9
47
(K9n
7
)
A knot diagram
1
Linearized knot diagam
8 9 8 2 3 4 2 6 5
Solving Sequence
3,9 2,6
5 4 7 8 1
c
2
c
5
c
4
c
6
c
8
c
1
c
3
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, a − 1, u
4
+ 2u
3
+ 3u
2
+ u + 1i
I
u
2
= hb + u, a + 1, u
3
− u
2
+ 1i
I
u
3
= hb + u, −u
3
+ 2u
2
+ a − 2u + 1, u
4
− u
3
+ u
2
+ 1i
I
u
4
= h−u
3
− 3u
2
+ b − 4u − 1, −u
3
− 2u
2
+ 2a − u + 3, u
4
+ 4u
3
+ 7u
2
+ 5u + 2i
I
u
5
= h−u
3
+ u
2
+ b − u − 1, a − 1, u
4
− u
3
+ u
2
+ 1i
* 5 irreducible components of dim
C
= 0, with total 19 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, a − 1, u
4
+ 2u
3
+ 3u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
u
2
a
6
=
1
−u
a
5
=
u + 1
−u
a
4
=
−u
3
− u
2
+ 1
−2u
2
− u − 1
a
7
=
−u
3
− u
2
−3u
2
− u − 1
a
8
=
−u
u
2
+ u
a
1
=
−u
3
− 2u
2
− u
u
3
+ u
2
+ u
a
1
=
−u
3
− 2u
2
− u
u
3
+ u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −3u
3
− 9u
2
− 9u − 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
7
u
4
− 3u
3
+ u
2
+ 2u + 1
c
2
, c
5
, c
8
u
4
+ 2u
3
+ 3u
2
+ u + 1
c
3
, c
9
u
4
+ 4u
3
+ 7u
2
+ 5u + 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
7
y
4
− 7y
3
+ 15y
2
− 2y + 1
c
2
, c
5
, c
8
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
c
3
, c
9
y
4
− 2y
3
+ 13y
2
+ 3y + 4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.043315 + 0.641200I
a = 1.00000
b = 0.043315 − 0.641200I
−0.858683 + 1.068330I −5.08685 − 4.49083I
u = −0.043315 − 0.641200I
a = 1.00000
b = 0.043315 + 0.641200I
−0.858683 − 1.068330I −5.08685 + 4.49083I
u = −0.95668 + 1.22719I
a = 1.00000
b = 0.95668 − 1.22719I
−8.18845 − 10.05000I −5.41315 + 5.52365I
u = −0.95668 − 1.22719I
a = 1.00000
b = 0.95668 + 1.22719I
−8.18845 + 10.05000I −5.41315 − 5.52365I
5
II. I
u
2
= hb + u, a + 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
u
2
a
6
=
−1
−u
a
5
=
u − 1
−u
a
4
=
0
−u + 1
a
7
=
−1
−u
2
+ u − 1
a
8
=
−u
−u
2
+ u
a
1
=
u
2
− u + 1
u − 1
a
1
=
u
2
− u + 1
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6u
2
+ 3u
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
3
+ 2u
2
+ u + 1
c
2
, c
5
, c
8
u
3
− u
2
+ 1
c
3
, c
9
u
3
− u + 1
c
7
u
3
− 2u
2
+ u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
7
y
3
− 2y
2
− 3y −1
c
2
, c
5
, c
8
y
3
− y
2
+ 2y −1
c
3
, c
9
y
3
− 2y
2
+ y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −1.00000
b = −0.877439 − 0.744862I
1.45094 + 3.77083I 1.34184 − 5.60826I
u = 0.877439 − 0.744862I
a = −1.00000
b = −0.877439 + 0.744862I
1.45094 − 3.77083I 1.34184 + 5.60826I
u = −0.754878
a = −1.00000
b = 0.754878
−6.19175 −5.68370
9
III. I
u
3
= hb + u, −u
3
+ 2u
2
+ a − 2u + 1, u
4
− u
3
+ u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
u
2
a
6
=
u
3
− 2u
2
+ 2u − 1
−u
a
5
=
u
3
− 2u
2
+ 3u − 1
−u
a
4
=
−2u
2
+ 3u − 2
1
a
7
=
u
3
+ u
2
− 3u + 4
−u
3
+ u
2
− u − 1
a
8
=
−2u
2
+ 3u − 3
1
a
1
=
−u
3
− 2u
2
+ 5u − 5
u
3
+ 1
a
1
=
−u
3
− 2u
2
+ 5u − 5
u
3
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
+ 4u − 10
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
u
4
+ 3u
3
+ u
2
− 2u + 1
c
2
, c
5
u
4
− u
3
+ u
2
+ 1
c
3
(u − 1)
4
c
4
u
4
− 2u
3
+ u
2
− 3u + 4
c
8
u
4
+ 4u
3
+ 7u
2
+ 5u + 2
c
9
u
4
+ 5u
3
+ 12u
2
+ 12u + 8
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
7
y
4
− 7y
3
+ 15y
2
− 2y + 1
c
2
, c
5
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
3
(y −1)
4
c
4
y
4
− 2y
3
− 3y
2
− y + 16
c
8
y
4
− 2y
3
+ 13y
2
+ 3y + 4
c
9
y
4
− y
3
+ 40y
2
+ 48y + 64
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.351808 + 0.720342I
a = −0.40926 + 2.34806I
b = 0.351808 − 0.720342I
−6.79074 − 1.41510I −9.82674 + 4.90874I
u = −0.351808 − 0.720342I
a = −0.40926 − 2.34806I
b = 0.351808 + 0.720342I
−6.79074 + 1.41510I −9.82674 − 4.90874I
u = 0.851808 + 0.911292I
a = −0.590739 − 0.055548I
b = −0.851808 − 0.911292I
0.21101 + 3.16396I −6.17326 − 2.56480I
u = 0.851808 − 0.911292I
a = −0.590739 + 0.055548I
b = −0.851808 + 0.911292I
0.21101 − 3.16396I −6.17326 + 2.56480I
13
IV.
I
u
4
= h−u
3
− 3u
2
+ b − 4u − 1, −u
3
− 2u
2
+ 2a − u + 3, u
4
+ 4u
3
+ 7u
2
+ 5u + 2i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
u
2
a
6
=
1
2
u
3
+ u
2
+
1
2
u −
3
2
u
3
+ 3u
2
+ 4u + 1
a
5
=
−
1
2
u
3
− 2u
2
−
7
2
u −
5
2
u
3
+ 3u
2
+ 4u + 1
a
4
=
1
2
u
3
+ u
2
−
1
2
u −
3
2
u
3
+ 6u
2
+ 7u + 3
a
7
=
5
2
u
3
+ 8u
2
+
19
2
u +
5
2
u
3
+ 7u
2
+ 8u + 5
a
8
=
−
3
2
u
3
− 5u
2
−
13
2
u −
3
2
−u
3
− 4u
2
− 5u − 3
a
1
=
1
2
u
3
+ 2u
2
+
7
2
u +
5
2
−u
3
− 3u
2
− 3u − 1
a
1
=
1
2
u
3
+ 2u
2
+
7
2
u +
5
2
−u
3
− 3u
2
− 3u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
2
+ 8u − 2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
u
4
+ 3u
3
+ u
2
− 2u + 1
c
2
u
4
+ 4u
3
+ 7u
2
+ 5u + 2
c
3
u
4
+ 5u
3
+ 12u
2
+ 12u + 8
c
5
, c
8
u
4
− u
3
+ u
2
+ 1
c
6
u
4
− 2u
3
+ u
2
− 3u + 4
c
9
(u − 1)
4
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
y
4
− 7y
3
+ 15y
2
− 2y + 1
c
2
y
4
− 2y
3
+ 13y
2
+ 3y + 4
c
3
y
4
− y
3
+ 40y
2
+ 48y + 64
c
5
, c
8
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
6
y
4
− 2y
3
− 3y
2
− y + 16
c
9
(y −1)
4
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.452576 + 0.585652I
a = −1.67796 − 0.15778I
b = −0.851808 + 0.911292I
0.21101 − 3.16396I −6.17326 + 2.56480I
u = −0.452576 − 0.585652I
a = −1.67796 + 0.15778I
b = −0.851808 − 0.911292I
0.21101 + 3.16396I −6.17326 − 2.56480I
u = −1.54742 + 1.12087I
a = −0.072042 + 0.413327I
b = 0.351808 + 0.720342I
−6.79074 + 1.41510I −9.82674 − 4.90874I
u = −1.54742 − 1.12087I
a = −0.072042 − 0.413327I
b = 0.351808 − 0.720342I
−6.79074 − 1.41510I −9.82674 + 4.90874I
17
V. I
u
5
= h−u
3
+ u
2
+ b − u − 1, a − 1, u
4
− u
3
+ u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
u
2
a
6
=
1
u
3
− u
2
+ u + 1
a
5
=
−u
3
+ u
2
− u
u
3
− u
2
+ u + 1
a
4
=
−u + 1
1
a
7
=
−u
3
+ u − 1
u
a
8
=
−u
1
a
1
=
u
3
− u
2
+ u
−u
3
+ u
2
− 1
a
1
=
u
3
− u
2
+ u
−u
3
+ u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
+ 4u − 10
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
4
− 2u
3
+ u
2
− 3u + 4
c
2
, c
8
u
4
− u
3
+ u
2
+ 1
c
3
, c
9
(u − 1)
4
c
4
, c
6
u
4
+ 3u
3
+ u
2
− 2u + 1
c
5
u
4
+ 4u
3
+ 7u
2
+ 5u + 2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
4
− 2y
3
− 3y
2
− y + 16
c
2
, c
8
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
3
, c
9
(y −1)
4
c
4
, c
6
y
4
− 7y
3
+ 15y
2
− 2y + 1
c
5
y
4
− 2y
3
+ 13y
2
+ 3y + 4
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.351808 + 0.720342I
a = 1.00000
b = 1.54742 + 1.12087I
−6.79074 − 1.41510I −9.82674 + 4.90874I
u = −0.351808 − 0.720342I
a = 1.00000
b = 1.54742 − 1.12087I
−6.79074 + 1.41510I −9.82674 − 4.90874I
u = 0.851808 + 0.911292I
a = 1.00000
b = 0.452576 + 0.585652I
0.21101 + 3.16396I −6.17326 − 2.56480I
u = 0.851808 − 0.911292I
a = 1.00000
b = 0.452576 − 0.585652I
0.21101 − 3.16396I −6.17326 + 2.56480I
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
(u
3
+ 2u
2
+ u + 1)(u
4
− 3u
3
+ u
2
+ 2u + 1)(u
4
− 2u
3
+ u
2
− 3u + 4)
· (u
4
+ 3u
3
+ u
2
− 2u + 1)
2
c
2
, c
5
, c
8
(u
3
− u
2
+ 1)(u
4
− u
3
+ u
2
+ 1)
2
(u
4
+ 2u
3
+ 3u
2
+ u + 1)
· (u
4
+ 4u
3
+ 7u
2
+ 5u + 2)
c
3
, c
9
((u − 1)
8
)(u
3
− u + 1)(u
4
+ 4u
3
+ ··· + 5u + 2)(u
4
+ 5u
3
+ ··· + 12u + 8)
c
7
(u
3
− 2u
2
+ u − 1)(u
4
− 3u
3
+ u
2
+ 2u + 1)(u
4
− 2u
3
+ u
2
− 3u + 4)
· (u
4
+ 3u
3
+ u
2
− 2u + 1)
2
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
7
(y
3
− 2y
2
− 3y −1)(y
4
− 7y
3
+ ··· − 2y + 1)
3
(y
4
− 2y
3
+ ··· − y + 16)
c
2
, c
5
, c
8
(y
3
− y
2
+ 2y −1)(y
4
− 2y
3
+ ··· + 3y + 4)(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
4
+ 2y
3
+ 7y
2
+ 5y + 1)
c
3
, c
9
(y −1)
8
(y
3
− 2y
2
+ y −1)(y
4
− 2y
3
+ 13y
2
+ 3y + 4)
· (y
4
− y
3
+ 40y
2
+ 48y + 64)
23
