8
1
(K8a
11
)
A knot diagram
1
Linearized knot diagam
5 4 7 2 1 8 3 6
Solving Sequence
1,5
2 6 4 3 8 7
c
1
c
5
c
4
c
2
c
8
c
7
c
3
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 3u + 1i
* 1 irreducible components of dim
C
= 0, with total 6 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
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 3u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
−u
u
a
4
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
u
2
+ 1
−u
2
a
7
=
−u
3
− 2u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
5
− 4u
4
− 20u
3
− 16u
2
− 24u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
u
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 3u + 1
c
3
, c
7
u
6
− u
5
+ u
4
+ 2u
2
− u + 1
3
(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
6
+ 9y
5
+ 29y
4
+ 40y
3
+ 22y
2
+ 3y + 1
c
3
, c
7
y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.142924 + 1.159520I
4.64282 + 2.65597I 1.58115 − 3.39809I
u = −0.142924 − 1.159520I
4.64282 − 2.65597I 1.58115 + 3.39809I
u = −0.321608 + 0.359079I
−0.258090 + 1.108710I −3.53615 − 6.18117I
u = −0.321608 − 0.359079I
−0.258090 − 1.108710I −3.53615 + 6.18117I
u = −0.03547 + 1.77530I
15.3545 + 3.4272I 1.95500 − 2.25224I
u = −0.03547 − 1.77530I
15.3545 − 3.4272I 1.95500 + 2.25224I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
u
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 3u + 1
c
3
, c
7
u
6
− u
5
+ u
4
+ 2u
2
− u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
y
6
+ 9y
5
+ 29y
4
+ 40y
3
+ 22y
2
+ 3y + 1
c
3
, c
7
y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1
7
