12n
0655
(K12n
0655
)
A knot diagram
1
Linearized knot diagam
3 8 12 11 9 10 2 6 3 6 4 9
Solving Sequence
3,9 6,10
11 5 4 8 2 1 7 12
c
9
c
10
c
5
c
4
c
8
c
2
c
1
c
7
c
12
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h8082770070u
10
+ 12874278132u
9
+ ··· + 78579513703b + 583530199807,
− 10533832913358u
10
− 20498499647469u
9
+ ··· + 43925948159977a − 453136624250477,
u
11
+ u
10
− 25u
9
+ 84u
8
− 4u
7
− 295u
6
+ 294u
5
+ 266u
4
− 592u
3
+ 258u
2
+ 53u − 43i
I
u
2
= h−13u
11
+ 11u
10
− 7u
9
+ 26u
8
+ 50u
7
− 47u
6
+ 8u
5
− 54u
4
− 44u
3
+ 10u
2
+ 46b + 21u + 53,
8u
11
+ 18u
10
− u
9
+ 30u
8
− 52u
7
− 33u
6
+ 11u
5
− 57u
4
+ 66u
3
+ 54u
2
+ 23a + 26u + 1,
u
12
+ 2u
10
− u
9
− 2u
8
− u
7
− 3u
6
+ 2u
5
+ 2u
4
+ 4u
3
+ u
2
− 1i
I
u
3
= hb
2
− b + 1, a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 25 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
= h8.08 × 10
9
u
10
+ 1.29 × 10
10
u
9
+ · · · + 7.86 × 10
10
b + 5.84 × 10
11
, −1.05 ×
10
13
u
10
−2.05×10
13
u
9
+· · ·+4.39×10
13
a−4.53×10
14
, u
11
+u
10
+· · ·+53u−43i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
6
=
0.239809u
10
+ 0.466660u
9
+ ··· − 5.07748u + 10.3159
−0.102861u
10
− 0.163838u
9
+ ··· − 2.44421u − 7.42598
a
10
=
1
u
2
a
11
=
1.00685u
10
+ 1.73463u
9
+ ··· + 5.81239u + 59.4569
0.149991u
10
+ 0.294158u
9
+ ··· − 2.04642u + 6.58542
a
5
=
0.136948u
10
+ 0.302823u
9
+ ··· − 7.52169u + 2.88994
−0.102861u
10
− 0.163838u
9
+ ··· − 2.44421u − 7.42598
a
4
=
−0.0526849u
10
− 0.162132u
9
+ ··· + 7.57873u + 2.62004
0.214365u
10
+ 0.374779u
9
+ ··· + 1.03850u + 11.9120
a
8
=
−0.240845u
10
− 0.364372u
9
+ ··· − 6.91721u − 16.6179
−0.192976u
10
− 0.323582u
9
+ ··· − 1.76300u − 11.8971
a
2
=
0.672190u
10
+ 1.17359u
9
+ ··· + 5.50874u + 38.9931
0.241840u
10
+ 0.437101u
9
+ ··· + 1.35348u + 12.9629
a
1
=
0.672190u
10
+ 1.17359u
9
+ ··· + 5.50874u + 38.9931
−0.145871u
10
− 0.264230u
9
+ ··· − 0.976455u − 8.59734
a
7
=
0.539583u
10
+ 0.987055u
9
+ ··· − 4.34462u + 27.4965
0.0516713u
10
+ 0.0842453u
9
+ ··· − 1.24682u + 2.06071
a
12
=
0.526320u
10
+ 0.909361u
9
+ ··· + 4.53228u + 30.3957
−0.145871u
10
− 0.264230u
9
+ ··· − 0.976455u − 8.59734
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
1137457766308
1021533678139
u
10
+
2052116979712
1021533678139
u
9
+ ··· +
13195254097138
1021533678139
u +
47127968197766
1021533678139
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
+ 30u
10
+ ··· + 46180u + 6241
c
2
, c
7
u
11
+ 2u
10
+ ··· + 290u + 79
c
3
, c
4
, c
11
u
11
− 3u
10
+ ··· − 4u + 1
c
5
, c
8
u
11
− 6u
10
+ ··· + 59u + 14
c
6
, c
10
u
11
+ 16u
10
+ ··· − 1205u − 239
c
9
u
11
+ u
10
+ ··· + 53u − 43
c
12
u
11
+ 2u
10
+ ··· − 9u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 270y
10
+ ··· + 1320014200y −38950081
c
2
, c
7
y
11
− 30y
10
+ ··· + 46180y −6241
c
3
, c
4
, c
11
y
11
+ 9y
10
+ ··· + 78y −1
c
5
, c
8
y
11
− 24y
10
+ ··· + 4069y −196
c
6
, c
10
y
11
− 94y
10
+ ··· + 639425y −57121
c
9
y
11
− 51y
10
+ ··· + 24997y −1849
c
12
y
11
− 28y
10
+ ··· + 289y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.14783
a = −0.625800
b = 0.599073
−7.73425 −2.27460
u = 0.961384 + 0.663629I
a = 0.896764 − 0.517452I
b = 0.196134 + 1.286040I
1.65294 + 2.05058I −10.88896 − 3.66206I
u = 0.961384 − 0.663629I
a = 0.896764 + 0.517452I
b = 0.196134 − 1.286040I
1.65294 − 2.05058I −10.88896 + 3.66206I
u = 0.729816 + 0.078717I
a = −2.86559 − 0.69887I
b = −0.674831 + 0.650678I
9.56370 + 2.81490I −11.11373 − 3.91531I
u = 0.729816 − 0.078717I
a = −2.86559 + 0.69887I
b = −0.674831 − 0.650678I
9.56370 − 2.81490I −11.11373 + 3.91531I
u = −1.47245 + 0.38796I
a = −0.199156 − 0.028059I
b = −1.09827 − 0.95568I
−1.22023 − 2.91334I −10.21854 + 2.00386I
u = −1.47245 − 0.38796I
a = −0.199156 + 0.028059I
b = −1.09827 + 0.95568I
−1.22023 + 2.91334I −10.21854 − 2.00386I
u = −0.357190
a = 0.642478
b = −0.230598
−0.556349 −18.0810
u = 2.17516 + 2.14936I
a = 0.426144 − 0.535385I
b = 1.92336 + 1.67643I
−15.4000 − 9.3794I −9.97556 + 3.09821I
u = 2.17516 − 2.14936I
a = 0.426144 + 0.535385I
b = 1.92336 − 1.67643I
−15.4000 + 9.3794I −9.97556 − 3.09821I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −6.57845
a = 0.327461
b = 4.93875
17.4529 −11.2510
6
II. I
u
2
= h−13u
11
+ 11u
10
+ · · · + 46b + 53, 8u
11
+ 18u
10
+ · · · + 23a +
1, u
12
+ 2u
10
+ · · · + u
2
− 1i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
6
=
−0.347826u
11
− 0.782609u
10
+ ··· − 1.13043u − 0.0434783
0.282609u
11
− 0.239130u
10
+ ··· − 0.456522u − 1.15217
a
10
=
1
u
2
a
11
=
−1.02174u
11
+ 1.32609u
10
+ ··· + 2.80435u + 0.934783
u
2
+ 1
a
5
=
−0.0652174u
11
− 1.02174u
10
+ ··· − 1.58696u − 1.19565
0.282609u
11
− 0.239130u
10
+ ··· − 0.456522u − 1.15217
a
4
=
0.891304u
11
− 0.369565u
10
+ ··· − 1.97826u − 1.32609
0.586957u
11
− 0.804348u
10
+ ··· − 0.717391u − 0.239130
a
8
=
−0.456522u
11
− 0.152174u
10
+ ··· − 0.108696u − 0.369565
0.282609u
11
− 0.239130u
10
+ ··· − 0.456522u − 1.15217
a
2
=
0.630435u
11
− 0.456522u
10
+ ··· − 0.326087u − 0.108696
0.239130u
11
+ 0.413043u
10
+ ··· + 2.15217u − 0.282609
a
1
=
0.630435u
11
− 0.456522u
10
+ ··· − 0.326087u − 0.108696
0.391304u
11
+ 0.130435u
10
+ ··· + 1.52174u + 0.173913
a
7
=
0.108696u
11
− 0.630435u
10
+ ··· − 1.02174u + 0.326087
−1
a
12
=
1.02174u
11
− 0.326087u
10
+ ··· + 1.19565u + 0.0652174
0.391304u
11
+ 0.130435u
10
+ ··· + 1.52174u + 0.173913
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
18
23
u
11
−
63
23
u
10
+
84
23
u
9
−
151
23
u
8
+
113
23
u
7
+
58
23
u
6
−
50
23
u
5
+
188
23
u
4
−
139
23
u
3
−
74
23
u
2
−
160
23
u−
153
23
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− 12u
11
+ ··· − 14u + 1
c
2
u
12
− 6u
10
+ u
9
+ 15u
8
− 4u
7
− 21u
6
+ 5u
5
+ 17u
4
− 3u
3
− 7u
2
+ 1
c
3
, c
4
u
12
+ u
11
+ ··· − 2u
2
− 1
c
5
u
12
− 3u
11
+ 5u
10
− 5u
9
+ 5u
7
− 5u
6
+ 2u
5
+ 3u
4
− 3u
3
− 1
c
6
u
12
+ 3u
9
− 3u
8
− 2u
7
+ 5u
6
− 5u
5
+ 5u
3
− 5u
2
+ 3u − 1
c
7
u
12
− 6u
10
− u
9
+ 15u
8
+ 4u
7
− 21u
6
− 5u
5
+ 17u
4
+ 3u
3
− 7u
2
+ 1
c
8
u
12
+ 3u
11
+ 5u
10
+ 5u
9
− 5u
7
− 5u
6
− 2u
5
+ 3u
4
+ 3u
3
− 1
c
9
u
12
+ 2u
10
− u
9
− 2u
8
− u
7
− 3u
6
+ 2u
5
+ 2u
4
+ 4u
3
+ u
2
− 1
c
10
u
12
− 3u
9
− 3u
8
+ 2u
7
+ 5u
6
+ 5u
5
− 5u
3
− 5u
2
− 3u − 1
c
11
u
12
− u
11
+ ··· − 2u
2
− 1
c
12
u
12
− 4u
11
+ ··· + 14u + 13
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 12y
11
+ ··· − 30y + 1
c
2
, c
7
y
12
− 12y
11
+ ··· − 14y + 1
c
3
, c
4
, c
11
y
12
+ 15y
11
+ ··· + 4y + 1
c
5
, c
8
y
12
+ y
11
+ ··· − 6y
2
+ 1
c
6
, c
10
y
12
− 6y
10
+ ··· + y + 1
c
9
y
12
+ 4y
11
+ ··· − 2y + 1
c
12
y
12
− 18y
11
+ ··· + 298y + 169
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.290321 + 0.812321I
a = 1.60962 + 2.31665I
b = −0.148186 − 0.670337I
10.30340 − 2.52674I −1.40798 + 1.03113I
u = −0.290321 − 0.812321I
a = 1.60962 − 2.31665I
b = −0.148186 + 0.670337I
10.30340 + 2.52674I −1.40798 − 1.03113I
u = −1.14608
a = −0.443256
b = 0.750402
−8.04457 −28.6700
u = 1.126080 + 0.271306I
a = −0.106545 + 0.615518I
b = 0.727354 + 0.248918I
−3.20078 − 2.30167I −14.2286 + 2.6049I
u = 1.126080 − 0.271306I
a = −0.106545 − 0.615518I
b = 0.727354 − 0.248918I
−3.20078 + 2.30167I −14.2286 − 2.6049I
u = 0.184248 + 1.148890I
a = 0.47497 − 1.34357I
b = −0.296082 + 1.038070I
2.61823 + 1.74326I −3.69799 − 2.86039I
u = 0.184248 − 1.148890I
a = 0.47497 + 1.34357I
b = −0.296082 − 1.038070I
2.61823 − 1.74326I −3.69799 + 2.86039I
u = −0.564251 + 0.474715I
a = −1.270550 − 0.606648I
b = −1.199700 + 0.382821I
0.75595 + 3.94751I −8.05344 − 4.79157I
u = −0.564251 − 0.474715I
a = −1.270550 + 0.606648I
b = −1.199700 − 0.382821I
0.75595 − 3.94751I −8.05344 + 4.79157I
u = 0.524737
a = −1.48305
b = −1.22481
−3.37867 −11.4520
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.14508 + 1.49712I
a = 0.255664 + 0.886834I
b = −0.34618 − 1.41208I
5.10447 − 1.33905I −8.55072 + 1.28965I
u = −0.14508 − 1.49712I
a = 0.255664 − 0.886834I
b = −0.34618 + 1.41208I
5.10447 + 1.33905I −8.55072 − 1.28965I
11
III. I
u
3
= hb
2
− b + 1, a − 1, u − 1i
(i) Arc colorings
a
3
=
0
1
a
9
=
1
0
a
6
=
1
b
a
10
=
1
1
a
11
=
b
0
a
5
=
b + 1
b
a
4
=
b
b
a
8
=
−b + 1
−b + 1
a
2
=
−b
−b + 1
a
1
=
−b
1
a
7
=
−b + 2
1
a
12
=
−b + 1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4b − 7
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
8
u
2
− u + 1
c
3
, c
4
, c
6
c
10
, c
11
u
2
+ u + 1
c
9
, c
12
(u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
10
c
11
y
2
+ y + 1
c
9
, c
12
(y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0.500000 + 0.866025I
1.64493 + 2.02988I −9.00000 − 3.46410I
u = 1.00000
a = 1.00000
b = 0.500000 − 0.866025I
1.64493 − 2.02988I −9.00000 + 3.46410I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
11
+ 30u
10
+ ··· + 46180u + 6241)
· (u
12
− 12u
11
+ ··· − 14u + 1)
c
2
(u
2
− u + 1)(u
11
+ 2u
10
+ ··· + 290u + 79)
· (u
12
− 6u
10
+ u
9
+ 15u
8
− 4u
7
− 21u
6
+ 5u
5
+ 17u
4
− 3u
3
− 7u
2
+ 1)
c
3
, c
4
(u
2
+ u + 1)(u
11
− 3u
10
+ ··· − 4u + 1)(u
12
+ u
11
+ ··· − 2u
2
− 1)
c
5
(u
2
− u + 1)(u
11
− 6u
10
+ ··· + 59u + 14)
· (u
12
− 3u
11
+ 5u
10
− 5u
9
+ 5u
7
− 5u
6
+ 2u
5
+ 3u
4
− 3u
3
− 1)
c
6
(u
2
+ u + 1)(u
11
+ 16u
10
+ ··· − 1205u − 239)
· (u
12
+ 3u
9
− 3u
8
− 2u
7
+ 5u
6
− 5u
5
+ 5u
3
− 5u
2
+ 3u − 1)
c
7
(u
2
− u + 1)(u
11
+ 2u
10
+ ··· + 290u + 79)
· (u
12
− 6u
10
− u
9
+ 15u
8
+ 4u
7
− 21u
6
− 5u
5
+ 17u
4
+ 3u
3
− 7u
2
+ 1)
c
8
(u
2
− u + 1)(u
11
− 6u
10
+ ··· + 59u + 14)
· (u
12
+ 3u
11
+ 5u
10
+ 5u
9
− 5u
7
− 5u
6
− 2u
5
+ 3u
4
+ 3u
3
− 1)
c
9
((u − 1)
2
)(u
11
+ u
10
+ ··· + 53u − 43)
· (u
12
+ 2u
10
− u
9
− 2u
8
− u
7
− 3u
6
+ 2u
5
+ 2u
4
+ 4u
3
+ u
2
− 1)
c
10
(u
2
+ u + 1)(u
11
+ 16u
10
+ ··· − 1205u − 239)
· (u
12
− 3u
9
− 3u
8
+ 2u
7
+ 5u
6
+ 5u
5
− 5u
3
− 5u
2
− 3u − 1)
c
11
(u
2
+ u + 1)(u
11
− 3u
10
+ ··· − 4u + 1)(u
12
− u
11
+ ··· − 2u
2
− 1)
c
12
((u − 1)
2
)(u
11
+ 2u
10
+ ··· − 9u − 4)(u
12
− 4u
11
+ ··· + 14u + 13)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
11
− 270y
10
+ ··· + 1.32001 × 10
9
y −3.89501 × 10
7
)
· (y
12
− 12y
11
+ ··· − 30y + 1)
c
2
, c
7
(y
2
+ y + 1)(y
11
− 30y
10
+ ··· + 46180y −6241)
· (y
12
− 12y
11
+ ··· − 14y + 1)
c
3
, c
4
, c
11
(y
2
+ y + 1)(y
11
+ 9y
10
+ ··· + 78y −1)(y
12
+ 15y
11
+ ··· + 4y + 1)
c
5
, c
8
(y
2
+ y + 1)(y
11
− 24y
10
+ ··· + 4069y −196)(y
12
+ y
11
+ ··· − 6y
2
+ 1)
c
6
, c
10
(y
2
+ y + 1)(y
11
− 94y
10
+ ··· + 639425y −57121)
· (y
12
− 6y
10
+ ··· + y + 1)
c
9
((y −1)
2
)(y
11
− 51y
10
+ ··· + 24997y −1849)
· (y
12
+ 4y
11
+ ··· − 2y + 1)
c
12
((y −1)
2
)(y
11
− 28y
10
+ ··· + 289y −16)
· (y
12
− 18y
11
+ ··· + 298y + 169)
17
