8
8
(K8a
4
)
A knot diagram
1
Linearized knot diagam
4 7 5 2 1 8 3 6
Solving Sequence
1,4
2 5 6 3 8 7
c
1
c
4
c
5
c
3
c
8
c
7
c
2
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
12
− u
11
− 3u
10
+ 4u
9
+ 3u
8
− 6u
7
+ 2u
6
+ 2u
5
− 4u
4
+ 3u
3
+ u
2
− 2u + 1i
* 1 irreducible components of dim
C
= 0, with total 12 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
12
− u
11
− 3u
10
+ 4u
9
+ 3u
8
− 6u
7
+ 2u
6
+ 2u
5
− 4u
4
+ 3u
3
+ u
2
− 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
6
=
−u
3
−u
3
+ u
a
3
=
u
3
u
5
− u
3
+ u
a
8
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
a
7
=
−u
9
+ 2u
7
− u
5
− 2u
3
+ u
−u
9
+ 3u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
10
− 12u
8
+ 4u
7
+ 16u
6
− 8u
5
+ 8u
3
− 8u
2
+ 4u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
12
− u
11
+ ··· − 2u + 1
c
2
, c
7
u
12
− u
11
− u
10
+ 2u
9
+ 3u
8
− 4u
7
− 2u
6
+ 4u
5
+ 2u
4
− 3u
3
− u
2
+ 1
c
3
u
12
+ 7u
11
+ ··· + 2u + 1
c
5
, c
6
, c
8
u
12
− 3u
11
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
12
− 7y
11
+ ··· − 2y + 1
c
2
, c
7
y
12
− 3y
11
+ ··· − 2y + 1
c
3
y
12
− 3y
11
+ ··· + 6y + 1
c
5
, c
6
, c
8
y
12
+ 13y
11
+ ··· + 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.961384 + 0.208970I
−1.73974 + 0.71593I −3.95647 − 0.64874I
u = −0.961384 − 0.208970I
−1.73974 − 0.71593I −3.95647 + 0.64874I
u = 0.958024 + 0.460561I
0.07674 − 4.24921I 2.17649 + 6.98310I
u = 0.958024 − 0.460561I
0.07674 + 4.24921I 2.17649 − 6.98310I
u = 0.049813 + 0.844037I
−4.04018 + 3.01307I 0.63175 − 2.63251I
u = 0.049813 − 0.844037I
−4.04018 − 3.01307I 0.63175 + 2.63251I
u = −1.238640 + 0.435356I
−7.91518 + 1.48234I −3.15258 − 0.67542I
u = −1.238640 − 0.435356I
−7.91518 − 1.48234I −3.15258 + 0.67542I
u = 1.228550 + 0.484706I
−7.55816 − 7.80134I −2.36611 + 5.63981I
u = 1.228550 − 0.484706I
−7.55816 + 7.80134I −2.36611 − 5.63981I
u = 0.463636 + 0.458719I
1.43731 + 0.35310I 6.66692 − 0.62981I
u = 0.463636 − 0.458719I
1.43731 − 0.35310I 6.66692 + 0.62981I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
u
12
− u
11
+ ··· − 2u + 1
c
2
, c
7
u
12
− u
11
− u
10
+ 2u
9
+ 3u
8
− 4u
7
− 2u
6
+ 4u
5
+ 2u
4
− 3u
3
− u
2
+ 1
c
3
u
12
+ 7u
11
+ ··· + 2u + 1
c
5
, c
6
, c
8
u
12
− 3u
11
+ ··· − 2u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
12
− 7y
11
+ ··· − 2y + 1
c
2
, c
7
y
12
− 3y
11
+ ··· − 2y + 1
c
3
y
12
− 3y
11
+ ··· + 6y + 1
c
5
, c
6
, c
8
y
12
+ 13y
11
+ ··· + 6y + 1
7
