12n
0574
(K12n
0574
)
A knot diagram
1
Linearized knot diagam
3 7 9 10 11 2 12 3 4 5 7 8
Solving Sequence
3,9
4 10 5
8,12
1 7 2 6 11
c
3
c
9
c
4
c
8
c
12
c
7
c
2
c
6
c
11
c
1
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
+ u
5
− 2u
4
+ u
3
+ 3u
2
+ b − 3u − 1, u
6
+ u
5
− 2u
4
+ 3u
2
+ 2a − u − 2,
u
7
+ 3u
6
− 4u
4
+ 3u
3
+ 3u
2
− 6u − 2i
I
u
2
= hb − u + 1, 3a − 2u + 3, u
2
− 3i
I
u
3
= hb, a + 1, u + 1i
I
u
4
= hb + 2, a + 1, u − 1i
I
u
5
= hb + 1, a, u − 1i
I
u
6
= hb + 1, a + 1, u − 1i
I
v
1
= ha, b + 1, v + 1i
* 7 irreducible components of dim
C
= 0, with total 14 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
6
+ u
5
− 2u
4
+ u
3
+ 3u
2
+ b − 3u − 1, u
6
+ u
5
− 2u
4
+ 3u
2
+ 2a −
u − 2, u
7
+ 3u
6
− 4u
4
+ 3u
3
+ 3u
2
− 6u − 2i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
10
=
−u
−u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
8
=
u
u
a
12
=
−
1
2
u
6
−
1
2
u
5
+ ··· +
1
2
u + 1
−u
6
− u
5
+ 2u
4
− u
3
− 3u
2
+ 3u + 1
a
1
=
3
2
u
6
+
5
2
u
5
+ ··· −
9
2
u − 1
u
6
+ 2u
5
− 2u
4
− 2u
3
+ 3u
2
− 2u − 1
a
7
=
−
1
2
u
6
−
1
2
u
5
+ ··· −
1
2
u
2
+
3
2
u
−u
6
− u
5
+ 3u
4
− 4u
2
+ 3u + 1
a
2
=
1
2
u
6
+
1
2
u
5
+ ··· +
3
2
u
2
−
5
2
u
u
6
+ 2u
5
− 2u
4
− 2u
3
+ 3u
2
− 2u − 1
a
6
=
−u
4
+ 3u
2
− 1
−u
6
+ 4u
4
− 3u
2
a
11
=
u
3
− 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
3
+ 4u − 20
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
7
− 5u
6
+ 17u
5
− 37u
4
+ 59u
3
+ 73u
2
+ 19u + 1
c
2
, c
6
, c
7
c
11
, c
12
u
7
− u
6
+ 3u
5
− 3u
4
+ 7u
3
+ 5u
2
− 3u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
7
+ 3u
6
− 4u
4
+ 3u
3
+ 3u
2
− 6u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
7
+ 9y
6
+ 37y
5
+ 1405y
4
+ 9539y
3
− 3013y
2
+ 215y −1
c
2
, c
6
, c
7
c
11
, c
12
y
7
+ 5y
6
+ 17y
5
+ 37y
4
+ 59y
3
− 73y
2
+ 19y −1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
7
− 9y
6
+ 30y
5
− 46y
4
+ 45y
3
− 61y
2
+ 48y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.08587
a = −0.409925
b = −0.928471
−4.90710 −18.2170
u = 0.650401 + 0.883152I
a = 0.010004 − 0.769994I
b = 1.56753 − 0.20564I
4.08163 − 2.95233I −14.9050 + 2.6687I
u = 0.650401 − 0.883152I
a = 0.010004 + 0.769994I
b = 1.56753 + 0.20564I
4.08163 + 2.95233I −14.9050 − 2.6687I
u = −1.66573 + 0.28903I
a = 0.95395 + 1.19109I
b = 1.78081 + 0.57849I
−3.67990 + 7.39754I −18.2542 − 3.6074I
u = −1.66573 − 0.28903I
a = 0.95395 − 1.19109I
b = 1.78081 − 0.57849I
−3.67990 − 7.39754I −18.2542 + 3.6074I
u = −0.306290
a = 0.715870
b = −0.152106
−0.466669 −21.1680
u = −1.74892
a = −1.23386
b = −1.61611
−15.1689 −16.2970
5
II. I
u
2
= hb − u + 1, 3a − 2u + 3, u
2
− 3i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
3
a
10
=
−u
−2u
a
5
=
−2
−3
a
8
=
u
u
a
12
=
2
3
u − 1
u − 1
a
1
=
−
1
3
u − 1
−1
a
7
=
1
3
u + 1
1
a
2
=
−
1
3
u
−1
a
6
=
1
0
a
11
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
6
(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
7
(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
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.73205
a = 0.154701
b = 0.732051
−16.4493 −24.0000
u = −1.73205
a = −2.15470
b = −2.73205
−16.4493 −24.0000
9
III. I
u
3
= hb, a + 1, u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
−1
a
4
=
1
1
a
10
=
1
0
a
5
=
0
1
a
8
=
−1
−1
a
12
=
−1
0
a
1
=
−2
−1
a
7
=
−2
−1
a
2
=
−1
−1
a
6
=
−1
0
a
11
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
c
9
, c
10
, c
11
c
12
u − 1
c
2
, c
3
, c
4
c
5
, c
7
u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−6.57974 −24.0000
13
IV. I
u
4
= hb + 2, a + 1, u − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
1
a
4
=
1
1
a
10
=
−1
0
a
5
=
0
1
a
8
=
1
1
a
12
=
−1
−2
a
1
=
0
−1
a
7
=
0
−1
a
2
=
1
−1
a
6
=
−1
0
a
11
=
−1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
11
c
12
u − 1
c
2
, c
7
, c
8
c
9
, c
10
u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −2.00000
−6.57974 −24.0000
17
V. I
u
5
= hb + 1, a, u − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
1
a
4
=
1
1
a
10
=
−1
0
a
5
=
0
1
a
8
=
1
1
a
12
=
0
−1
a
1
=
1
0
a
7
=
1
0
a
2
=
1
0
a
6
=
1
0
a
11
=
−1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −18
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
c
3
, c
4
, c
5
c
7
, c
8
, c
9
c
10
, c
11
, c
12
u − 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
c
3
, c
4
, c
5
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −1.00000
−4.93480 −18.0000
21
VI. I
u
6
= hb + 1, a + 1, u − 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
1
a
4
=
1
1
a
10
=
−1
0
a
5
=
0
1
a
8
=
1
1
a
12
=
−1
−1
a
1
=
−1
−1
a
7
=
1
1
a
2
=
0
−1
a
6
=
1
0
a
11
=
−1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −18
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u + 1
c
2
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
10
u − 1
c
7
, c
11
, c
12
u
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
y −1
c
7
, c
11
, c
12
y
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −1.00000
−4.93480 −18.0000
25
VII. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
−1
0
a
4
=
1
0
a
10
=
−1
0
a
5
=
1
0
a
8
=
−1
0
a
12
=
0
−1
a
1
=
1
−1
a
7
=
−1
1
a
2
=
2
−1
a
6
=
1
0
a
11
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
26
(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
27
(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
28
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
29
VIII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u − 1)
5
(u + 1)(u
7
− 5u
6
+ ··· + 19u + 1)
c
2
, c
7
u(u − 1)
4
(u + 1)
2
(u
7
− u
6
+ 3u
5
− 3u
4
+ 7u
3
+ 5u
2
− 3u − 1)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u(u − 1)
3
(u + 1)(u
2
− 3)(u
7
+ 3u
6
− 4u
4
+ 3u
3
+ 3u
2
− 6u − 2)
c
6
, c
11
, c
12
u(u − 1)
3
(u + 1)
3
(u
7
− u
6
+ 3u
5
− 3u
4
+ 7u
3
+ 5u
2
− 3u − 1)
30
IX. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y(y −1)
6
(y
7
+ 9y
6
+ ··· + 215y −1)
c
2
, c
6
, c
7
c
11
, c
12
y(y −1)
6
(y
7
+ 5y
6
+ 17y
5
+ 37y
4
+ 59y
3
− 73y
2
+ 19y −1)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y(y −3)
2
(y −1)
4
(y
7
− 9y
6
+ ··· + 48y −4)
31
