12n
0457
(K12n
0457
)
A knot diagram
1
Linearized knot diagam
3 6 8 7 2 11 4 3 12 6 10 9
Solving Sequence
6,10
11 7
3,12
2 1 5 4 9 8
c
10
c
6
c
11
c
2
c
1
c
5
c
4
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−35u
13
+ 632u
12
+ ··· + 1322b − 1940, −98u
13
+ 712u
12
+ ··· + 1983a − 3449,
u
14
− 2u
13
− 2u
12
+ 8u
11
− 2u
10
− 12u
9
+ 12u
8
+ 5u
7
− 16u
6
+ u
5
+ 22u
4
− 20u
3
+ 7u − 3i
I
u
2
= h−2u
2
b + b
2
+ u
2
− 3u + 1, −u
2
+ a + u, u
3
− u
2
+ 1i
I
u
3
= h−u
2
+ b, −u
2
+ a − u, u
3
+ u
2
− 1i
* 3 irreducible components of dim
C
= 0, with total 23 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−35u
13
+ 632u
12
+ · · · + 1322b − 1940, −98u
13
+ 712u
12
+ · · · +
1983a − 3449, u
14
− 2u
13
+ · · · + 7u − 3i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
3
=
0.0494201u
13
− 0.359052u
12
+ ··· − 1.11346u + 1.73928
0.0264750u
13
− 0.478064u
12
+ ··· − 0.667927u + 1.46747
a
12
=
−u
2
+ 1
−u
2
a
2
=
0.0494201u
13
− 0.359052u
12
+ ··· − 1.11346u + 1.73928
0.391074u
13
− 1.00454u
12
+ ··· − 2.63767u + 2.24811
a
1
=
−u
6
+ u
4
− 2u
2
+ 1
−u
6
− u
2
a
5
=
0.893091u
13
− 1.06001u
12
+ ··· − 4.26475u + 2.50277
−0.0264750u
13
+ 0.478064u
12
+ ··· + 1.66793u − 1.46747
a
4
=
0.654060u
13
− 0.986636u
12
+ ··· − 3.70575u + 2.82501
0.164902u
13
− 0.0347958u
12
+ ··· + 0.111195u + 0.0688351
a
9
=
u
4
− u
2
+ 1
u
4
a
8
=
0.0229450u
13
+ 0.119012u
12
+ ··· − 0.445537u + 0.271810
−0.581694u
13
+ 0.746596u
12
+ ··· + 2.14675u − 1.81392
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1575
661
u
13
−
2000
661
u
12
+ ··· −
16600
661
u +
9963
661
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
− 6u
13
+ ··· − 4u + 1
c
2
, c
5
u
14
+ 4u
13
+ ··· + 4u − 1
c
3
, c
4
, c
7
c
8
u
14
+ u
13
+ ··· − 32u + 8
c
6
, c
10
u
14
+ 2u
13
+ ··· − 7u − 3
c
9
, c
11
, c
12
u
14
− 8u
13
+ ··· − 49u + 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 34y
13
+ ··· − 908y + 1
c
2
, c
5
y
14
+ 6y
13
+ ··· + 4y + 1
c
3
, c
4
, c
7
c
8
y
14
+ 7y
13
+ ··· − 256y + 64
c
6
, c
10
y
14
− 8y
13
+ ··· − 49y + 9
c
9
, c
11
, c
12
y
14
+ 16y
12
+ ··· + 263y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.331897 + 1.038650I
a = 1.49914 − 0.30941I
b = 0.042314 − 0.355178I
1.13414 − 3.62470I 1.53739 + 1.98303I
u = 0.331897 − 1.038650I
a = 1.49914 + 0.30941I
b = 0.042314 + 0.355178I
1.13414 + 3.62470I 1.53739 − 1.98303I
u = 0.948812 + 0.550000I
a = −1.055470 + 0.686434I
b = −0.17948 + 1.87576I
−6.22033 + 2.16614I 0.60547 − 2.67775I
u = 0.948812 − 0.550000I
a = −1.055470 − 0.686434I
b = −0.17948 − 1.87576I
−6.22033 − 2.16614I 0.60547 + 2.67775I
u = −0.902807 + 0.737867I
a = 0.166043 − 0.126427I
b = 0.950914 − 0.969597I
−7.92408 − 2.80343I 2.49909 + 2.82255I
u = −0.902807 − 0.737867I
a = 0.166043 + 0.126427I
b = 0.950914 + 0.969597I
−7.92408 + 2.80343I 2.49909 − 2.82255I
u = 1.24269
a = 1.46455
b = 1.28241
0.813631 6.43730
u = 0.525421 + 0.402657I
a = 0.303592 − 0.966121I
b = 0.204868 − 0.509751I
−1.00992 + 1.33356I −0.10273 − 5.72522I
u = 0.525421 − 0.402657I
a = 0.303592 + 0.966121I
b = 0.204868 + 0.509751I
−1.00992 − 1.33356I −0.10273 + 5.72522I
u = −0.624400
a = 0.326568
b = −0.335778
0.793364 13.8290
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.23913 + 0.70216I
a = 0.336015 − 1.316350I
b = 0.02959 − 2.64674I
3.85381 + 9.90530I 2.86734 − 5.00880I
u = 1.23913 − 0.70216I
a = 0.336015 + 1.316350I
b = 0.02959 + 2.64674I
3.85381 − 9.90530I 2.86734 + 5.00880I
u = −1.45159 + 0.36092I
a = −0.31154 + 1.47612I
b = −0.52152 + 2.50638I
6.89548 − 1.15921I 5.46011 + 0.65565I
u = −1.45159 − 0.36092I
a = −0.31154 − 1.47612I
b = −0.52152 − 2.50638I
6.89548 + 1.15921I 5.46011 − 0.65565I
6
II. I
u
2
= h−2u
2
b + b
2
+ u
2
− 3u + 1, −u
2
+ a + u, u
3
− u
2
+ 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
2
+ u + 1
a
3
=
u
2
− u
b
a
12
=
−u
2
+ 1
−u
2
a
2
=
u
2
− u
b + u
a
1
=
0
u
a
5
=
−u
2
+ u
−b
a
4
=
u
2
b − 2u
2
+ 2u + 1
bu − 2u
2
+ 1
a
9
=
−u
u
2
− u − 1
a
8
=
−u
2
+ b − u
−u
2
b − u
2
+ b − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
7
c
8
(u
2
+ 2)
3
c
6
(u
3
+ u
2
− 1)
2
c
9
(u
3
+ u
2
+ 2u + 1)
2
c
10
(u
3
− u
2
+ 1)
2
c
11
, c
12
(u
3
− u
2
+ 2u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
7
c
8
(y + 2)
6
c
6
, c
10
(y
3
− y
2
+ 2y −1)
2
c
9
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.662359 + 0.562280I
b = −0.580103 + 0.370424I
−9.60386 + 2.82812I −3.50976 − 2.97945I
u = 0.877439 + 0.744862I
a = −0.662359 + 0.562280I
b = 1.01026 + 2.24386I
−9.60386 + 2.82812I −3.50976 − 2.97945I
u = 0.877439 − 0.744862I
a = −0.662359 − 0.562280I
b = −0.580103 − 0.370424I
−9.60386 − 2.82812I −3.50976 + 2.97945I
u = 0.877439 − 0.744862I
a = −0.662359 − 0.562280I
b = 1.01026 − 2.24386I
−9.60386 − 2.82812I −3.50976 + 2.97945I
u = −0.754878
a = 1.32472
b = 0.56984 + 1.87343I
−5.46628 3.01950
u = −0.754878
a = 1.32472
b = 0.56984 − 1.87343I
−5.46628 3.01950
10
III. I
u
3
= h−u
2
+ b, −u
2
+ a − u, u
3
+ u
2
− 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
u
2
+ u − 1
a
3
=
u
2
+ u
u
2
a
12
=
−u
2
+ 1
−u
2
a
2
=
u
2
+ u
u
2
− u
a
1
=
0
−u
a
5
=
u
2
+ u
u
2
a
4
=
u
2
+ u
u
2
a
9
=
u
u
2
+ u − 1
a
8
=
u
u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 2u + 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
7
c
8
u
3
c
5
(u + 1)
3
c
6
u
3
− u
2
+ 1
c
9
u
3
+ u
2
+ 2u + 1
c
10
u
3
+ u
2
− 1
c
11
, c
12
u
3
− u
2
+ 2u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
7
c
8
y
3
c
6
, c
10
y
3
− y
2
+ 2y −1
c
9
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.662359 − 0.562280I
b = 0.215080 − 1.307140I
−4.66906 − 2.82812I 4.89456 + 3.73884I
u = −0.877439 − 0.744862I
a = −0.662359 + 0.562280I
b = 0.215080 + 1.307140I
−4.66906 + 2.82812I 4.89456 − 3.73884I
u = 0.754878
a = 1.32472
b = 0.569840
−0.531480 0.210880
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
14
− 6u
13
+ ··· − 4u + 1)
c
2
((u − 1)
3
)(u + 1)
6
(u
14
+ 4u
13
+ ··· + 4u − 1)
c
3
, c
4
, c
7
c
8
u
3
(u
2
+ 2)
3
(u
14
+ u
13
+ ··· − 32u + 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
14
+ 4u
13
+ ··· + 4u − 1)
c
6
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
(u
14
+ 2u
13
+ ··· − 7u − 3)
c
9
((u
3
+ u
2
+ 2u + 1)
3
)(u
14
− 8u
13
+ ··· − 49u + 9)
c
10
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
14
+ 2u
13
+ ··· − 7u − 3)
c
11
, c
12
((u
3
− u
2
+ 2u − 1)
3
)(u
14
− 8u
13
+ ··· − 49u + 9)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
14
− 34y
13
+ ··· − 908y + 1)
c
2
, c
5
((y −1)
9
)(y
14
+ 6y
13
+ ··· + 4y + 1)
c
3
, c
4
, c
7
c
8
y
3
(y + 2)
6
(y
14
+ 7y
13
+ ··· − 256y + 64)
c
6
, c
10
((y
3
− y
2
+ 2y −1)
3
)(y
14
− 8y
13
+ ··· − 49y + 9)
c
9
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
14
+ 16y
12
+ ··· + 263y + 81)
16
