8
19
(K8n
3
)
A knot diagram
1
Linearized knot diagam
7 5 7 2 3 1 4 2
Solving Sequence
2,4 5,7
1 3 6 8
c
4
c
1
c
3
c
5
c
8
c
2
, c
6
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u + 1, a + 1, u
2
− 2u − 1i
I
u
2
= hb, a + 1, u + 1i
* 2 irreducible components of dim
C
= 0, with total 3 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
= hb + u + 1, a + 1, u
2
− 2u − 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
2u + 1
a
7
=
−1
−u − 1
a
1
=
u
4u + 1
a
3
=
−u
−4u − 2
a
6
=
−2u
−8u − 3
a
8
=
u
−u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
u
2
− 2u − 1
c
3
, c
7
u
2
+ 4u + 2
c
8
u
2
+ 6u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
y
2
− 6y + 1
c
3
, c
7
y
2
− 12y + 4
c
8
y
2
− 34y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.414214
a = −1.00000
b = −0.585786
−0.822467 −12.0000
u = 2.41421
a = −1.00000
b = −3.41421
18.9167 −12.0000
5
II. I
u
2
= hb, a + 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
4
=
1
0
a
5
=
1
1
a
7
=
−1
0
a
1
=
−1
−1
a
3
=
1
0
a
6
=
0
1
a
8
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
u + 1
c
2
, c
6
, c
8
u − 1
c
3
, c
7
u
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
y −1
c
3
, c
7
y
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
(u + 1)(u
2
− 2u − 1)
c
2
, c
6
(u − 1)(u
2
− 2u − 1)
c
3
, c
7
u(u
2
+ 4u + 2)
c
8
(u − 1)(u
2
+ 6u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
(y −1)(y
2
− 6y + 1)
c
3
, c
7
y(y
2
− 12y + 4)
c
8
(y −1)(y
2
− 34y + 1)
11
