8
4
(K8a
17
)
A knot diagram
1
Linearized knot diagam
6 5 8 7 1 2 4 3
Solving Sequence
2,7
6 1 5 3 4 8
c
6
c
1
c
5
c
2
c
4
c
8
c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
+ u
8
− 4u
7
− 3u
6
+ 5u
5
+ u
4
− 2u
3
+ 2u
2
+ u + 1i
* 1 irreducible components of dim
C
= 0, with total 9 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
− 4u
7
− 3u
6
+ 5u
5
+ u
4
− 2u
3
+ 2u
2
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
1
=
−u
−u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
4
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
8
=
u
8
− 3u
6
+ u
4
+ 2u
2
+ 1
u
8
− 4u
6
+ 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
+ 12u
4
− 4u
3
− 8u
2
+ 8u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u
9
− u
8
− 4u
7
+ 3u
6
+ 5u
5
− u
4
− 2u
3
− 2u
2
+ u − 1
c
2
u
9
+ 3u
8
+ 2u
7
− 5u
6
− u
5
+ 13u
4
+ 10u
3
− 2u
2
+ u + 3
c
3
, c
4
, c
7
c
8
u
9
− u
8
+ 6u
7
− 5u
6
+ 11u
5
− 7u
4
+ 6u
3
− 2u
2
+ u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
y
9
− 9y
8
+ 32y
7
− 55y
6
+ 45y
5
− 19y
4
+ 16y
3
− 10y
2
− 3y −1
c
2
y
9
− 5y
8
+ 32y
7
− 87y
6
+ 185y
5
− 223y
4
+ 180y
3
− 62y
2
+ 13y −9
c
3
, c
4
, c
7
c
8
y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
− 6y
2
− 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.482242 + 0.666986I
7.06362 + 2.21388I 4.24115 − 3.04598I
u = 0.482242 − 0.666986I
7.06362 − 2.21388I 4.24115 + 3.04598I
u = −1.28056
2.83680 1.66670
u = 1.380230 + 0.162431I
5.16280 + 3.41073I 5.88238 − 4.39642I
u = 1.380230 − 0.162431I
5.16280 − 3.41073I 5.88238 + 4.39642I
u = −0.230908 + 0.456719I
0.035384 − 1.109690I 0.55374 + 6.23947I
u = −0.230908 − 0.456719I
0.035384 + 1.109690I 0.55374 − 6.23947I
u = −1.49128 + 0.23430I
13.4612 − 5.5005I 7.48937 + 2.97298I
u = −1.49128 − 0.23430I
13.4612 + 5.5005I 7.48937 − 2.97298I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u
9
− u
8
− 4u
7
+ 3u
6
+ 5u
5
− u
4
− 2u
3
− 2u
2
+ u − 1
c
2
u
9
+ 3u
8
+ 2u
7
− 5u
6
− u
5
+ 13u
4
+ 10u
3
− 2u
2
+ u + 3
c
3
, c
4
, c
7
c
8
u
9
− u
8
+ 6u
7
− 5u
6
+ 11u
5
− 7u
4
+ 6u
3
− 2u
2
+ u − 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
y
9
− 9y
8
+ 32y
7
− 55y
6
+ 45y
5
− 19y
4
+ 16y
3
− 10y
2
− 3y −1
c
2
y
9
− 5y
8
+ 32y
7
− 87y
6
+ 185y
5
− 223y
4
+ 180y
3
− 62y
2
+ 13y −9
c
3
, c
4
, c
7
c
8
y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
− 6y
2
− 3y −1
7
