8
9
(K8a
16
)
A knot diagram
1
Linearized knot diagam
7 6 1 8 2 3 4 5
Solving Sequence
3,7
6 2 1 4 5 8
c
6
c
2
c
1
c
3
c
5
c
8
c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
12
+ u
11
− 5u
10
− 4u
9
+ 9u
8
+ 4u
7
− 6u
6
+ 2u
5
− 3u
3
+ u
2
+ 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
− 5u
10
− 4u
9
+ 9u
8
+ 4u
7
− 6u
6
+ 2u
5
− 3u
3
+ u
2
+ 1i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
2
=
−u
−u
3
+ u
a
1
=
u
3
− 2u
−u
3
+ u
a
4
=
−u
7
+ 4u
5
− 4u
3
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
8
=
−u
9
+ 4u
7
− 5u
5
+ 2u
3
− u
−u
11
+ 5u
9
− 8u
7
+ 3u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
9
+ 16u
7
− 4u
6
− 20u
5
+ 12u
4
+ 4u
3
− 8u
2
+ 4u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 3u
11
+ ··· + 4u + 1
c
2
, c
5
, c
6
u
12
− u
11
− 5u
10
+ 4u
9
+ 9u
8
− 4u
7
− 6u
6
− 2u
5
+ 3u
3
+ u
2
+ 1
c
3
u
12
− 3u
11
+ ··· − 4u + 1
c
4
, c
7
, c
8
u
12
+ u
11
− 5u
10
− 4u
9
+ 9u
8
+ 4u
7
− 6u
6
+ 2u
5
− 3u
3
+ u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
12
+ y
11
+ ··· − 2y + 1
c
2
, c
4
, c
5
c
6
, c
7
, c
8
y
12
− 11y
11
+ ··· + 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.851576 + 0.246566I
−3.11509 − 0.09361I −1.99088 − 0.76204I
u = 0.851576 − 0.246566I
−3.11509 + 0.09361I −1.99088 + 0.76204I
u = 0.227035 + 0.729376I
−5.13898 + 3.88480I −4.80561 − 4.17140I
u = 0.227035 − 0.729376I
−5.13898 − 3.88480I −4.80561 + 4.17140I
u = −1.343200 + 0.063939I
3.11509 − 0.09361I 1.99088 − 0.76204I
u = −1.343200 − 0.063939I
3.11509 + 0.09361I 1.99088 + 0.76204I
u = 1.383160 + 0.208829I
5.13898 + 3.88480I 4.80561 − 4.17140I
u = 1.383160 − 0.208829I
5.13898 − 3.88480I 4.80561 + 4.17140I
u = −1.39026 + 0.29206I
− 7.58818I 0. + 5.13539I
u = −1.39026 − 0.29206I
7.58818I 0. − 5.13539I
u = −0.228302 + 0.503204I
− 1.20211I 0. + 5.63740I
u = −0.228302 − 0.503204I
1.20211I 0. − 5.63740I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
12
+ 3u
11
+ ··· + 4u + 1
c
2
, c
5
, c
6
u
12
− u
11
− 5u
10
+ 4u
9
+ 9u
8
− 4u
7
− 6u
6
− 2u
5
+ 3u
3
+ u
2
+ 1
c
3
u
12
− 3u
11
+ ··· − 4u + 1
c
4
, c
7
, c
8
u
12
+ u
11
− 5u
10
− 4u
9
+ 9u
8
+ 4u
7
− 6u
6
+ 2u
5
− 3u
3
+ u
2
+ 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
12
+ y
11
+ ··· − 2y + 1
c
2
, c
4
, c
5
c
6
, c
7
, c
8
y
12
− 11y
11
+ ··· + 2y + 1
7
