8
3
(K8a
18
)
A knot diagram
1
Linearized knot diagam
7 6 1 8 3 2 5 4
Solving Sequence
3,6
2 7 1 5 8 4
c
2
c
6
c
1
c
5
c
7
c
4
c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1i
* 1 irreducible components of dim
C
= 0, with total 8 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
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
4
+ 2u
2
a
5
=
−u
u
a
8
=
−u
5
− 2u
3
+ u
u
5
+ 3u
3
+ u
a
4
=
u
6
+ 3u
4
+ 2u
2
+ 1
u
7
− u
6
+ 4u
5
− 3u
4
+ 4u
3
− 2u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
+ 4u
5
− 16u
4
+ 12u
3
− 16u
2
+ 8u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
8
+ u
7
+ 5u
6
+ 4u
5
+ 7u
4
+ 4u
3
+ 2u
2
+ 1
c
3
, c
4
, c
7
c
8
u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1
3
(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
y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 39y
4
+ 22y
3
+ 18y
2
+ 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.647085 + 0.502738I
6.60959 + 2.18536I 3.58319 − 3.14055I
u = 0.647085 − 0.502738I
6.60959 − 2.18536I 3.58319 + 3.14055I
u = −0.283060 + 0.443755I
− 1.04600I 0. + 6.68545I
u = −0.283060 − 0.443755I
1.04600I 0. − 6.68545I
u = −0.06382 + 1.51723I
−6.60959 − 2.18536I −3.58319 + 3.14055I
u = −0.06382 − 1.51723I
−6.60959 + 2.18536I −3.58319 − 3.14055I
u = 0.19980 + 1.51366I
5.23868I 0. − 3.04258I
u = 0.19980 − 1.51366I
− 5.23868I 0. + 3.04258I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
8
+ u
7
+ 5u
6
+ 4u
5
+ 7u
4
+ 4u
3
+ 2u
2
+ 1
c
3
, c
4
, c
7
c
8
u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 39y
4
+ 22y
3
+ 18y
2
+ 4y + 1
7
