12n
0725
(K12n
0725
)
A knot diagram
1
Linearized knot diagam
4 6 7 10 2 3 11 12 4 7 8 9
Solving Sequence
3,7 4,11
8 12 6 2 1 5 10 9
c
3
c
7
c
11
c
6
c
2
c
1
c
5
c
10
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
3
+ 2u
2
+ b + 2u − 1, a + 1, u
4
− 2u
3
− u
2
+ 4u − 1i
I
u
2
= hb − 2u + 2, a + 1, u
2
+ u − 1i
I
u
3
= hb − 2, a − u − 2, u
2
+ u − 1i
I
u
4
= hb, a − u, u
2
− u + 1i
* 4 irreducible components of dim
C
= 0, with total 10 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−u
3
+ 2u
2
+ b + 2u − 1, a + 1, u
4
− 2u
3
− u
2
+ 4u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
11
=
−1
u
3
− 2u
2
− 2u + 1
a
8
=
−u
−u
2
− 2u + 1
a
12
=
u
2
− 1
2u
3
− 3u + 1
a
6
=
u
u
a
2
=
−u
2
+ 1
−u
2
a
1
=
2u
3
− 2u
2
− 4u + 2
4u
3
− 5u
2
− 10u + 3
a
5
=
−u
3
+ 2u
−u
3
+ u
a
10
=
−1
u
3
− u
2
− 2u + 1
a
9
=
u
3
− 2u
4u
3
− 2u
2
− 9u + 3
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
3
− 2u
2
− 2u − 16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
+ 6u
3
+ 23u
2
+ 10u + 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
10
, c
11
, c
12
u
4
+ 2u
3
− u
2
− 4u − 1
c
4
, c
9
u
4
+ 2u
3
+ 8u
2
+ 12u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
4
+ 10y
3
+ 411y
2
− 54y + 1
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
4
− 6y
3
+ 15y
2
− 14y + 1
c
4
, c
9
y
4
+ 12y
3
+ 24y
2
− 80y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.34859
a = −1.00000
b = −2.39292
−11.3144 −21.8460
u = 1.53492 + 0.55154I
a = −1.00000
b = −3.95793 − 0.75887I
−3.91702 − 5.91675I −18.7424 + 2.9716I
u = 1.53492 − 0.55154I
a = −1.00000
b = −3.95793 + 0.75887I
−3.91702 + 5.91675I −18.7424 − 2.9716I
u = 0.278744
a = −1.00000
b = 0.308773
−0.590771 −16.6700
5
II. I
u
2
= hb − 2u + 2, a + 1, u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u + 1
a
11
=
−1
2u − 2
a
8
=
−u
−3u + 2
a
12
=
−u
−3u + 1
a
6
=
u
u
a
2
=
u
u − 1
a
1
=
0
−u
a
5
=
1
−u + 1
a
10
=
−1
u − 1
a
9
=
−1
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −20
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
, c
10
c
11
, c
12
u
2
− u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
4
, c
9
y
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.00000
b = −0.763932
−1.97392 −20.0000
u = −1.61803
a = −1.00000
b = −5.23607
−17.7653 −20.0000
9
III. I
u
3
= hb − 2, a − u − 2, u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u + 1
a
11
=
u + 2
2
a
8
=
−2u − 3
−u − 2
a
12
=
−2u − 3
−2u − 1
a
6
=
u
u
a
2
=
u
u − 1
a
1
=
0
−u
a
5
=
1
−u + 1
a
10
=
u + 2
1
a
9
=
u + 2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −15
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
, c
10
c
11
, c
12
u
2
− u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
4
, c
9
y
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 2.61803
b = 2.00000
−9.86960 −15.0000
u = −1.61803
a = 0.381966
b = 2.00000
−9.86960 −15.0000
13
IV. I
u
4
= hb, a − u, u
2
− u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u − 1
a
11
=
u
0
a
8
=
1
u
a
12
=
2u − 1
−1
a
6
=
u
u
a
2
=
−u + 2
−u + 1
a
1
=
−4u + 4
u + 2
a
5
=
2u + 1
u + 1
a
10
=
u
1
a
9
=
u + 2
2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −15
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
2
− u + 7
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
10
, c
11
, c
12
u
2
+ u + 1
c
4
, c
9
(u + 2)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
2
+ 13y + 49
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
2
+ y + 1
c
4
, c
9
(y −4)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0
0 −15.0000
u = 0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = 0
0 −15.0000
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 7)(u
2
+ u − 1)
2
(u
4
+ 6u
3
+ 23u
2
+ 10u + 1)
c
2
, c
3
, c
7
c
8
(u
2
+ u − 1)
2
(u
2
+ u + 1)(u
4
+ 2u
3
− u
2
− 4u − 1)
c
4
, c
9
u
4
(u + 2)
2
(u
4
+ 2u
3
+ 8u
2
+ 12u + 4)
c
5
, c
6
, c
10
c
11
, c
12
(u
2
− u − 1)
2
(u
2
+ u + 1)(u
4
+ 2u
3
− u
2
− 4u − 1)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
− 3y + 1)
2
(y
2
+ 13y + 49)(y
4
+ 10y
3
+ 411y
2
− 54y + 1)
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
10
, c
11
, c
12
(y
2
− 3y + 1)
2
(y
2
+ y + 1)(y
4
− 6y
3
+ 15y
2
− 14y + 1)
c
4
, c
9
y
4
(y −4)
2
(y
4
+ 12y
3
+ 24y
2
− 80y + 16)
19
