12n
0581
(K12n
0581
)
A knot diagram
1
Linearized knot diagam
3 7 11 10 9 2 12 3 5 4 7 8
Solving Sequence
7,11
12
4,8
3 2 1 6 10 5 9
c
11
c
7
c
3
c
2
c
1
c
6
c
10
c
4
c
9
c
5
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
− u
8
− 7u
7
+ 6u
6
+ 14u
5
− 7u
4
− 5u
3
− 10u
2
+ 4b + u,
− u
9
+ u
8
+ 7u
7
− 6u
6
− 14u
5
+ 7u
4
+ 5u
3
+ 10u
2
+ 4a − u − 4,
u
10
− u
9
− 8u
8
+ 7u
7
+ 21u
6
− 13u
5
− 19u
4
+ u
3
+ 6u
2
− 2u − 1i
I
u
2
= h2u
7
+ u
6
− 6u
5
− 6u
4
+ 5u
3
+ 11u
2
+ 2b − 3u − 10,
− 5u
7
− 3u
6
+ 16u
5
+ 18u
4
− 15u
3
− 37u
2
+ 8a + 9u + 39, u
8
− u
7
− 4u
6
+ 2u
5
+ 7u
4
+ u
3
− 9u
2
− 3u + 8i
I
u
3
= hb + a + 1, a
2
+ 2a + 4, u + 1i
I
u
4
= hb, a + 1, u + 1i
I
u
5
= hb + a + 1, a
2
+ 2a + 2, u − 1i
* 5 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
= hu
9
− u
8
+ · · · + 4b + u, −u
9
+ u
8
+ · · · + 4a − 4, u
10
− u
9
+ · · · − 2u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
4
=
1
4
u
9
−
1
4
u
8
+ ··· +
1
4
u + 1
−
1
4
u
9
+
1
4
u
8
+ ··· +
5
2
u
2
−
1
4
u
a
8
=
−u
−u
3
+ u
a
3
=
1
−
1
4
u
9
+
1
4
u
8
+ ··· +
5
2
u
2
−
1
4
u
a
2
=
1
−
1
4
u
9
+
1
4
u
8
+ ··· +
3
2
u
2
−
1
4
u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
u
−
1
4
u
8
+
1
4
u
7
+ ··· +
1
2
u −
1
4
a
10
=
−
1
4
u
9
+
1
2
u
8
+ ··· −
1
4
u +
5
4
1
2
u
9
−
3
4
u
8
+ ··· +
1
2
u −
1
4
a
5
=
−
1
4
u
9
+
1
2
u
8
+ ··· +
1
4
u +
3
4
1
2
u
7
−
7
2
u
5
+ ··· −
1
2
u +
1
2
a
9
=
1
4
u
8
−
1
4
u
7
+ ··· −
3
2
u +
1
4
−
1
4
u
9
+
1
4
u
8
+ ··· +
7
4
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
9
−
7
2
u
8
−
31
2
u
7
+
53
2
u
6
+ 38u
5
−60u
4
−
61
2
u
3
+
63
2
u
2
+ 14u −
37
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
+ 17u
9
+ ··· + 16u + 1
c
2
, c
6
, c
7
c
11
, c
12
u
10
− u
9
− 8u
8
+ 7u
7
+ 21u
6
− 13u
5
− 19u
4
+ u
3
+ 6u
2
− 2u − 1
c
3
, c
4
, c
5
c
9
, c
10
u
10
+ 3u
9
+ ··· + 12u + 2
c
8
u
10
+ 3u
9
+ ··· + 108u + 58
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 49y
9
+ ··· − 100y + 1
c
2
, c
6
, c
7
c
11
, c
12
y
10
− 17y
9
+ ··· − 16y + 1
c
3
, c
4
, c
5
c
9
, c
10
y
10
+ 13y
9
+ ··· − 36y + 4
c
8
y
10
− 51y
9
+ ··· − 9460y + 3364
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.624283 + 0.413630I
a = 0.99995 + 1.69388I
b = 0.00005 − 1.69388I
11.16200 + 1.46971I −6.43725 − 4.71631I
u = −0.624283 − 0.413630I
a = 0.99995 − 1.69388I
b = 0.00005 + 1.69388I
11.16200 − 1.46971I −6.43725 + 4.71631I
u = 0.425863 + 0.318105I
a = 0.919105 − 0.860258I
b = 0.080895 + 0.860258I
2.03579 − 1.27062I −6.43196 + 5.78765I
u = 0.425863 − 0.318105I
a = 0.919105 + 0.860258I
b = 0.080895 − 0.860258I
2.03579 + 1.27062I −6.43196 − 5.78765I
u = −0.299023
a = 0.714196
b = 0.285804
−0.505065 −19.6030
u = 1.74982 + 0.35246I
a = 0.79690 − 1.70108I
b = 0.20310 + 1.70108I
−4.80759 − 8.37238I −11.74586 + 3.25359I
u = 1.74982 − 0.35246I
a = 0.79690 + 1.70108I
b = 0.20310 − 1.70108I
−4.80759 + 8.37238I −11.74586 − 3.25359I
u = −1.85465 + 0.19086I
a = 0.362420 + 0.933540I
b = 0.637580 − 0.933540I
−13.8130 + 4.9888I −13.52702 − 3.37301I
u = −1.85465 − 0.19086I
a = 0.362420 − 0.933540I
b = 0.637580 + 0.933540I
−13.8130 − 4.9888I −13.52702 + 3.37301I
u = 1.90554
a = 0.129056
b = 0.870944
−16.6134 −16.1130
5
II. I
u
2
= h2u
7
+ u
6
+ · · · + 2b − 10, −5u
7
− 3u
6
+ · · · + 8a + 39, u
8
− u
7
−
4u
6
+ 2u
5
+ 7u
4
+ u
3
− 9u
2
− 3u + 8i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
4
=
5
8
u
7
+
3
8
u
6
+ ··· −
9
8
u −
39
8
−u
7
−
1
2
u
6
+ ··· +
3
2
u + 5
a
8
=
−u
−u
3
+ u
a
3
=
−
3
8
u
7
−
1
8
u
6
+ ··· +
3
8
u +
1
8
−u
7
−
1
2
u
6
+ ··· +
3
2
u + 5
a
2
=
−
3
8
u
7
−
1
8
u
6
+ ··· +
3
8
u +
1
8
1
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
3
8
u
7
+
5
8
u
6
+ ··· +
9
8
u −
21
8
−
1
2
u
7
−
1
2
u
6
+ ··· − 3u
2
+ 3
a
10
=
−
1
8
u
7
−
3
8
u
6
+ ··· +
5
8
u +
7
8
−
1
2
u
7
+
3
2
u
5
+ ··· +
1
2
u + 3
a
5
=
1
4
u
7
−
1
4
u
6
+ ··· −
5
4
u −
3
4
u
7
+ u
6
− 3u
5
− 4u
4
+ 2u
3
+ 6u
2
− u − 6
a
9
=
1
8
u
7
−
1
8
u
6
+ ··· −
9
8
u −
3
8
1
2
u
7
+
1
2
u
6
+ ··· + 3u
2
− 3
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
7
− 6u
5
− 4u
4
+ 8u
3
+ 10u
2
− 6u − 22
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 9u
7
+ 34u
6
+ 76u
5
+ 127u
4
+ 179u
3
+ 199u
2
+ 153u + 64
c
2
, c
6
, c
7
c
11
, c
12
u
8
− u
7
− 4u
6
+ 2u
5
+ 7u
4
+ u
3
− 9u
2
− 3u + 8
c
3
, c
4
, c
5
c
9
, c
10
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
8
(u
4
− u
3
+ u
2
+ 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
− 13y
7
+ ··· + 2063y + 4096
c
2
, c
6
, c
7
c
11
, c
12
y
8
− 9y
7
+ 34y
6
− 76y
5
+ 127y
4
− 179y
3
+ 199y
2
− 153y + 64
c
3
, c
4
, c
5
c
9
, c
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
8
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.974589 + 0.525375I
a = −0.575851 + 1.230320I
b = −0.395123 − 0.506844I
−3.50087 − 1.41510I −13.8267 + 4.9087I
u = 0.974589 − 0.525375I
a = −0.575851 − 1.230320I
b = −0.395123 + 0.506844I
−3.50087 + 1.41510I −13.8267 − 4.9087I
u = −0.728625 + 0.959908I
a = −0.72016 − 2.57269I
b = −0.10488 + 1.55249I
3.50087 + 3.16396I −10.17326 − 2.56480I
u = −0.728625 − 0.959908I
a = −0.72016 + 2.57269I
b = −0.10488 − 1.55249I
3.50087 − 3.16396I −10.17326 + 2.56480I
u = −1.326400 + 0.194967I
a = −0.267111 + 0.013410I
b = −0.395123 − 0.506844I
−3.50087 − 1.41510I −13.8267 + 4.9087I
u = −1.326400 − 0.194967I
a = −0.267111 − 0.013410I
b = −0.395123 + 0.506844I
−3.50087 + 1.41510I −13.8267 − 4.9087I
u = 1.58043 + 0.04862I
a = −0.374382 + 0.959864I
b = −0.10488 − 1.55249I
3.50087 − 3.16396I −10.17326 + 2.56480I
u = 1.58043 − 0.04862I
a = −0.374382 − 0.959864I
b = −0.10488 + 1.55249I
3.50087 + 3.16396I −10.17326 − 2.56480I
9
III. I
u
3
= hb + a + 1, a
2
+ 2a + 4, u + 1i
(i) Arc colorings
a
7
=
0
−1
a
11
=
1
0
a
12
=
1
1
a
4
=
a
−a − 1
a
8
=
1
0
a
3
=
−1
−a − 1
a
2
=
−1
−a
a
1
=
0
1
a
6
=
−1
−a − 1
a
10
=
−a − 3
3
a
5
=
−a + 1
2a + 2
a
9
=
a + 2
−3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u − 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
+ 3
c
6
, c
11
, c
12
(u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 3)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000 + 1.73205I
b = − 1.73205I
9.86960 −12.0000
u = −1.00000
a = −1.00000 − 1.73205I
b = 1.73205I
9.86960 −12.0000
13
IV. I
u
4
= hb, a + 1, u + 1i
(i) Arc colorings
a
7
=
0
−1
a
11
=
1
0
a
12
=
1
1
a
4
=
−1
0
a
8
=
1
0
a
3
=
−1
0
a
2
=
−1
1
a
1
=
0
1
a
6
=
−1
0
a
10
=
1
0
a
5
=
−1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
11
, c
12
u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
y −1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
17
V. I
u
5
= hb + a + 1, a
2
+ 2a + 2, u − 1i
(i) Arc colorings
a
7
=
0
1
a
11
=
1
0
a
12
=
1
1
a
4
=
a
−a − 1
a
8
=
−1
0
a
3
=
−1
−a − 1
a
2
=
−1
−a
a
1
=
0
1
a
6
=
1
a + 1
a
10
=
−a − 1
1
a
5
=
a + 1
0
a
9
=
−a − 2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
11
c
12
(u − 1)
2
c
2
, c
7
(u + 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
+ 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000 + 1.00000I
b = − 1.000000I
0 −12.0000
u = 1.00000
a = −1.00000 − 1.00000I
b = 1.000000I
0 −12.0000
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
· (u
8
+ 9u
7
+ 34u
6
+ 76u
5
+ 127u
4
+ 179u
3
+ 199u
2
+ 153u + 64)
· (u
10
+ 17u
9
+ ··· + 16u + 1)
c
2
, c
7
(u − 1)
3
(u + 1)
2
(u
8
− u
7
− 4u
6
+ 2u
5
+ 7u
4
+ u
3
− 9u
2
− 3u + 8)
· (u
10
− u
9
− 8u
8
+ 7u
7
+ 21u
6
− 13u
5
− 19u
4
+ u
3
+ 6u
2
− 2u − 1)
c
3
, c
4
, c
5
c
9
, c
10
u(u
2
+ 1)(u
2
+ 3)(u
4
− u
3
+ ··· − 2u + 1)
2
(u
10
+ 3u
9
+ ··· + 12u + 2)
c
6
, c
11
, c
12
(u − 1)
2
(u + 1)
3
(u
8
− u
7
− 4u
6
+ 2u
5
+ 7u
4
+ u
3
− 9u
2
− 3u + 8)
· (u
10
− u
9
− 8u
8
+ 7u
7
+ 21u
6
− 13u
5
− 19u
4
+ u
3
+ 6u
2
− 2u − 1)
c
8
u(u
2
+ 1)(u
2
+ 3)(u
4
− u
3
+ u
2
+ 1)
2
(u
10
+ 3u
9
+ ··· + 108u + 58)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
8
− 13y
7
+ ··· + 2063y + 4096)
· (y
10
− 49y
9
+ ··· − 100y + 1)
c
2
, c
6
, c
7
c
11
, c
12
(y −1)
5
· (y
8
− 9y
7
+ 34y
6
− 76y
5
+ 127y
4
− 179y
3
+ 199y
2
− 153y + 64)
· (y
10
− 17y
9
+ ··· − 16y + 1)
c
3
, c
4
, c
5
c
9
, c
10
y(y + 1)
2
(y + 3)
2
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
10
+ 13y
9
+ ··· − 36y + 4)
c
8
y(y + 1)
2
(y + 3)
2
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
10
− 51y
9
+ ··· − 9460y + 3364)
23
