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
