9
48
(K9n
6
)
A knot diagram
1
Linearized knot diagam
5 7 9 2 4 3 6 4 3
Solving Sequence
3,9 4,7
2 6 5 1 8
c
3
c
2
c
6
c
5
c
1
c
8
c
4
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, −u
2
+ a, u
3
− u
2
+ u + 1i
I
u
2
= hb − u, −u
3
+ a + 1, u
4
+ u
3
+ u
2
+ 1i
I
u
3
= hu
2
+ b + u, u
3
+ 2u
2
+ a + 2u, u
4
+ u
3
+ u
2
+ 1i
I
u
4
= h−u
3
+ u
2
+ b − u + 1, −u
3
+ 2a + u − 1, u
4
− 2u
3
+ 3u
2
− 3u + 2i
I
u
5
= hb + u, a + 2u + 1, u
2
+ 1i
* 5 irreducible components of dim
C
= 0, with total 17 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
= hb − u, −u
2
+ a, u
3
− u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
7
=
u
2
u
a
2
=
u
2
− u
u
2
a
6
=
u
2
− u
u
a
5
=
−u
−u − 1
a
1
=
−u
−u
a
8
=
−u
u
2
− 1
a
8
=
−u
u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6u + 8
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
c
9
u
3
+ u
2
+ u − 1
c
5
, c
7
u
3
+ u
2
+ 3u − 1
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
c
9
y
3
+ y
2
+ 3y −1
c
5
, c
7
y
3
+ 5y
2
+ 11y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.771845 + 1.115140I
a = −0.64780 + 1.72143I
b = 0.771845 + 1.115140I
2.02941 + 9.53188I 3.36893 − 6.69086I
u = 0.771845 − 1.115140I
a = −0.64780 − 1.72143I
b = 0.771845 − 1.115140I
2.02941 − 9.53188I 3.36893 + 6.69086I
u = −0.543689
a = 0.295598
b = −0.543689
0.875992 11.2620
5
II. I
u
2
= hb − u, −u
3
+ a + 1, u
4
+ u
3
+ u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
7
=
u
3
− 1
u
a
2
=
−u
3
− u
2
− u
u
2
a
6
=
u
3
− u − 1
u
a
5
=
−u
1
a
1
=
−u
−u
a
8
=
−u
u
3
+ u
a
8
=
−u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
− 4u + 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
4
+ 2u
3
+ 3u
2
+ 3u + 2
c
2
, c
3
, c
6
c
8
, c
9
u
4
− u
3
+ u
2
+ 1
c
5
u
4
+ 2u
3
+ u
2
+ 3u + 4
c
7
u
4
+ u
3
+ 3u
2
+ 2u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
4
+ 2y
3
+ y
2
+ 3y + 4
c
2
, c
3
, c
6
c
8
, c
9
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
5
y
4
− 2y
3
− 3y
2
− y + 16
c
7
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = −1.50411 − 0.10631I
b = 0.351808 + 0.720342I
−3.50087 + 1.41510I 2.17326 − 4.90874I
u = 0.351808 − 0.720342I
a = −1.50411 + 0.10631I
b = 0.351808 − 0.720342I
−3.50087 − 1.41510I 2.17326 + 4.90874I
u = −0.851808 + 0.911292I
a = 0.504108 + 1.226850I
b = −0.851808 + 0.911292I
3.50087 − 3.16396I 5.82674 + 2.56480I
u = −0.851808 − 0.911292I
a = 0.504108 − 1.226850I
b = −0.851808 − 0.911292I
3.50087 + 3.16396I 5.82674 − 2.56480I
9
III. I
u
3
= hu
2
+ b + u, u
3
+ 2u
2
+ a + 2u, u
4
+ u
3
+ u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
7
=
−u
3
− 2u
2
− 2u
−u
2
− u
a
2
=
u
3
− u − 1
u
3
− 1
a
6
=
−u
3
− u
2
− u
−u
2
− u
a
5
=
−u
3
− 2u
2
− u
−u
3
− 2u
2
− u − 1
a
1
=
−u
−u
a
8
=
−u
u
3
+ u
a
8
=
−u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
− 4u + 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
8
, c
9
u
4
− u
3
+ u
2
+ 1
c
2
, c
6
u
4
+ 2u
3
+ 3u
2
+ 3u + 2
c
5
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
7
u
4
+ 2u
3
+ u
2
+ 3u + 4
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
8
, c
9
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
2
, c
6
y
4
+ 2y
3
+ y
2
+ 3y + 4
c
5
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
7
y
4
− 2y
3
− 3y
2
− y + 16
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = 0.59074 − 2.34806I
b = 0.043315 − 1.227190I
−3.50087 + 1.41510I 2.17326 − 4.90874I
u = 0.351808 − 0.720342I
a = 0.59074 + 2.34806I
b = 0.043315 + 1.227190I
−3.50087 − 1.41510I 2.17326 + 4.90874I
u = −0.851808 + 0.911292I
a = 0.409261 + 0.055548I
b = 0.956685 + 0.641200I
3.50087 − 3.16396I 5.82674 + 2.56480I
u = −0.851808 − 0.911292I
a = 0.409261 − 0.055548I
b = 0.956685 − 0.641200I
3.50087 + 3.16396I 5.82674 − 2.56480I
13
IV. I
u
4
= h−u
3
+ u
2
+ b − u + 1, −u
3
+ 2a + u − 1, u
4
− 2u
3
+ 3u
2
− 3u + 2i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
7
=
1
2
u
3
−
1
2
u +
1
2
u
3
− u
2
+ u − 1
a
2
=
−
1
2
u
3
+ u
2
−
3
2
u +
3
2
−u
3
+ u
2
− 2u + 1
a
6
=
−
1
2
u
3
+ u
2
−
3
2
u +
3
2
u
3
− u
2
+ u − 1
a
5
=
1
2
u
3
+
1
2
u +
1
2
1
a
1
=
−u
−u
a
8
=
−u
u
3
+ u
a
8
=
−u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u + 2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
6
u
4
− u
3
+ u
2
+ 1
c
3
, c
8
, c
9
u
4
+ 2u
3
+ 3u
2
+ 3u + 2
c
5
, c
7
u
4
+ u
3
+ 3u
2
+ 2u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
3
, c
8
, c
9
y
4
+ 2y
3
+ y
2
+ 3y + 4
c
5
, c
7
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.956685 + 0.641200I
a = −0.130534 + 0.427872I
b = −0.851808 + 0.911292I
3.50087 − 3.16396I 5.82674 + 2.56480I
u = 0.956685 − 0.641200I
a = −0.130534 − 0.427872I
b = −0.851808 − 0.911292I
3.50087 + 3.16396I 5.82674 − 2.56480I
u = 0.043315 + 1.227190I
a = 0.38053 − 1.53420I
b = 0.351808 − 0.720342I
−3.50087 − 1.41510I 2.17326 + 4.90874I
u = 0.043315 − 1.227190I
a = 0.38053 + 1.53420I
b = 0.351808 + 0.720342I
−3.50087 + 1.41510I 2.17326 − 4.90874I
17
V. I
u
5
= hb + u, a + 2u + 1, u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
1
a
7
=
−2u − 1
−u
a
2
=
u − 1
−1
a
6
=
−u − 1
−u
a
5
=
−u
−u + 1
a
1
=
u
u
a
8
=
−u
0
a
8
=
−u
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
c
9
u
2
+ 1
c
5
, c
7
(u + 1)
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
c
9
(y + 1)
2
c
5
, c
7
(y − 1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −1.00000 − 2.00000I
b = − 1.000000I
−4.93480 −4.00000
u = − 1.000000I
a = −1.00000 + 2.00000I
b = 1.000000I
−4.93480 −4.00000
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
c
9
(u
2
+ 1)(u
3
+ u
2
+ u − 1)(u
4
− u
3
+ u
2
+ 1)
2
(u
4
+ 2u
3
+ ··· + 3u + 2)
c
5
, c
7
(u + 1)
2
(u
3
+ u
2
+ 3u − 1)(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
· (u
4
+ 2u
3
+ u
2
+ 3u + 4)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
c
9
(y + 1)
2
(y
3
+ y
2
+ 3y − 1)(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
4
+ 2y
3
+ y
2
+ 3y + 4)
c
5
, c
7
(y − 1)
2
(y
3
+ 5y
2
+ 11y − 1)(y
4
− 2y
3
− 3y
2
− y + 16)
· (y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
23
