12n
0577
(K12n
0577
)
A knot diagram
1
Linearized knot diagam
3 7 12 11 10 2 12 1 3 4 5 8
Solving Sequence
5,11 8,12
1 4 3 7 2 6 10 9
c
11
c
12
c
4
c
3
c
7
c
2
c
6
c
10
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
18
+ 3u
17
+ ··· + 4b + 4, 2u
18
+ 3u
17
+ ··· + 4a − 2, u
19
+ 2u
18
+ ··· − 2u + 2i
I
u
2
= hu
3
− 3u
2
+ b − 2u + 2, 2u
3
− 3u
2
+ 3a − 3u, u
4
− 3u
2
+ 3i
I
u
3
= hu
3
+ u
2
+ b, −u
2
+ a + u + 2, u
4
− u
2
− 1i
I
u
4
= h−a
2
+ b + a + 2, a
3
− 2a
2
− 3a − 1, u − 1i
I
v
1
= ha, b − 1, v −1i
* 5 irreducible components of dim
C
= 0, with total 31 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
= hu
18
+3u
17
+· · ·+4b+4, 2u
18
+3u
17
+· · ·+4a−2, u
19
+2u
18
+· · ·−2u+2i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
0
a
8
=
−
1
2
u
18
−
3
4
u
17
+ ··· + u +
1
2
−
1
4
u
18
−
3
4
u
17
+ ··· +
3
2
u − 1
a
12
=
1
u
2
a
1
=
1
2
u
18
−
13
4
u
16
+ ··· −
1
2
u +
3
2
3
2
u
18
+
3
2
u
17
+ ··· − 3u + 2
a
4
=
u
u
a
3
=
−u
3
+ 2u
−u
5
+ u
3
+ u
a
7
=
−
1
4
u
17
+
3
2
u
15
+ ··· + u + 1
−
1
4
u
17
+
3
2
u
15
+ ··· −
3
2
u
2
−
1
2
u
a
2
=
1
2
u
12
−
5
2
u
10
+ ··· +
3
2
u + 1
1
4
u
17
−
3
2
u
15
+ ··· +
3
2
u
2
+
1
2
u
a
6
=
u
5
− 2u
3
+ u
u
5
− u
3
− u
a
10
=
−u
2
+ 1
−u
2
a
9
=
u
10
− 5u
8
+ 8u
6
− 3u
4
− 3u
2
+ 1
u
12
− 4u
10
+ 4u
8
+ 2u
6
− 3u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
18
+ 14u
16
+ 2u
15
−40u
14
−12u
13
+ 46u
12
+ 28u
11
+ 18u
10
−
26u
9
− 98u
8
− 4u
7
+ 64u
6
+ 22u
5
+ 36u
4
− 12u
3
− 38u
2
+ 8u − 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
19
− 4u
18
+ ··· + 1887u + 49
c
2
, c
6
u
19
− 2u
18
+ ··· − 37u − 7
c
3
, c
5
u
19
− 3u
18
+ ··· − 122u − 46
c
4
, c
10
, c
11
u
19
− 2u
18
+ ··· − 2u − 2
c
7
, c
8
, c
12
u
19
+ 2u
18
+ ··· + 63u − 7
c
9
u
19
− 4u
18
+ ··· + 13446u − 5482
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
19
+ 64y
18
+ ··· + 3231195y −2401
c
2
, c
6
y
19
+ 4y
18
+ ··· + 1887y −49
c
3
, c
5
y
19
+ 35y
18
+ ··· + 9456y −2116
c
4
, c
10
, c
11
y
19
− 14y
18
+ ··· + 8y −4
c
7
, c
8
, c
12
y
19
− 36y
18
+ ··· + 6895y −49
c
9
y
19
+ 110y
18
+ ··· − 1950902712y −30052324
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.057514 + 0.997076I
a = 4.41674 + 0.12405I
b = −2.30270 − 0.31667I
−19.3196 + 4.9982I −1.30896 − 2.07714I
u = −0.057514 − 0.997076I
a = 4.41674 − 0.12405I
b = −2.30270 + 0.31667I
−19.3196 − 4.9982I −1.30896 + 2.07714I
u = −0.142968 + 0.865479I
a = −3.01025 + 1.29423I
b = 1.029780 + 0.322487I
7.61295 + 0.58148I −0.023432 − 0.755964I
u = −0.142968 − 0.865479I
a = −3.01025 − 1.29423I
b = 1.029780 − 0.322487I
7.61295 − 0.58148I −0.023432 + 0.755964I
u = 1.24695
a = −0.418171
b = −1.55733
−2.34368 −4.04380
u = −1.223500 + 0.244957I
a = −0.177243 + 0.379170I
b = −0.340423 + 0.077135I
−2.81621 + 4.28308I −9.76275 − 6.60090I
u = −1.223500 − 0.244957I
a = −0.177243 − 0.379170I
b = −0.340423 − 0.077135I
−2.81621 − 4.28308I −9.76275 + 6.60090I
u = −1.178360 + 0.467565I
a = 2.39170 − 0.15913I
b = 3.99920 − 2.07726I
4.46238 + 4.21258I −3.28610 − 3.54866I
u = −1.178360 − 0.467565I
a = 2.39170 + 0.15913I
b = 3.99920 + 2.07726I
4.46238 − 4.21258I −3.28610 + 3.54866I
u = −1.28379
a = −1.10593
b = −1.73530
−5.57169 −16.6430
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.282330 + 0.529998I
a = −2.87135 + 1.25440I
b = −4.30847 + 5.80802I
16.3839 + 0.4170I −3.95888 − 0.90643I
u = −1.282330 − 0.529998I
a = −2.87135 − 1.25440I
b = −4.30847 − 5.80802I
16.3839 − 0.4170I −3.95888 + 0.90643I
u = 1.358030 + 0.344537I
a = 1.32857 + 1.36569I
b = 2.46464 + 3.90915I
2.84646 − 4.89611I −3.78070 + 3.50098I
u = 1.358030 − 0.344537I
a = 1.32857 − 1.36569I
b = 2.46464 − 3.90915I
2.84646 + 4.89611I −3.78070 − 3.50098I
u = 1.35479 + 0.47305I
a = −2.92129 − 1.22251I
b = −5.11425 − 5.33647I
15.7390 − 10.2337I −4.55408 + 4.64263I
u = 1.35479 − 0.47305I
a = −2.92129 + 1.22251I
b = −5.11425 + 5.33647I
15.7390 + 10.2337I −4.55408 − 4.64263I
u = 0.021436 + 0.497066I
a = 0.993630 − 0.078348I
b = 0.348156 − 0.347395I
0.87785 − 1.46275I −1.25478 + 5.25309I
u = 0.021436 − 0.497066I
a = 0.993630 + 0.078348I
b = 0.348156 + 0.347395I
0.87785 + 1.46275I −1.25478 − 5.25309I
u = 0.337682
a = 1.22310
b = −0.259248
−0.889902 −13.4540
6
II. I
u
2
= hu
3
− 3u
2
+ b − 2u + 2, 2u
3
− 3u
2
+ 3a − 3u, u
4
− 3u
2
+ 3i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
0
a
8
=
−
2
3
u
3
+ u
2
+ u
−u
3
+ 3u
2
+ 2u − 2
a
12
=
1
u
2
a
1
=
2
3
u
3
− u
2
− u + 1
u
3
− 2u
2
− 2u + 2
a
4
=
u
u
a
3
=
−u
3
+ 2u
−2u
3
+ 4u
a
7
=
−
2
3
u
3
+ u
2
+ u − 1
−u
3
+ 2u
2
+ 2u − 2
a
2
=
−
1
3
u
3
− u
2
+ u + 1
−u
3
− 2u
2
+ 2u + 2
a
6
=
−u
3
+ 2u
−2u
3
+ 4u
a
10
=
−u
2
+ 1
−u
2
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 12
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
(u − 1)
4
c
3
, c
5
, c
9
u
4
+ 3u
2
+ 3
c
4
, c
10
, c
11
u
4
− 3u
2
+ 3
c
6
, c
7
, c
8
(u + 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
(y −1)
4
c
3
, c
5
, c
9
(y
2
+ 3y + 3)
2
c
4
, c
10
, c
11
(y
2
− 3y + 3)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.271230 + 0.340625I
a = 1.69666 + 0.13208I
b = 3.43060 + 1.66747I
− 4.05977I −6.00000 + 3.46410I
u = 1.271230 − 0.340625I
a = 1.69666 − 0.13208I
b = 3.43060 − 1.66747I
4.05977I −6.00000 − 3.46410I
u = −1.271230 + 0.340625I
a = 1.30334 − 1.59997I
b = 1.56940 − 3.52868I
4.05977I −6.00000 − 3.46410I
u = −1.271230 − 0.340625I
a = 1.30334 + 1.59997I
b = 1.56940 + 3.52868I
− 4.05977I −6.00000 + 3.46410I
10
III. I
u
3
= hu
3
+ u
2
+ b, −u
2
+ a + u + 2, u
4
− u
2
− 1i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
0
a
8
=
u
2
− u − 2
−u
3
− u
2
a
12
=
1
u
2
a
1
=
u
2
− u − 1
−u
3
a
4
=
u
u
a
3
=
−u
3
+ 2u
0
a
7
=
u
2
− u − 1
−u
3
a
2
=
−u
3
+ u
2
+ u − 1
−u
3
a
6
=
u
3
− 2u
0
a
10
=
−u
2
+ 1
−u
2
a
9
=
−1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
c
8
(u − 1)
4
c
2
, c
12
(u + 1)
4
c
3
, c
5
, c
9
u
4
+ u
2
− 1
c
4
, c
10
, c
11
u
4
− u
2
− 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
(y −1)
4
c
3
, c
5
, c
9
(y
2
+ y −1)
2
c
4
, c
10
, c
11
(y
2
− y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.786151I
a = −2.61803 − 0.78615I
b = 0.618034 + 0.485868I
3.94784 −1.52790
u = − 0.786151I
a = −2.61803 + 0.78615I
b = 0.618034 − 0.485868I
3.94784 −1.52790
u = 1.27202
a = −1.65399
b = −3.67621
−3.94784 −10.4720
u = −1.27202
a = 0.890054
b = 0.440137
−3.94784 −10.4720
14
IV. I
u
4
= h−a
2
+ b + a + 2, a
3
− 2a
2
− 3a − 1, u − 1i
(i) Arc colorings
a
5
=
0
1
a
11
=
1
0
a
8
=
a
a
2
− a − 2
a
12
=
1
1
a
1
=
−a
a
2
− 4a − 2
a
4
=
1
1
a
3
=
1
1
a
7
=
−a
2
+ 3a + 2
a
a
2
=
−a
2
+ 2a + 2
−a
a
6
=
0
−1
a
10
=
0
−1
a
9
=
−1
−2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ 2u
2
+ u + 1
c
2
, c
6
, c
7
c
8
, c
12
u
3
− u − 1
c
3
, c
5
u
3
c
4
, c
9
, c
10
c
11
(u + 1)
3
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
3
− 2y
2
− 3y − 1
c
2
, c
6
, c
7
c
8
, c
12
y
3
− 2y
2
+ y − 1
c
3
, c
5
y
3
c
4
, c
9
, c
10
c
11
(y − 1)
3
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.539798 + 0.182582I
b = −1.202160 − 0.379697I
−1.64493 −6.00000
u = 1.00000
a = −0.539798 − 0.182582I
b = −1.202160 + 0.379697I
−1.64493 −6.00000
u = 1.00000
a = 3.07960
b = 4.40431
−1.64493 −6.00000
18
V. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
1
0
a
8
=
0
1
a
12
=
1
0
a
1
=
1
−1
a
4
=
1
0
a
3
=
1
0
a
7
=
−1
1
a
2
=
2
−1
a
6
=
1
0
a
10
=
1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
u − 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
c
6
, c
7
, c
8
u + 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
y − 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
3
+ 2u
2
+ u + 1)(u
19
− 4u
18
+ ··· + 1887u + 49)
c
2
((u − 1)
5
)(u + 1)
4
(u
3
− u − 1)(u
19
− 2u
18
+ ··· − 37u − 7)
c
3
, c
5
u
4
(u
4
+ u
2
− 1)(u
4
+ 3u
2
+ 3)(u
19
− 3u
18
+ ··· − 122u − 46)
c
4
, c
10
, c
11
u(u + 1)
3
(u
4
− 3u
2
+ 3)(u
4
− u
2
− 1)(u
19
− 2u
18
+ ··· − 2u − 2)
c
6
((u − 1)
4
)(u + 1)
5
(u
3
− u − 1)(u
19
− 2u
18
+ ··· − 37u − 7)
c
7
, c
8
((u − 1)
4
)(u + 1)
5
(u
3
− u − 1)(u
19
+ 2u
18
+ ··· + 63u − 7)
c
9
u(u + 1)
3
(u
4
+ u
2
− 1)(u
4
+ 3u
2
+ 3)(u
19
− 4u
18
+ ··· + 13446u − 5482)
c
12
((u − 1)
5
)(u + 1)
4
(u
3
− u − 1)(u
19
+ 2u
18
+ ··· + 63u − 7)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
9
)(y
3
− 2y
2
− 3y − 1)(y
19
+ 64y
18
+ ··· + 3231195y − 2401)
c
2
, c
6
((y − 1)
9
)(y
3
− 2y
2
+ y − 1)(y
19
+ 4y
18
+ ··· + 1887y − 49)
c
3
, c
5
y
4
(y
2
+ y − 1)
2
(y
2
+ 3y + 3)
2
(y
19
+ 35y
18
+ ··· + 9456y − 2116)
c
4
, c
10
, c
11
y(y − 1)
3
(y
2
− 3y + 3)
2
(y
2
− y − 1)
2
(y
19
− 14y
18
+ ··· + 8y − 4)
c
7
, c
8
, c
12
((y − 1)
9
)(y
3
− 2y
2
+ y − 1)(y
19
− 36y
18
+ ··· + 6895y − 49)
c
9
y(y − 1)
3
(y
2
+ y − 1)
2
(y
2
+ 3y + 3)
2
· (y
19
+ 110y
18
+ ··· − 1950902712y − 30052324)
24
