12n
0467
(K12n
0467
)
A knot diagram
1
Linearized knot diagam
3 6 9 8 2 11 12 3 4 1 7 8
Solving Sequence
3,8
9 4 5
10,12
1 2 7 11 6
c
8
c
3
c
4
c
9
c
12
c
1
c
7
c
11
c
6
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h140613342u
26
+ 233504827u
25
+ ··· + 1306898116b + 2324027520,
315767111u
26
+ 587740472u
25
+ ··· + 1306898116a − 1648431188, u
27
+ u
26
+ ··· − 4u − 4i
I
u
2
= h2b − 2a + u, 2a
2
− 2au − 2a + u − 1, u
2
− 2i
I
v
1
= ha, b + v + 1, v
2
+ v −1i
* 3 irreducible components of dim
C
= 0, with total 33 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
= h1.41 × 10
8
u
26
+ 2 .34 × 10
8
u
25
+ · · · + 1.31 × 10
9
b + 2.32 × 10
9
, 3.16 ×
10
8
u
26
+ 5.88 × 10
8
u
25
+ · · · + 1.31 × 10
9
a − 1.65 × 10
9
, u
27
+ u
26
+ · · · − 4u − 4i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
4
=
u
−u
3
+ u
a
5
=
−u
3
+ 2u
−u
3
+ u
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
12
=
−0.241616u
26
− 0.449722u
25
+ ··· − 13.1892u + 1.26133
−0.107593u
26
− 0.178671u
25
+ ··· − 2.19364u − 1.77828
a
1
=
−0.134023u
26
− 0.271051u
25
+ ··· − 10.9955u + 3.03961
−0.107593u
26
− 0.178671u
25
+ ··· − 2.19364u − 1.77828
a
2
=
−0.134023u
26
− 0.271051u
25
+ ··· − 10.9955u + 3.03961
0.0945547u
26
− 0.183568u
25
+ ··· − 3.27784u − 2.32639
a
7
=
−0.673320u
26
− 0.574468u
25
+ ··· − 15.4301u + 0.529920
−0.102694u
26
+ 0.0537208u
25
+ ··· − 3.65606u − 0.825481
a
11
=
−0.00809889u
26
+ 0.000418223u
25
+ ··· + 2.05151u + 3.50212
0.355054u
26
+ 0.372385u
25
+ ··· + 7.65668u + 1.09099
a
6
=
−0.409090u
26
+ 0.0395121u
25
+ ··· − 4.08989u + 7.12801
−0.180513u
26
+ 0.126995u
25
+ ··· + 3.62778u + 1.76201
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
9879130
326724529
u
26
+
101640265
326724529
u
25
+ ··· −
4373584718
326724529
u +
3242559308
326724529
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
27
+ 7u
26
+ ··· + 73u + 1
c
2
, c
5
u
27
+ 3u
26
+ ··· + 7u + 1
c
3
, c
8
, c
9
u
27
+ u
26
+ ··· − 4u − 4
c
4
u
27
− 3u
26
+ ··· + 612u + 220
c
6
, c
7
, c
11
c
12
u
27
− 2u
26
+ ··· − 6u + 1
c
10
u
27
− 2u
26
+ ··· − 20u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
27
+ 33y
26
+ ··· + 4113y −1
c
2
, c
5
y
27
− 7y
26
+ ··· + 73y −1
c
3
, c
8
, c
9
y
27
− 37y
26
+ ··· + 336y −16
c
4
y
27
− 97y
26
+ ··· + 1370704y −48400
c
6
, c
7
, c
11
c
12
y
27
− 30y
26
+ ··· + 32y −1
c
10
y
27
+ 42y
26
+ ··· + 160y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.910258 + 0.590456I
a = 2.00698 + 1.21820I
b = 1.50366 − 0.20859I
−3.25530 + 7.25549I −5.95539 − 6.08777I
u = 0.910258 − 0.590456I
a = 2.00698 − 1.21820I
b = 1.50366 + 0.20859I
−3.25530 − 7.25549I −5.95539 + 6.08777I
u = −1.017290 + 0.392893I
a = 1.59907 − 0.86702I
b = 1.384080 + 0.097780I
−2.42951 − 1.33319I −4.55270 + 1.10070I
u = −1.017290 − 0.392893I
a = 1.59907 + 0.86702I
b = 1.384080 − 0.097780I
−2.42951 + 1.33319I −4.55270 − 1.10070I
u = −1.028180 + 0.401751I
a = −0.842937 + 0.402129I
b = −0.466581 − 0.606665I
3.20449 − 4.27323I −1.36721 + 6.77417I
u = −1.028180 − 0.401751I
a = −0.842937 − 0.402129I
b = −0.466581 + 0.606665I
3.20449 + 4.27323I −1.36721 − 6.77417I
u = 1.132760 + 0.119465I
a = −0.399911 + 0.146187I
b = −0.444802 − 0.471118I
3.11947 + 0.51181I −0.345493 + 0.472617I
u = 1.132760 − 0.119465I
a = −0.399911 − 0.146187I
b = −0.444802 + 0.471118I
3.11947 − 0.51181I −0.345493 − 0.472617I
u = 0.068539 + 0.776104I
a = −2.94508 − 0.05064I
b = −1.42888 − 0.10362I
−5.79751 − 2.66305I −8.58849 + 2.68063I
u = 0.068539 − 0.776104I
a = −2.94508 + 0.05064I
b = −1.42888 + 0.10362I
−5.79751 + 2.66305I −8.58849 − 2.68063I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.701128
a = −0.723697
b = 1.54174
−9.07206 −6.92100
u = 1.38081
a = −0.782809
b = −0.202634
3.22160 2.65910
u = −1.43515
a = 1.63913
b = 1.62828
−3.96458 −1.96340
u = 0.093206 + 0.505933I
a = 0.909549 − 0.188312I
b = 0.279075 − 0.365424I
−0.272367 + 1.017390I −4.41879 − 6.56996I
u = 0.093206 − 0.505933I
a = 0.909549 + 0.188312I
b = 0.279075 + 0.365424I
−0.272367 − 1.017390I −4.41879 + 6.56996I
u = −0.462517
a = 1.41263
b = 0.790251
−1.60060 −3.53340
u = −1.60836
a = −0.0902962
b = −1.36994
−1.00091 −6.04390
u = 0.389158
a = −3.32064
b = −1.64070
−10.0949 1.45710
u = −0.327289
a = 3.18758
b = −0.436653
−2.27386 5.71450
u = −1.70507 + 0.18041I
a = −1.29573 + 0.99047I
b = −1.58969 − 0.26870I
5.77830 − 10.34990I −4.51654 + 4.83744I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.70507 − 0.18041I
a = −1.29573 − 0.99047I
b = −1.58969 + 0.26870I
5.77830 + 10.34990I −4.51654 − 4.83744I
u = 1.74296 + 0.07415I
a = −1.029860 − 0.595198I
b = −1.47829 + 0.32375I
7.48038 + 3.07161I −3.06454 − 1.05246I
u = 1.74296 − 0.07415I
a = −1.029860 + 0.595198I
b = −1.47829 − 0.32375I
7.48038 − 3.07161I −3.06454 + 1.05246I
u = 1.74801 + 0.11240I
a = 0.591464 + 0.471611I
b = 0.634607 − 0.775759I
13.1108 + 6.4432I −1.45116 − 4.64591I
u = 1.74801 − 0.11240I
a = 0.591464 − 0.471611I
b = 0.634607 + 0.775759I
13.1108 − 6.4432I −1.45116 + 4.64591I
u = −1.76409 + 0.02868I
a = 0.245519 + 0.313706I
b = 0.451661 − 0.831495I
13.66050 − 1.13182I −0.424228 − 0.165787I
u = −1.76409 − 0.02868I
a = 0.245519 − 0.313706I
b = 0.451661 + 0.831495I
13.66050 + 1.13182I −0.424228 + 0.165787I
7
II. I
u
2
= h2b − 2a + u, 2a
2
− 2au − 2a + u − 1, u
2
− 2i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
−2
a
4
=
u
−u
a
5
=
0
−u
a
10
=
−1
0
a
12
=
a
a −
1
2
u
a
1
=
1
2
u
a −
1
2
u
a
2
=
1
2
u
a +
1
2
u
a
7
=
−
1
2
au − a +
1
2
u +
1
2
−a +
1
2
u − 1
a
11
=
−
1
2
au −
1
2
−a +
1
2
u − 1
a
6
=
1
2
u
a −
1
2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8
8
(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
8
c
9
(u
2
− 2)
2
c
6
, c
7
(u
2
− u − 1)
2
c
10
, c
11
, c
12
(u
2
+ u − 1)
2
9
(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
8
c
9
(y −2)
4
c
6
, c
7
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = 0.0890728
b = −0.618034
2.30291 −8.00000
u = 1.41421
a = 2.32514
b = 1.61803
−5.59278 −8.00000
u = −1.41421
a = 0.910927
b = 1.61803
−5.59278 −8.00000
u = −1.41421
a = −1.32514
b = −0.618034
2.30291 −8.00000
11
III. I
v
1
= ha, b + v + 1, v
2
+ v − 1i
(i) Arc colorings
a
3
=
v
0
a
8
=
1
0
a
9
=
1
0
a
4
=
v
0
a
5
=
v
0
a
10
=
1
0
a
12
=
0
−v − 1
a
1
=
v + 1
−v − 1
a
2
=
2v + 1
−v − 1
a
7
=
1
−v − 2
a
11
=
−v − 1
v + 2
a
6
=
−v − 1
v + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −18
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
8
c
9
u
2
c
5
(u + 1)
2
c
6
, c
7
, c
10
u
2
+ u − 1
c
11
, c
12
u
2
− u − 1
13
(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
8
c
9
y
2
c
6
, c
7
, c
10
c
11
, c
12
y
2
− 3y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.618034
a = 0
b = −1.61803
−10.5276 −18.0000
v = −1.61803
a = 0
b = 0.618034
−2.63189 −18.0000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
27
+ 7u
26
+ ··· + 73u + 1)
c
2
((u − 1)
2
)(u + 1)
4
(u
27
+ 3u
26
+ ··· + 7u + 1)
c
3
, c
8
, c
9
u
2
(u
2
− 2)
2
(u
27
+ u
26
+ ··· − 4u − 4)
c
4
u
2
(u
2
− 2)
2
(u
27
− 3u
26
+ ··· + 612u + 220)
c
5
((u − 1)
4
)(u + 1)
2
(u
27
+ 3u
26
+ ··· + 7u + 1)
c
6
, c
7
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
27
− 2u
26
+ ··· − 6u + 1)
c
10
((u
2
+ u − 1)
3
)(u
27
− 2u
26
+ ··· − 20u − 1)
c
11
, c
12
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
27
− 2u
26
+ ··· − 6u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
6
)(y
27
+ 33y
26
+ ··· + 4113y − 1)
c
2
, c
5
((y − 1)
6
)(y
27
− 7y
26
+ ··· + 73y − 1)
c
3
, c
8
, c
9
y
2
(y − 2)
4
(y
27
− 37y
26
+ ··· + 336y − 16)
c
4
y
2
(y − 2)
4
(y
27
− 97y
26
+ ··· + 1370704y − 48400)
c
6
, c
7
, c
11
c
12
((y
2
− 3y + 1)
3
)(y
27
− 30y
26
+ ··· + 32y − 1)
c
10
((y
2
− 3y + 1)
3
)(y
27
+ 42y
26
+ ··· + 160y − 1)
17
