12n
0552
(K12n
0552
)
A knot diagram
1
Linearized knot diagam
3 7 8 11 1 12 2 5 12 5 7 9
Solving Sequence
2,8
7 3
4,11
5 9 12 1 6 10
c
7
c
2
c
3
c
4
c
8
c
11
c
1
c
6
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h5u
20
+ 56u
19
+ ··· + 4b − 116, 9u
20
+ 75u
19
+ ··· + 16a + 72, u
21
+ 11u
20
+ ··· − 80u − 16i
I
u
2
= h−u
16
+ u
15
+ ··· + b + 5, −3u
16
− 9u
15
+ ··· + 2a − 4, u
17
+ u
16
+ ··· + 2u + 2i
* 2 irreducible components of dim
C
= 0, with total 38 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
= h5u
20
+ 56u
19
+ · · · + 4b − 116, 9u
20
+ 75u
19
+ · · · + 16a + 72, u
21
+
11u
20
+ · · · − 80u − 16i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
11
=
−0.562500u
20
− 4.68750u
19
+ ··· − 16.7500u − 4.50000
−
5
4
u
20
− 14u
19
+ ··· +
241
2
u + 29
a
5
=
3
16
u
20
+
31
16
u
19
+ ··· −
85
8
u
2
−
7
2
u
−
5
8
u
20
−
49
8
u
19
+ ··· + 29u + 7
a
9
=
3
8
u
20
+ 3u
19
+ ··· +
159
4
u +
23
2
11
8
u
20
+
121
8
u
19
+ ··· −
275
2
u − 34
a
12
=
−0.562500u
20
− 4.43750u
19
+ ··· − 7.25000u + 0.500000
−
7
4
u
20
−
39
2
u
19
+ ··· +
281
2
u + 33
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
3
16
u
20
+
31
16
u
19
+ ··· −
63
2
u − 8
9
8
u
20
+
85
8
u
19
+ ··· − 42u − 11
a
10
=
−
11
16
u
20
−
85
16
u
19
+ ··· −
213
4
u − 14
−
5
2
u
20
− 27u
19
+ ··· + 225u + 55
(ii) Obstruction class = −1
(iii) Cusp Shapes =
21
2
u
20
+
225
2
u
19
+
1289
2
u
18
+ 2524u
17
+
14905
2
u
16
+
34975
2
u
15
+
67375
2
u
14
+ 54537u
13
+
151085
2
u
12
+
181727
2
u
11
+
191943
2
u
10
+
179037
2
u
9
+ 73543u
8
+
52381u
7
+ 31237u
6
+
28777
2
u
5
+ 3949u
4
− 589u
3
− 1353u
2
− 726u − 166
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
21
+ 9u
20
+ ··· + 896u − 256
c
2
, c
7
u
21
+ 11u
20
+ ··· − 80u − 16
c
3
u
21
− 11u
20
+ ··· − 8656u − 2512
c
4
, c
6
, c
10
c
11
u
21
+ 27u
19
+ ··· + u − 1
c
5
, c
8
u
21
+ 2u
20
+ ··· + 28u
2
− 1
c
9
, c
12
u
21
+ 4u
20
+ ··· − 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
21
+ 5y
20
+ ··· + 2367488y −65536
c
2
, c
7
y
21
+ 9y
20
+ ··· + 896y −256
c
3
y
21
− 107y
20
+ ··· − 395058816y −6310144
c
4
, c
6
, c
10
c
11
y
21
+ 54y
20
+ ··· − 9y −1
c
5
, c
8
y
21
− 46y
20
+ ··· + 56y −1
c
9
, c
12
y
21
+ 2y
20
+ ··· − y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.472469 + 0.828155I
a = −0.488123 − 0.166277I
b = 0.188036 + 0.222130I
0.20225 + 1.98003I 2.28798 − 3.26174I
u = 0.472469 − 0.828155I
a = −0.488123 + 0.166277I
b = 0.188036 − 0.222130I
0.20225 − 1.98003I 2.28798 + 3.26174I
u = −0.771804 + 0.719603I
a = −1.319550 − 0.250548I
b = 0.47482 + 1.81639I
3.69054 + 0.94413I 3.84004 + 4.58061I
u = −0.771804 − 0.719603I
a = −1.319550 + 0.250548I
b = 0.47482 − 1.81639I
3.69054 − 0.94413I 3.84004 − 4.58061I
u = 0.036593 + 0.936717I
a = 0.078038 − 0.908614I
b = −0.472625 + 0.388644I
−1.84690 + 1.45685I −2.25321 − 5.17450I
u = 0.036593 − 0.936717I
a = 0.078038 + 0.908614I
b = −0.472625 − 0.388644I
−1.84690 − 1.45685I −2.25321 + 5.17450I
u = −0.710005 + 0.988627I
a = 1.54173 + 0.94297I
b = −0.24228 − 2.12126I
2.87016 − 6.56510I −1.92301 + 3.76598I
u = −0.710005 − 0.988627I
a = 1.54173 − 0.94297I
b = −0.24228 + 2.12126I
2.87016 + 6.56510I −1.92301 − 3.76598I
u = −0.769484 + 0.051011I
a = −0.141173 + 0.103619I
b = −0.347832 + 0.454102I
−2.61653 + 1.28499I 0.15587 − 3.07961I
u = −0.769484 − 0.051011I
a = −0.141173 − 0.103619I
b = −0.347832 − 0.454102I
−2.61653 − 1.28499I 0.15587 + 3.07961I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.435565 + 1.195780I
a = −0.004901 − 0.689508I
b = −0.468235 + 0.470276I
−6.21856 − 2.94897I −0.504582 + 0.049416I
u = −0.435565 − 1.195780I
a = −0.004901 + 0.689508I
b = −0.468235 − 0.470276I
−6.21856 + 2.94897I −0.504582 − 0.049416I
u = −0.472104 + 1.199130I
a = 0.644752 + 0.145048I
b = −0.345148 − 0.550837I
−5.96069 − 5.81849I −5.01229 + 8.57270I
u = −0.472104 − 1.199130I
a = 0.644752 − 0.145048I
b = −0.345148 + 0.550837I
−5.96069 + 5.81849I −5.01229 − 8.57270I
u = −1.49061 + 0.03728I
a = 0.034735 + 0.714146I
b = 0.00483 − 2.16677I
−15.8145 − 3.8268I 1.96641 + 1.96096I
u = −1.49061 − 0.03728I
a = 0.034735 − 0.714146I
b = 0.00483 + 2.16677I
−15.8145 + 3.8268I 1.96641 − 1.96096I
u = 0.423148
a = −1.20478
b = 0.358221
0.913214 11.0410
u = −0.76209 + 1.50016I
a = 1.21983 + 1.44613I
b = −0.15370 − 2.09019I
19.0348 − 11.6307I 0. + 4.72402I
u = −0.76209 − 1.50016I
a = 1.21983 − 1.44613I
b = −0.15370 + 2.09019I
19.0348 + 11.6307I 0. − 4.72402I
u = −0.80898 + 1.48155I
a = −1.21294 − 1.39486I
b = 0.18302 + 2.08076I
19.3623 − 4.1247I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.80898 − 1.48155I
a = −1.21294 + 1.39486I
b = 0.18302 − 2.08076I
19.3623 + 4.1247I 0
7
II.
I
u
2
= h−u
16
+u
15
+· · ·+b+5, −3u
16
−9u
15
+· · ·+2a−4, u
17
+u
16
+· · ·+2u+2i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
11
=
3
2
u
16
+
9
2
u
15
+ ··· +
19
2
u + 2
u
16
− u
15
+ ··· − 8u − 5
a
5
=
−
1
2
u
16
−
3
2
u
15
+ ··· +
3
2
u − 1
u
15
+ u
14
+ ··· + 2u + 1
a
9
=
5
2
u
16
+
11
2
u
15
+ ··· +
21
2
u + 8
u
16
− u
15
+ ··· − 4u − 5
a
12
=
5
2
u
16
+
13
2
u
15
+ ··· +
21
2
u + 3
−3u
15
− 5u
14
+ ··· − 12u − 7
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
−
3
2
u
16
−
5
2
u
15
+ ··· −
1
2
u − 1
u
16
+ 2u
15
+ ··· + 3u + 1
a
10
=
5u
16
+ 9u
15
+ ··· + 20u + 6
u
16
− 3u
15
+ ··· − 13u − 10
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
16
− 2u
15
− 3u
14
− 13u
13
− 16u
12
− 34u
11
− 48u
10
− 74u
9
−
85u
8
− 100u
7
− 105u
6
− 105u
5
− 96u
4
− 75u
3
− 56u
2
− 30u − 16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
− 9u
16
+ ··· − 32u + 4
c
2
u
17
− u
16
+ ··· + 2u − 2
c
3
u
17
+ u
16
+ ··· + 6u − 2
c
4
, c
11
u
17
+ u
16
+ ··· − u − 1
c
5
, c
8
u
17
− u
16
+ ··· + 2u + 1
c
6
, c
10
u
17
− u
16
+ ··· − u + 1
c
7
u
17
+ u
16
+ ··· + 2u + 2
c
9
u
17
+ 7u
16
+ ··· + 5u + 1
c
12
u
17
− 7u
16
+ ··· + 5u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
+ 5y
16
+ ··· − 8y −16
c
2
, c
7
y
17
+ 9y
16
+ ··· − 32y −4
c
3
y
17
+ y
16
+ ··· − 24y −4
c
4
, c
6
, c
10
c
11
y
17
+ 5y
16
+ ··· + 9y −1
c
5
, c
8
y
17
− 3y
16
+ ··· − 6y −1
c
9
, c
12
y
17
+ y
16
+ ··· + y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.281134 + 0.946456I
a = 1.24113 − 1.45623I
b = 0.36623 + 1.54673I
0.52851 − 1.83462I 0.29796 + 3.58847I
u = 0.281134 − 0.946456I
a = 1.24113 + 1.45623I
b = 0.36623 − 1.54673I
0.52851 + 1.83462I 0.29796 − 3.58847I
u = −0.969290
a = 0.776451
b = −0.382171
−1.53916 3.39910
u = 0.728050 + 0.766886I
a = 1.70484 − 0.42800I
b = −0.50880 + 2.22693I
3.93432 − 1.40561I 11.9334 + 8.1612I
u = 0.728050 − 0.766886I
a = 1.70484 + 0.42800I
b = −0.50880 − 2.22693I
3.93432 + 1.40561I 11.9334 − 8.1612I
u = 0.224277 + 0.858763I
a = −1.56206 + 0.82631I
b = 0.127397 − 1.349340I
0.93314 + 4.06440I 0.37552 − 3.75729I
u = 0.224277 − 0.858763I
a = −1.56206 − 0.82631I
b = 0.127397 + 1.349340I
0.93314 − 4.06440I 0.37552 + 3.75729I
u = −0.737443 + 0.842981I
a = 1.105660 − 0.250731I
b = −0.533874 + 0.383844I
−3.11346 − 2.81675I 2.33737 + 2.85701I
u = −0.737443 − 0.842981I
a = 1.105660 + 0.250731I
b = −0.533874 − 0.383844I
−3.11346 + 2.81675I 2.33737 − 2.85701I
u = −0.382757 + 1.099770I
a = −0.682974 − 0.515143I
b = 0.561927 + 0.046321I
−7.10806 − 3.64002I −7.66149 + 4.40489I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.382757 − 1.099770I
a = −0.682974 + 0.515143I
b = 0.561927 − 0.046321I
−7.10806 + 3.64002I −7.66149 − 4.40489I
u = 0.676951 + 0.964759I
a = −1.78717 + 1.20472I
b = 0.10658 − 2.49443I
3.30875 + 6.79114I 13.3760 − 11.7173I
u = 0.676951 − 0.964759I
a = −1.78717 − 1.20472I
b = 0.10658 + 2.49443I
3.30875 − 6.79114I 13.3760 + 11.7173I
u = −0.306582 + 0.677700I
a = −1.79619 + 0.12408I
b = 0.674089 − 0.611126I
−5.45769 + 0.67304I −4.27931 − 3.12764I
u = −0.306582 − 0.677700I
a = −1.79619 − 0.12408I
b = 0.674089 + 0.611126I
−5.45769 − 0.67304I −4.27931 + 3.12764I
u = −0.498985 + 1.260640I
a = 0.388541 − 0.408565I
b = −0.102456 + 0.376600I
−5.41541 − 5.16567I 1.42099 + 1.44225I
u = −0.498985 − 1.260640I
a = 0.388541 + 0.408565I
b = −0.102456 − 0.376600I
−5.41541 + 5.16567I 1.42099 − 1.44225I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
17
− 9u
16
+ ··· − 32u + 4)(u
21
+ 9u
20
+ ··· + 896u − 256)
c
2
(u
17
− u
16
+ ··· + 2u − 2)(u
21
+ 11u
20
+ ··· − 80u − 16)
c
3
(u
17
+ u
16
+ ··· + 6u − 2)(u
21
− 11u
20
+ ··· − 8656u − 2512)
c
4
, c
11
(u
17
+ u
16
+ ··· − u − 1)(u
21
+ 27u
19
+ ··· + u − 1)
c
5
, c
8
(u
17
− u
16
+ ··· + 2u + 1)(u
21
+ 2u
20
+ ··· + 28u
2
− 1)
c
6
, c
10
(u
17
− u
16
+ ··· − u + 1)(u
21
+ 27u
19
+ ··· + u − 1)
c
7
(u
17
+ u
16
+ ··· + 2u + 2)(u
21
+ 11u
20
+ ··· − 80u − 16)
c
9
(u
17
+ 7u
16
+ ··· + 5u + 1)(u
21
+ 4u
20
+ ··· − 5u − 1)
c
12
(u
17
− 7u
16
+ ··· + 5u − 1)(u
21
+ 4u
20
+ ··· − 5u − 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
17
+ 5y
16
+ ··· − 8y −16)(y
21
+ 5y
20
+ ··· + 2367488y −65536)
c
2
, c
7
(y
17
+ 9y
16
+ ··· − 32y −4)(y
21
+ 9y
20
+ ··· + 896y −256)
c
3
(y
17
+ y
16
+ ··· − 24y −4)
· (y
21
− 107y
20
+ ··· − 395058816y −6310144)
c
4
, c
6
, c
10
c
11
(y
17
+ 5y
16
+ ··· + 9y −1)(y
21
+ 54y
20
+ ··· − 9y −1)
c
5
, c
8
(y
17
− 3y
16
+ ··· − 6y −1)(y
21
− 46y
20
+ ··· + 56y −1)
c
9
, c
12
(y
17
+ y
16
+ ··· + y −1)(y
21
+ 2y
20
+ ··· − y −1)
14
