8
11
(K8a
9
)
A knot diagram
1
Linearized knot diagam
6 7 8 1 2 5 4 3
Solving Sequence
2,6
1 5 7 3 4 8
c
1
c
5
c
6
c
2
c
4
c
8
c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
10
− u
9
+ 3u
8
− 3u
7
+ 5u
6
− 5u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1i
I
u
2
= hu
3
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 13 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
10
− u
9
+ 3u
8
− 3u
7
+ 5u
6
− 5u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
5
=
−u
u
a
7
=
−u
3
u
3
+ u
a
3
=
−u
6
− u
4
+ 1
u
6
+ 2u
4
+ u
2
a
4
=
u
3
u
5
+ u
3
+ u
a
8
=
−u
5
− 2u
3
+ u
2
− u + 1
−u
9
− 3u
7
− 4u
5
+ u
4
− u
3
+ 2u
2
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
9
+ 4u
8
− 8u
7
+ 8u
6
− 8u
5
+ 12u
4
+ 4u
2
− 4u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
10
− u
9
+ 3u
8
− 3u
7
+ 5u
6
− 5u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1
c
2
, c
4
u
10
− 2u
9
− u
8
+ 5u
7
− 3u
6
− 4u
5
+ 12u
4
− 13u
3
+ 5u
2
− u + 2
c
3
, c
7
, c
8
u
10
− u
9
+ 5u
8
− 5u
7
+ 9u
6
− 9u
5
+ 6u
4
− 6u
3
+ u
2
+ 1
c
6
u
10
+ 5u
9
+ 13u
8
+ 19u
7
+ 17u
6
+ 7u
5
− 2u
3
+ u
2
+ 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
− 2y
3
+ y
2
+ 2y + 1
c
2
, c
4
y
10
− 6y
9
+ ··· + 19y + 4
c
3
, c
7
, c
8
y
10
+ 9y
9
+ 33y
8
+ 59y
7
+ 41y
6
− 21y
5
− 44y
4
− 6y
3
+ 13y
2
+ 2y + 1
c
6
y
10
+ y
9
+ 13y
8
+ 11y
7
+ 45y
6
+ 35y
5
+ 12y
4
+ 2y
3
+ 9y
2
− 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.584958 + 0.771492I
4.93719 − 2.31006I 0.86369 + 3.52133I
u = −0.584958 − 0.771492I
4.93719 + 2.31006I 0.86369 − 3.52133I
u = 0.248527 + 0.782547I
−0.448055 + 1.231690I −4.90177 − 5.44908I
u = 0.248527 − 0.782547I
−0.448055 − 1.231690I −4.90177 + 5.44908I
u = 0.761643 + 0.208049I
2.41360 − 3.47839I −0.80497 + 2.79515I
u = 0.761643 − 0.208049I
2.41360 + 3.47839I −0.80497 − 2.79515I
u = −0.449566 + 1.164790I
−4.87665 − 4.14585I −8.98134 + 3.97600I
u = −0.449566 − 1.164790I
−4.87665 + 4.14585I −8.98134 − 3.97600I
u = 0.524355 + 1.163410I
−0.38115 + 8.28632I −4.17560 − 6.14881I
u = 0.524355 − 1.163410I
−0.38115 − 8.28632I −4.17560 + 6.14881I
5
II. I
u
2
= hu
3
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
5
=
−u
u
a
7
=
u + 1
−1
a
3
=
−u
1
a
4
=
−u − 1
−u
2
+ u
a
8
=
−u
2
+ 1
u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
8
u
3
+ u + 1
c
2
, c
4
(u + 1)
3
c
6
u
3
+ 2u
2
+ u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
8
y
3
+ 2y
2
+ y −1
c
2
, c
4
(y −1)
3
c
6
y
3
− 2y
2
+ 5y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.341164 + 1.161540I
−1.64493 −6.00000
u = 0.341164 − 1.161540I
−1.64493 −6.00000
u = −0.682328
−1.64493 −6.00000
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
3
+ u + 1)(u
10
− u
9
+ ··· − 2u + 1)
c
2
, c
4
((u + 1)
3
)(u
10
− 2u
9
+ ··· − u + 2)
c
3
, c
7
, c
8
(u
3
+ u + 1)(u
10
− u
9
+ 5u
8
− 5u
7
+ 9u
6
− 9u
5
+ 6u
4
− 6u
3
+ u
2
+ 1)
c
6
(u
3
+ 2u
2
+ u − 1)
· (u
10
+ 5u
9
+ 13u
8
+ 19u
7
+ 17u
6
+ 7u
5
− 2u
3
+ u
2
+ 2u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
3
+ 2y
2
+ y −1)
· (y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
− 2y
3
+ y
2
+ 2y + 1)
c
2
, c
4
((y −1)
3
)(y
10
− 6y
9
+ ··· + 19y + 4)
c
3
, c
7
, c
8
(y
3
+ 2y
2
+ y −1)
· (y
10
+ 9y
9
+ 33y
8
+ 59y
7
+ 41y
6
− 21y
5
− 44y
4
− 6y
3
+ 13y
2
+ 2y + 1)
c
6
(y
3
− 2y
2
+ 5y −1)
· (y
10
+ y
9
+ 13y
8
+ 11y
7
+ 45y
6
+ 35y
5
+ 12y
4
+ 2y
3
+ 9y
2
− 2y + 1)
11
