
50
+ ··· + 9y −1)
c
6
, c
7
, c
8
c
11
, c
12
(y −1)(y
51
− 70y
50
+ ··· + 9y −1)
c
10
y(y
51
+ 9y
50
+ ··· + 43575y −2601)
16
