8
2
(K8a
8
)
A knot diagram
1
Linearized knot diagam
6 7 8 1 2 5 3 4
Solving Sequence
1,6
2 5 7 4 8 3
c
1
c
5
c
6
c
4
c
8
c
3
c
2
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
+ u
7
+ 3u
6
+ 2u
5
+ 3u
4
+ 2u
3
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
+ 3u
6
+ 2u
5
+ 3u
4
+ 2u
3
1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
7
=
u
3
u
5
+ u
3
+ u
a
4
=
u
3
u
3
+ u
a
8
=
u
6
u
4
+ 1
u
6
2u
4
u
2
a
3
=
u
6
u
4
+ 1
u
7
u
6
2u
5
u
4
2u
3
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
7
4u
6
8u
5
4u
4
4u
3
4u
2
+ 4u 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
8
u
7
+ 3u
6
2u
5
+ 3u
4
2u
3
1
c
2
, c
3
, c
4
c
7
, c
8
u
8
+ u
7
5u
6
4u
5
+ 7u
4
+ 4u
3
2u
2
2u 1
c
6
u
8
+ 5u
7
+ 11u
6
+ 10u
5
u
4
10u
3
6u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
8
+ 5y
7
+ 11y
6
+ 10y
5
y
4
10y
3
6y
2
+ 1
c
2
, c
3
, c
4
c
7
, c
8
y
8
11y
7
+ 47y
6
98y
5
+ 103y
4
50y
3
+ 6y
2
+ 1
c
6
y
8
3y
7
+ 19y
6
34y
5
+ 71y
4
66y
3
+ 34y
2
12y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.914675
10.1759 7.82210
u = 0.252896 + 0.819281I
0.491278 1.275320I 5.18053 + 5.08518I
u = 0.252896 0.819281I
0.491278 + 1.275320I 5.18053 5.08518I
u = 0.394459 + 1.112500I
4.34520 + 3.63283I 10.42240 4.51802I
u = 0.394459 1.112500I
4.34520 3.63283I 10.42240 + 4.51802I
u = 0.473514 + 1.273020I
14.0724 4.9352I 10.98443 + 2.99422I
u = 0.473514 1.273020I
14.0724 + 4.9352I 10.98443 2.99422I
u = 0.578577
1.35429 7.00320
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
8
u
7
+ 3u
6
2u
5
+ 3u
4
2u
3
1
c
2
, c
3
, c
4
c
7
, c
8
u
8
+ u
7
5u
6
4u
5
+ 7u
4
+ 4u
3
2u
2
2u 1
c
6
u
8
+ 5u
7
+ 11u
6
+ 10u
5
u
4
10u
3
6u
2
+ 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
8
+ 5y
7
+ 11y
6
+ 10y
5
y
4
10y
3
6y
2
+ 1
c
2
, c
3
, c
4
c
7
, c
8
y
8
11y
7
+ 47y
6
98y
5
+ 103y
4
50y
3
+ 6y
2
+ 1
c
6
y
8
3y
7
+ 19y
6
34y
5
+ 71y
4
66y
3
+ 34y
2
12y + 1
7