9
10
(K9a
39
)
A knot diagram
1
Linearized knot diagam
7 6 9 8 1 2 3 4 5
Solving Sequence
1,7
2 6 3 8 5 4 9
c
1
c
6
c
2
c
7
c
5
c
4
c
9
c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
+ 3u
4
+ u
3
+ 2u
2
+ 2u − 1i
I
u
2
= hu
10
− u
9
+ 4u
8
− 4u
7
+ 6u
6
− 6u
5
+ 3u
4
− 3u
3
+ 1i
* 2 irreducible components of dim
C
= 0, with total 16 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
+ 3u
4
+ u
3
+ 2u
2
+ 2u − 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
−u
5
− 2u
3
− u
u
4
+ 2u
2
a
5
=
u
3
+ 2u
u
3
+ u
a
4
=
−u
4
− u
2
+ 1
−u
5
− u
3
+ u
a
9
=
u
3
+ 2u
u
4
+ u
3
+ u
2
+ 2u − 1
a
9
=
u
3
+ 2u
u
4
+ u
3
+ u
2
+ 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
5
− 4u
4
− 8u
3
− 12u
2
− 4u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
u
6
+ 3u
4
− u
3
+ 2u
2
− 2u − 1
c
5
, c
7
, c
9
u
6
− 3u
5
+ 2u
4
− u
3
+ 5u
2
− 3u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
y
6
+ 6y
5
+ 13y
4
+ 9y
3
− 6y
2
− 8y + 1
c
5
, c
7
, c
9
y
6
− 5y
5
+ 8y
4
− 3y
3
+ 11y
2
− 29y + 4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.841864
−6.52764 −14.6820
u = −0.126468 + 1.352400I
8.36373 + 3.39374I −1.63982 − 3.51762I
u = −0.126468 − 1.352400I
8.36373 − 3.39374I −1.63982 + 3.51762I
u = 0.376468 + 1.319680I
1.76812 − 8.77346I −6.43784 + 5.90094I
u = 0.376468 − 1.319680I
1.76812 + 8.77346I −6.43784 − 5.90094I
u = 0.341865
−0.576591 −17.1630
5
II. I
u
2
= hu
10
− u
9
+ 4u
8
− 4u
7
+ 6u
6
− 6u
5
+ 3u
4
− 3u
3
+ 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
5
=
u
3
+ 2u
u
3
+ u
a
4
=
−2u
9
− 8u
7
− 11u
5
+ u
4
− 2u
3
+ 3u
2
+ 5u + 3
−2u
9
− 8u
7
+ u
6
− 11u
5
+ 4u
4
− 3u
3
+ 5u
2
+ 3u + 2
a
9
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
a
9
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
9
+ 12u
7
+ 12u
5
− 4u
3
− 8u − 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
u
10
+ u
9
+ 4u
8
+ 4u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 3u
3
+ 1
c
5
, c
7
, c
9
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
y
10
+ 7y
9
+ 20y
8
+ 26y
7
+ 6y
6
− 22y
5
− 19y
4
+ 3y
3
+ 6y
2
+ 1
c
5
, c
7
, c
9
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.839548 + 0.070481I
−2.58269 − 4.40083I −10.74431 + 3.49859I
u = 0.839548 − 0.070481I
−2.58269 + 4.40083I −10.74431 − 3.49859I
u = 0.090539 + 1.215350I
2.96077 − 1.53058I −6.51511 + 4.43065I
u = 0.090539 − 1.215350I
2.96077 + 1.53058I −6.51511 − 4.43065I
u = 0.383413 + 1.200420I
0.888787 −7.48114 + 0.I
u = 0.383413 − 1.200420I
0.888787 −7.48114 + 0.I
u = −0.383851 + 1.270630I
−2.58269 + 4.40083I −10.74431 − 3.49859I
u = −0.383851 − 1.270630I
−2.58269 − 4.40083I −10.74431 + 3.49859I
u = −0.429649 + 0.392970I
2.96077 + 1.53058I −6.51511 − 4.43065I
u = −0.429649 − 0.392970I
2.96077 − 1.53058I −6.51511 + 4.43065I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
(u
6
+ 3u
4
− u
3
+ 2u
2
− 2u − 1)
· (u
10
+ u
9
+ 4u
8
+ 4u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 3u
3
+ 1)
c
5
, c
7
, c
9
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
(u
6
− 3u
5
+ 2u
4
− u
3
+ 5u
2
− 3u − 2)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
(y
6
+ 6y
5
+ 13y
4
+ 9y
3
− 6y
2
− 8y + 1)
· (y
10
+ 7y
9
+ 20y
8
+ 26y
7
+ 6y
6
− 22y
5
− 19y
4
+ 3y
3
+ 6y
2
+ 1)
c
5
, c
7
, c
9
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
· (y
6
− 5y
5
+ 8y
4
− 3y
3
+ 11y
2
− 29y + 4)
11
