12n
0503
(K12n
0503
)
A knot diagram
1
Linearized knot diagam
3 6 12 9 2 11 10 3 6 7 4 8
Solving Sequence
6,11 4,7
12 3 2 1 5 10 9 8
c
6
c
11
c
3
c
2
c
1
c
5
c
10
c
9
c
8
c
4
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
7
+ u
6
+ 3u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ b, a − 1, u
8
+ u
7
+ 4u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ u
2
+ 3u − 1i
I
u
2
= h−u
2
+ b + 2u, a + 1, u
3
− u
2
+ 2u − 1i
I
u
3
= hu
15
+ 2u
14
+ 7u
13
+ 10u
12
+ 18u
11
+ 19u
10
+ 17u
9
+ 12u
8
− 2u
7
− 3u
6
− 9u
5
− 4u
4
+ 5u
3
+ u
2
+ 2b + 4u + 1,
2u
17
+ 5u
16
+ ··· + 2a + 19u, u
18
+ 3u
17
+ ··· + 4u + 1i
I
u
4
= hu
2
a − au + b + u, u
2
a + a
2
− au + 3u
2
+ a − u + 5, u
3
− u
2
+ 2u − 1i
* 4 irreducible components of dim
C
= 0, with total 35 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
7
+ u
6
+ 3u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ b, a − 1, u
8
+ u
7
+ 4u
6
+ 3u
5
+
5u
4
+ 4u
3
+ u
2
+ 3u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
4
=
1
−u
7
− u
6
− 3u
5
− 2u
4
− 2u
3
− 2u
2
a
7
=
1
u
2
a
12
=
−u
−u
6
− u
5
− 3u
4
− 2u
3
− u
2
− 2u + 1
a
3
=
u
2
+ 1
−u
3
− u
a
2
=
−u
3
+ u
2
− u + 1
−u
3
− u
a
1
=
u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u
u
6
+ u
5
+ u
4
+ 3u
3
− 2u
2
+ 3u − 1
a
5
=
−u
6
+ u
5
− 2u
4
+ 2u
3
− u
2
+ u + 1
−u
6
− 2u
4
− u
2
a
10
=
u
u
3
+ u
a
9
=
u
3
+ 2u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
7
− 4u
6
− 12u
5
− 10u
4
− 10u
3
− 12u
2
+ 2u − 16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 13u
7
+ 52u
6
+ 39u
5
− 95u
4
+ 44u
3
− 5u
2
+ 11u + 1
c
2
, c
5
, c
9
u
8
+ u
7
− 6u
6
− 3u
5
+ 5u
4
− 10u
3
+ 5u
2
− u − 1
c
3
, c
6
, c
7
c
10
, c
11
u
8
− u
7
+ 4u
6
− 3u
5
+ 5u
4
− 4u
3
+ u
2
− 3u − 1
c
4
u
8
− 3u
7
− 4u
6
+ 19u
5
− 33u
4
+ 100u
3
+ 91u
2
+ 35u + 7
c
8
u
8
+ 7u
7
+ 17u
6
+ 16u
5
+ 10u
4
+ 28u
3
+ 44u
2
+ 32u + 8
c
12
u
8
− 6u
7
+ 7u
6
+ 18u
5
− 36u
4
− 10u
3
+ 45u
2
− 14u − 12
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
− 65y
7
+ ··· − 131y + 1
c
2
, c
5
, c
9
y
8
− 13y
7
+ 52y
6
− 39y
5
− 95y
4
− 44y
3
− 5y
2
− 11y + 1
c
3
, c
6
, c
7
c
10
, c
11
y
8
+ 7y
7
+ 20y
6
+ 25y
5
+ y
4
− 32y
3
− 33y
2
− 11y + 1
c
4
y
8
− 17y
7
+ 64y
6
+ 685y
5
− 3215y
4
− 17392y
3
+ 819y
2
+ 49y + 49
c
8
y
8
− 15y
7
+ 85y
6
− 220y
5
+ 268y
4
− 656y
3
+ 304y
2
− 320y + 64
c
12
y
8
− 22y
7
+ ··· − 1276y + 144
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.327709 + 0.937994I
a = 1.00000
b = 0.142755 + 0.492603I
1.82186 − 2.79026I −10.46165 + 4.33295I
u = 0.327709 − 0.937994I
a = 1.00000
b = 0.142755 − 0.492603I
1.82186 + 2.79026I −10.46165 − 4.33295I
u = −1.04994
a = 1.00000
b = 1.57429
−16.6633 −16.3280
u = 0.051026 + 1.292250I
a = 1.00000
b = 2.39119 − 1.00724I
7.37589 − 2.10973I −4.67802 + 3.20330I
u = 0.051026 − 1.292250I
a = 1.00000
b = 2.39119 + 1.00724I
7.37589 + 2.10973I −4.67802 − 3.20330I
u = −0.48935 + 1.37392I
a = 1.00000
b = 1.78025 − 1.97737I
−7.96719 + 11.01890I −10.38982 − 5.43515I
u = −0.48935 − 1.37392I
a = 1.00000
b = 1.78025 + 1.97737I
−7.96719 − 11.01890I −10.38982 + 5.43515I
u = 0.271177
a = 1.00000
b = −0.202677
−0.602204 −16.6130
5
II. I
u
2
= h−u
2
+ b + 2u, a + 1, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
4
=
−1
u
2
− 2u
a
7
=
1
u
2
a
12
=
−u
−u
2
− u + 1
a
3
=
−u
2
− 1
−u
2
+ u − 1
a
2
=
−2u
2
+ u − 2
−u
2
+ u − 1
a
1
=
−u
2
− 1
−u
2
a
5
=
−u
−u
a
10
=
u
u
2
− u + 1
a
9
=
u
2
+ 1
u
2
− u + 1
a
8
=
u
2
+ 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
2
+ 8u − 20
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
c
11
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
3
, c
10
, c
12
u
3
+ u
2
+ 2u + 1
c
4
u
3
+ 3u
2
+ 2u − 1
c
5
, c
9
u
3
− u
2
+ 1
c
8
u
3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
10
, c
11
c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
, c
9
y
3
− y
2
+ 2y −1
c
4
y
3
− 5y
2
+ 10y −1
c
8
y
3
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.00000
b = −2.09252 − 2.05200I
6.04826 − 5.65624I −4.98049 + 5.95889I
u = 0.215080 − 1.307140I
a = −1.00000
b = −2.09252 + 2.05200I
6.04826 + 5.65624I −4.98049 − 5.95889I
u = 0.569840
a = −1.00000
b = −0.814963
−2.22691 −18.0390
9
III. I
u
3
=
hu
15
+2u
14
+· · ·+2b +1, 2u
17
+5u
16
+· · ·+2a +19u, u
18
+3u
17
+· · ·+4u +1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
4
=
−u
17
−
5
2
u
16
+ ··· − 4u
2
−
19
2
u
−
1
2
u
15
− u
14
+ ··· − 2u −
1
2
a
7
=
1
u
2
a
12
=
−
5
2
u
17
−
15
2
u
16
+ ··· −
31
2
u − 7
−
1
2
u
17
−
5
2
u
16
+ ··· −
7
2
u − 2
a
3
=
5
2
u
17
+ 8u
16
+ ··· + 17u + 4
−
1
2
u
17
− u
15
+ ··· + 6u +
5
2
a
2
=
2u
17
+ 8u
16
+ ··· + 23u +
13
2
−
1
2
u
17
− u
15
+ ··· + 6u +
5
2
a
1
=
2u
17
+
11
2
u
16
+ ··· +
23
2
u +
11
2
2u
17
+ 6u
16
+ ··· + 7u +
5
2
a
5
=
−
3
2
u
17
−
7
2
u
16
+ ··· −
23
2
u − 1
−
1
2
u
17
− u
16
+ ··· − 3u − 1
a
10
=
u
u
3
+ u
a
9
=
u
3
+ 2u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1
2
u
17
+ 4u
16
+ 13u
15
+ 37u
14
+
143
2
u
13
+
251
2
u
12
+
331
2
u
11
+
375
2
u
10
+
315
2
u
9
+
169
2
u
8
+
15
2
u
7
−
125
2
u
6
− 62u
5
−
73
2
u
4
+
13
2
u
3
+ 28u
2
+ 18u −
3
2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
18
+ 31u
17
+ ··· − 4u + 1
c
2
, c
5
, c
9
u
18
+ 3u
17
+ ··· + 4u + 1
c
3
, c
6
, c
7
c
10
, c
11
u
18
− 3u
17
+ ··· − 4u + 1
c
4
u
18
− 21u
16
+ ··· + 640u + 1709
c
8
(u
9
− 3u
8
− 4u
7
+ 17u
6
− 8u
5
+ u
4
− 9u
3
+ 20u
2
− 12u + 8)
2
c
12
(u
9
+ 2u
8
− 4u
7
− 5u
6
+ u
5
− 11u
4
+ u
3
− 2u
2
+ u − 3)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
18
− 103y
17
+ ··· + 100y + 1
c
2
, c
5
, c
9
y
18
− 31y
17
+ ··· + 4y + 1
c
3
, c
6
, c
7
c
10
, c
11
y
18
+ 13y
17
+ ··· + 4y + 1
c
4
y
18
− 42y
17
+ ··· − 5810040y + 2920681
c
8
(y
9
− 17y
8
+ ··· − 176y −64)
2
c
12
(y
9
− 12y
8
+ 38y
7
+ 13y
6
− 107y
5
− 135y
4
− 71y
3
− 68y
2
− 11y −9)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.030260 + 0.064097I
a = 0.593567 − 1.271510I
b = 1.29698 − 0.73855I
−12.47200 + 5.60959I −13.58318 − 2.91483I
u = −1.030260 − 0.064097I
a = 0.593567 + 1.271510I
b = 1.29698 + 0.73855I
−12.47200 − 5.60959I −13.58318 + 2.91483I
u = −0.210201 + 1.054780I
a = −1.310130 + 0.002220I
b = −1.54000 + 1.83413I
4.64765 + 4.49302I −9.27331 − 2.63055I
u = −0.210201 − 1.054780I
a = −1.310130 − 0.002220I
b = −1.54000 − 1.83413I
4.64765 − 4.49302I −9.27331 + 2.63055I
u = −0.132410 + 0.848357I
a = −0.345719 − 0.790410I
b = −0.749423 + 0.610778I
−0.376754 + 0.892025I −13.26164 − 1.57550I
u = −0.132410 − 0.848357I
a = −0.345719 + 0.790410I
b = −0.749423 − 0.610778I
−0.376754 − 0.892025I −13.26164 + 1.57550I
u = 0.716326 + 0.188635I
a = −0.464508 − 1.061990I
b = −0.749423 − 0.610778I
−0.376754 − 0.892025I −13.26164 + 1.57550I
u = 0.716326 − 0.188635I
a = −0.464508 + 1.061990I
b = −0.749423 + 0.610778I
−0.376754 + 0.892025I −13.26164 − 1.57550I
u = 0.227734 + 1.247250I
a = 0.186877 + 0.242619I
b = −0.116635 − 0.608467I
2.67018 − 2.29545I −5.81910 + 1.31175I
u = 0.227734 − 1.247250I
a = 0.186877 − 0.242619I
b = −0.116635 + 0.608467I
2.67018 + 2.29545I −5.81910 − 1.31175I
13
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.273050 + 1.382370I
a = −0.763279 + 0.001294I
b = −1.54000 − 1.83413I
4.64765 − 4.49302I −9.27331 + 2.63055I
u = 0.273050 − 1.382370I
a = −0.763279 − 0.001294I
b = −1.54000 + 1.83413I
4.64765 + 4.49302I −9.27331 − 2.63055I
u = −0.554172 + 1.295970I
a = −0.690827 + 0.723020I
b = 0.218157
−8.67732 −11.12553 + 0.I
u = −0.554172 − 1.295970I
a = −0.690827 − 0.723020I
b = 0.218157
−8.67732 −11.12553 + 0.I
u = −0.53003 + 1.34802I
a = 0.301448 + 0.645745I
b = 1.29698 − 0.73855I
−12.47200 + 5.60959I −13.58318 − 2.91483I
u = −0.53003 − 1.34802I
a = 0.301448 − 0.645745I
b = 1.29698 + 0.73855I
−12.47200 − 5.60959I −13.58318 + 2.91483I
u = −0.260047 + 0.288335I
a = 1.99257 − 2.58691I
b = −0.116635 − 0.608467I
2.67018 − 2.29545I −5.81910 + 1.31175I
u = −0.260047 − 0.288335I
a = 1.99257 + 2.58691I
b = −0.116635 + 0.608467I
2.67018 + 2.29545I −5.81910 − 1.31175I
14
IV. I
u
4
= hu
2
a − au + b + u, u
2
a + a
2
− au + 3u
2
+ a − u + 5, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
4
=
a
−u
2
a + au − u
a
7
=
1
u
2
a
12
=
−au + 2u
2
+ a − u + 3
−au + u
2
+ a + 1
a
3
=
−au + u
2
+ 1
−u
2
a + u
2
− u + 1
a
2
=
−u
2
a − au + 2u
2
− u + 2
−u
2
a + u
2
− u + 1
a
1
=
−u
2
a − au + 3u
2
+ a − 2u + 4
−u
2
a + au + u
2
− u + 2
a
5
=
au + a + 1
au
a
10
=
u
u
2
− u + 1
a
9
=
u
2
+ 1
u
2
− u + 1
a
8
=
u
2
+ 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
2
a − 2au + 2u − 15
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
c
11
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
3
, c
10
(u
3
+ u
2
+ 2u + 1)
2
c
4
u
6
− u
5
+ 4u
4
− u
3
+ 2u
2
+ 2u + 1
c
5
, c
9
(u
3
− u
2
+ 1)
2
c
8
u
6
c
12
(u
3
− u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
10
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
, c
9
(y
3
− y
2
+ 2y −1)
2
c
4
y
6
+ 7y
5
+ 18y
4
+ 21y
3
+ 16y
2
+ 1
c
8
y
6
c
12
(y
3
− 2y
2
+ y −1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.947279 + 0.320410I
b = 1.32472
6.04826 −8.87505 + 0.I
u = 0.215080 + 1.307140I
a = −0.069840 + 0.424452I
b = −0.662359 − 0.562280I
1.91067 − 2.82812I −13.06248 + 4.84887I
u = 0.215080 − 1.307140I
a = 0.947279 − 0.320410I
b = 1.32472
6.04826 −8.87505 + 0.I
u = 0.215080 − 1.307140I
a = −0.069840 − 0.424452I
b = −0.662359 + 0.562280I
1.91067 + 2.82812I −13.06248 − 4.84887I
u = 0.569840
a = −0.37744 + 2.29387I
b = −0.662359 + 0.562280I
1.91067 + 2.82812I −13.06248 − 4.84887I
u = 0.569840
a = −0.37744 − 2.29387I
b = −0.662359 − 0.562280I
1.91067 − 2.82812I −13.06248 + 4.84887I
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 2u − 1)
3
· (u
8
+ 13u
7
+ 52u
6
+ 39u
5
− 95u
4
+ 44u
3
− 5u
2
+ 11u + 1)
· (u
18
+ 31u
17
+ ··· − 4u + 1)
c
2
(u
3
+ u
2
− 1)
3
(u
8
+ u
7
− 6u
6
− 3u
5
+ 5u
4
− 10u
3
+ 5u
2
− u − 1)
· (u
18
+ 3u
17
+ ··· + 4u + 1)
c
3
, c
10
(u
3
+ u
2
+ 2u + 1)
3
(u
8
− u
7
+ 4u
6
− 3u
5
+ 5u
4
− 4u
3
+ u
2
− 3u − 1)
· (u
18
− 3u
17
+ ··· − 4u + 1)
c
4
(u
3
+ 3u
2
+ 2u − 1)(u
6
− u
5
+ 4u
4
− u
3
+ 2u
2
+ 2u + 1)
· (u
8
− 3u
7
− 4u
6
+ 19u
5
− 33u
4
+ 100u
3
+ 91u
2
+ 35u + 7)
· (u
18
− 21u
16
+ ··· + 640u + 1709)
c
5
, c
9
(u
3
− u
2
+ 1)
3
(u
8
+ u
7
− 6u
6
− 3u
5
+ 5u
4
− 10u
3
+ 5u
2
− u − 1)
· (u
18
+ 3u
17
+ ··· + 4u + 1)
c
6
, c
7
, c
11
(u
3
− u
2
+ 2u − 1)
3
(u
8
− u
7
+ 4u
6
− 3u
5
+ 5u
4
− 4u
3
+ u
2
− 3u − 1)
· (u
18
− 3u
17
+ ··· − 4u + 1)
c
8
u
9
(u
8
+ 7u
7
+ 17u
6
+ 16u
5
+ 10u
4
+ 28u
3
+ 44u
2
+ 32u + 8)
· (u
9
− 3u
8
− 4u
7
+ 17u
6
− 8u
5
+ u
4
− 9u
3
+ 20u
2
− 12u + 8)
2
c
12
(u
3
− u + 1)
2
(u
3
+ u
2
+ 2u + 1)
· (u
8
− 6u
7
+ 7u
6
+ 18u
5
− 36u
4
− 10u
3
+ 45u
2
− 14u − 12)
· (u
9
+ 2u
8
− 4u
7
− 5u
6
+ u
5
− 11u
4
+ u
3
− 2u
2
+ u − 3)
2
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
+ 3y
2
+ 2y −1)
3
)(y
8
− 65y
7
+ ··· − 131y + 1)
· (y
18
− 103y
17
+ ··· + 100y + 1)
c
2
, c
5
, c
9
(y
3
− y
2
+ 2y −1)
3
· (y
8
− 13y
7
+ 52y
6
− 39y
5
− 95y
4
− 44y
3
− 5y
2
− 11y + 1)
· (y
18
− 31y
17
+ ··· + 4y + 1)
c
3
, c
6
, c
7
c
10
, c
11
(y
3
+ 3y
2
+ 2y −1)
3
· (y
8
+ 7y
7
+ 20y
6
+ 25y
5
+ y
4
− 32y
3
− 33y
2
− 11y + 1)
· (y
18
+ 13y
17
+ ··· + 4y + 1)
c
4
(y
3
− 5y
2
+ 10y −1)(y
6
+ 7y
5
+ 18y
4
+ 21y
3
+ 16y
2
+ 1)
· (y
8
− 17y
7
+ 64y
6
+ 685y
5
− 3215y
4
− 17392y
3
+ 819y
2
+ 49y + 49)
· (y
18
− 42y
17
+ ··· − 5810040y + 2920681)
c
8
y
9
(y
8
− 15y
7
+ ··· − 320y + 64)
· (y
9
− 17y
8
+ ··· − 176y −64)
2
c
12
(y
3
− 2y
2
+ y −1)
2
(y
3
+ 3y
2
+ 2y −1)
· (y
8
− 22y
7
+ ··· − 1276y + 144)
· (y
9
− 12y
8
+ 38y
7
+ 13y
6
− 107y
5
− 135y
4
− 71y
3
− 68y
2
− 11y −9)
2
20
