12n
0345
(K12n
0345
)
A knot diagram
1
Linearized knot diagam
3 6 8 7 2 11 3 4 12 1 7 10
Solving Sequence
3,7
8 4
9,11
12 6 2 1 5 10
c
7
c
3
c
8
c
11
c
6
c
2
c
1
c
5
c
10
c
4
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−4.30590 × 10
29
u
38
+ 1.40716 × 10
30
u
37
+ ··· + 3.25262 × 10
28
b − 5.80343 × 10
30
,
− 1.92426 × 10
30
u
38
+ 6.35210 × 10
30
u
37
+ ··· + 3.25262 × 10
28
a − 2.53835 × 10
31
,
u
39
− 3u
38
+ ··· + 36u + 4i
I
u
2
= h−au + b − 2a − 1, 2a
2
− au + 2a + 2u − 3, u
2
− 2i
I
v
1
= ha, b + v + 2, v
2
+ 3v + 1i
* 3 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−4.31 × 10
29
u
38
+ 1.41 × 10
30
u
37
+ · · · + 3.25 × 10
28
b − 5.80 ×
10
30
, −1.92 × 10
30
u
38
+ 6.35 × 10
30
u
37
+ · · · + 3.25 × 10
28
a − 2.54 ×
10
31
, u
39
− 3u
38
+ · · · + 36u + 4i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
−u
2
a
4
=
u
−u
3
+ u
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
11
=
59.1604u
38
− 195.292u
37
+ ··· + 4436.04u + 780.403
13.2382u
38
− 43.2623u
37
+ ··· + 993.748u + 178.423
a
12
=
45.9221u
38
− 152.030u
37
+ ··· + 3442.29u + 601.980
13.2382u
38
− 43.2623u
37
+ ··· + 993.748u + 178.423
a
6
=
51.1260u
38
− 169.150u
37
+ ··· + 3814.68u + 661.729
23.7145u
38
− 78.2230u
37
+ ··· + 1784.77u + 313.995
a
2
=
−1.07361u
38
+ 3.79195u
37
+ ··· − 41.8527u + 1.03040
26.3378u
38
− 87.1349u
37
+ ··· + 1988.06u + 348.765
a
1
=
−1.07361u
38
+ 3.79195u
37
+ ··· − 41.8527u + 1.03040
26.7461u
38
− 88.2756u
37
+ ··· + 2004.32u + 351.049
a
5
=
−u
3
+ 2u
−u
3
+ u
a
10
=
7.08403u
38
− 23.4120u
37
+ ··· + 528.155u + 95.4828
−26.7461u
38
+ 88.2756u
37
+ ··· − 2004.32u − 351.049
(ii) Obstruction class = −1
(iii) Cusp Shapes = −56.2608u
38
+ 183.733u
37
+ ··· − 4395.07u − 821.737
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
39
+ 15u
38
+ ··· + 5679u + 81
c
2
, c
5
u
39
+ 3u
38
+ ··· − 45u + 9
c
3
, c
7
, c
8
u
39
− 3u
38
+ ··· + 36u + 4
c
4
u
39
+ 9u
38
+ ··· − 12340u − 380
c
6
, c
11
u
39
− 2u
38
+ ··· − 10u + 1
c
9
, c
10
, c
12
u
39
− 4u
38
+ ··· − 6u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
39
+ 25y
38
+ ··· + 19679031y −6561
c
2
, c
5
y
39
− 15y
38
+ ··· + 5679y −81
c
3
, c
7
, c
8
y
39
− 49y
38
+ ··· + 560y −16
c
4
y
39
− 109y
38
+ ··· + 28757360y −144400
c
6
, c
11
y
39
− 6y
38
+ ··· + 46y −1
c
9
, c
10
, c
12
y
39
− 30y
38
+ ··· + 38y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.278366 + 0.955359I
a = −0.591023 + 0.828865I
b = −0.623478 + 0.614458I
−3.35402 + 4.18126I −7.78277 − 6.99514I
u = −0.278366 − 0.955359I
a = −0.591023 − 0.828865I
b = −0.623478 − 0.614458I
−3.35402 − 4.18126I −7.78277 + 6.99514I
u = −0.887627 + 0.471243I
a = −1.50907 + 0.17598I
b = −0.886394 − 0.797483I
2.35149 − 5.07575I −1.77054 + 6.40036I
u = −0.887627 − 0.471243I
a = −1.50907 − 0.17598I
b = −0.886394 + 0.797483I
2.35149 + 5.07575I −1.77054 − 6.40036I
u = −0.957964
a = −0.302923
b = 1.25836
−8.12538 −10.0210
u = 1.065650 + 0.085902I
a = −0.718491 + 0.083392I
b = −0.515563 + 0.600875I
2.77568 − 0.03264I 0
u = 1.065650 − 0.085902I
a = −0.718491 − 0.083392I
b = −0.515563 − 0.600875I
2.77568 + 0.03264I 0
u = −0.796295 + 0.736026I
a = 1.42339 + 0.01369I
b = 0.950413 + 0.884348I
−1.74573 − 9.69427I 0. + 8.10788I
u = −0.796295 − 0.736026I
a = 1.42339 − 0.01369I
b = 0.950413 − 0.884348I
−1.74573 + 9.69427I 0. − 8.10788I
u = 0.851097 + 0.768811I
a = 0.636558 − 0.270946I
b = 0.379215 − 0.675835I
0.64343 + 2.93674I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.851097 − 0.768811I
a = 0.636558 + 0.270946I
b = 0.379215 + 0.675835I
0.64343 − 2.93674I 0
u = 0.756459 + 0.321115I
a = 0.848073 + 0.323450I
b = 0.687338 + 0.927457I
−1.16384 + 3.37058I −4.85431 − 4.93903I
u = 0.756459 − 0.321115I
a = 0.848073 − 0.323450I
b = 0.687338 − 0.927457I
−1.16384 − 3.37058I −4.85431 + 4.93903I
u = −0.729654 + 0.114009I
a = 1.94616 − 0.30625I
b = 0.877049 + 0.529169I
−0.720527 − 0.677496I −4.45492 + 3.52872I
u = −0.729654 − 0.114009I
a = 1.94616 + 0.30625I
b = 0.877049 − 0.529169I
−0.720527 + 0.677496I −4.45492 − 3.52872I
u = 1.36027
a = −1.02723
b = −0.392618
3.15342 0
u = 0.025858 + 0.596227I
a = 1.28935 − 0.78910I
b = 0.476410 − 0.516668I
−0.39053 + 1.36769I −3.78544 − 4.37417I
u = 0.025858 − 0.596227I
a = 1.28935 + 0.78910I
b = 0.476410 + 0.516668I
−0.39053 − 1.36769I −3.78544 + 4.37417I
u = −1.42742
a = −10.9292
b = −0.157920
1.67196 0
u = 1.47236
a = 0.751761
b = 1.74885
−4.21706 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.111960 + 0.411759I
a = −4.18092 − 0.61496I
b = −0.394904 + 0.488828I
−3.05808 − 0.73206I −13.3386 − 6.5570I
u = 0.111960 − 0.411759I
a = −4.18092 + 0.61496I
b = −0.394904 − 0.488828I
−3.05808 + 0.73206I −13.3386 + 6.5570I
u = 1.64569 + 0.02330I
a = −0.940304 − 0.261465I
b = −1.33024 + 0.90609I
7.65877 + 1.14141I 0
u = 1.64569 − 0.02330I
a = −0.940304 + 0.261465I
b = −1.33024 − 0.90609I
7.65877 − 1.14141I 0
u = −1.64517 + 0.08077I
a = −0.403452 − 0.032549I
b = −0.98617 + 1.32010I
7.20723 − 4.84674I 0
u = −1.64517 − 0.08077I
a = −0.403452 + 0.032549I
b = −0.98617 − 1.32010I
7.20723 + 4.84674I 0
u = 1.65274 + 0.23220I
a = −1.008000 − 0.471693I
b = −1.18235 + 1.09377I
6.4748 + 13.4066I 0
u = 1.65274 − 0.23220I
a = −1.008000 + 0.471693I
b = −1.18235 − 1.09377I
6.4748 − 13.4066I 0
u = 1.66262 + 0.24913I
a = −0.241455 − 0.016965I
b = −0.136254 + 0.574075I
2.81425 + 0.97653I 0
u = 1.66262 − 0.24913I
a = −0.241455 + 0.016965I
b = −0.136254 − 0.574075I
2.81425 − 0.97653I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.315278
a = 4.23572
b = −0.575388
−2.39339 9.22220
u = 1.67924 + 0.13585I
a = 0.987276 + 0.371552I
b = 1.25098 − 1.02158I
11.24360 + 7.46891I 0
u = 1.67924 − 0.13585I
a = 0.987276 − 0.371552I
b = 1.25098 + 1.02158I
11.24360 − 7.46891I 0
u = −0.301204
a = −2.36999
b = −1.68053
−10.3072 10.7730
u = −1.69258 + 0.19776I
a = −0.675700 + 0.148591I
b = −0.787702 − 1.170530I
9.38751 − 6.54529I 0
u = −1.69258 − 0.19776I
a = −0.675700 − 0.148591I
b = −0.787702 + 1.170530I
9.38751 + 6.54529I 0
u = −1.70589 + 0.04672I
a = 0.538540 − 0.089371I
b = 0.85440 + 1.24646I
12.52260 − 0.71174I 0
u = −1.70589 − 0.04672I
a = 0.538540 + 0.089371I
b = 0.85440 − 1.24646I
12.52260 + 0.71174I 0
u = −0.262239
a = 2.84003
b = 0.533756
−1.18388 −7.92960
8
II. I
u
2
= h−au + b − 2a − 1, 2a
2
− au + 2a + 2u − 3, u
2
− 2i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
−2
a
4
=
u
−u
a
9
=
−1
0
a
11
=
a
au + 2a + 1
a
12
=
−au − a − 1
au + 2a + 1
a
6
=
1
2
u
−au − 2a − 2
a
2
=
1
2
u
−au − 2a + u − 2
a
1
=
1
2
u
−au − 2a − 2
a
5
=
0
−u
a
10
=
au + 2a +
1
2
u
−au − 2a − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
7
c
8
(u
2
− 2)
2
c
6
, c
12
(u
2
− u − 1)
2
c
9
, c
10
, c
11
(u
2
+ u − 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
7
c
8
(y −2)
4
c
6
, c
9
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −0.473911
b = −0.618034
2.30291 −8.00000
u = 1.41421
a = 0.181018
b = 1.61803
−5.59278 −8.00000
u = −1.41421
a = 1.05505
b = 1.61803
−5.59278 −8.00000
u = −1.41421
a = −2.76216
b = −0.618034
2.30291 −8.00000
12
III. I
v
1
= ha, b + v + 2, v
2
+ 3v + 1i
(i) Arc colorings
a
3
=
v
0
a
7
=
1
0
a
8
=
1
0
a
4
=
v
0
a
9
=
1
0
a
11
=
0
−v − 2
a
12
=
v + 2
−v − 2
a
6
=
1
−v − 3
a
2
=
v − 1
v + 3
a
1
=
−1
v + 3
a
5
=
v
0
a
10
=
−v − 2
v + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −26
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
7
c
8
u
2
c
5
(u + 1)
2
c
6
, c
9
, c
10
u
2
+ u − 1
c
11
, c
12
u
2
− u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
2
c
3
, c
4
, c
7
c
8
y
2
c
6
, c
9
, c
10
c
11
, c
12
y
2
− 3y + 1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.381966
a = 0
b = −1.61803
−10.5276 −26.0000
v = −2.61803
a = 0
b = 0.618034
−2.63189 −26.0000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
39
+ 15u
38
+ ··· + 5679u + 81)
c
2
((u − 1)
2
)(u + 1)
4
(u
39
+ 3u
38
+ ··· − 45u + 9)
c
3
, c
7
, c
8
u
2
(u
2
− 2)
2
(u
39
− 3u
38
+ ··· + 36u + 4)
c
4
u
2
(u
2
− 2)
2
(u
39
+ 9u
38
+ ··· − 12340u − 380)
c
5
((u − 1)
4
)(u + 1)
2
(u
39
+ 3u
38
+ ··· − 45u + 9)
c
6
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
39
− 2u
38
+ ··· − 10u + 1)
c
9
, c
10
((u
2
+ u − 1)
3
)(u
39
− 4u
38
+ ··· − 6u − 1)
c
11
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
39
− 2u
38
+ ··· − 10u + 1)
c
12
((u
2
− u − 1)
3
)(u
39
− 4u
38
+ ··· − 6u − 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
6
)(y
39
+ 25y
38
+ ··· + 19679031y − 6561)
c
2
, c
5
((y − 1)
6
)(y
39
− 15y
38
+ ··· + 5679y − 81)
c
3
, c
7
, c
8
y
2
(y − 2)
4
(y
39
− 49y
38
+ ··· + 560y − 16)
c
4
y
2
(y − 2)
4
(y
39
− 109y
38
+ ··· + 2.87574 × 10
7
y − 144400)
c
6
, c
11
((y
2
− 3y + 1)
3
)(y
39
− 6y
38
+ ··· + 46y − 1)
c
9
, c
10
, c
12
((y
2
− 3y + 1)
3
)(y
39
− 30y
38
+ ··· + 38y − 1)
18
