9
46
(K9n
5
)
A knot diagram
1
Linearized knot diagam
5 7 9 6 1 3 6 4 3
Solving Sequence
3,6
7
1,2
5 4 9 8
c
6
c
2
c
5
c
4
c
9
c
8
c
1
, c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
3
− u
2
+ 2b − 3u + 1, a + 1, u
4
+ 4u
2
− 2u + 1i
I
u
2
= hb − 1, 2a + u − 1, u
2
+ u + 2i
I
u
3
= hb − u, a + 1, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 8 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
= h−u
3
− u
2
+ 2b − 3u + 1, a + 1, u
4
+ 4u
2
− 2u + 1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
−u
2
a
1
=
−1
1
2
u
3
+
1
2
u
2
+
3
2
u −
1
2
a
2
=
−u
u
3
+ u
a
5
=
1
2
u
3
+
1
2
u
2
+
3
2
u +
1
2
1
2
u
3
+
1
2
u
2
+
1
2
u +
1
2
a
4
=
u
1
2
u
3
+
1
2
u
2
+
1
2
u +
1
2
a
9
=
−1
1
2
u
3
−
1
2
u
2
+
3
2
u −
1
2
a
8
=
−u
2
− 1
u
2
a
8
=
−u
2
− 1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
− 14u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
4
+ 3u
3
+ 5u
2
+ 3u + 2
c
2
, c
3
, c
6
c
8
, c
9
u
4
+ 4u
2
− 2u + 1
c
4
u
4
− u
3
+ 11u
2
− 11u + 4
c
7
u
4
+ 8u
3
+ 18u
2
+ 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
4
+ y
3
+ 11y
2
+ 11y + 4
c
2
, c
3
, c
6
c
8
, c
9
y
4
+ 8y
3
+ 18y
2
+ 4y + 1
c
4
y
4
+ 21y
3
+ 107y
2
− 33y + 16
c
7
y
4
− 28y
3
+ 262y
2
+ 20y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.264316 + 0.422125I
a = −1.00000
b = −0.219104 + 0.751390I
0.426736 + 1.175630I 4.79089 − 5.96277I
u = 0.264316 − 0.422125I
a = −1.00000
b = −0.219104 − 0.751390I
0.426736 − 1.175630I 4.79089 + 5.96277I
u = −0.26432 + 1.99036I
a = −1.00000
b = −1.28090 − 1.27441I
−16.8761 − 4.7517I −0.79089 + 2.00586I
u = −0.26432 − 1.99036I
a = −1.00000
b = −1.28090 + 1.27441I
−16.8761 + 4.7517I −0.79089 − 2.00586I
5
II. I
u
2
= hb − 1, 2a + u − 1, u
2
+ u + 2i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
u + 2
a
1
=
−
1
2
u +
1
2
1
a
2
=
−u
2
a
5
=
1
2
u +
1
2
−1
a
4
=
1
2
u +
3
2
−1
a
9
=
−
1
2
u +
1
2
−1
a
8
=
u + 1
−u − 2
a
8
=
u + 1
−u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
2
c
2
, c
3
, c
6
c
8
, c
9
u
2
+ u + 2
c
4
(u + 1)
2
c
7
u
2
+ 3u + 4
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
(y −1)
2
c
2
, c
3
, c
6
c
8
, c
9
y
2
+ 3y + 4
c
7
y
2
− y + 16
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.50000 + 1.32288I
a = 0.750000 − 0.661438I
b = 1.00000
−4.93480 −2.00000
u = −0.50000 − 1.32288I
a = 0.750000 + 0.661438I
b = 1.00000
−4.93480 −2.00000
9
III. I
u
3
= hb − u, a + 1, u
2
+ 1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
1
a
1
=
−1
u
a
2
=
−u
0
a
5
=
u + 1
1
a
4
=
u
1
a
9
=
−1
u + 1
a
8
=
0
1
a
8
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
8
c
9
u
2
+ 1
c
4
, c
7
(u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
8
c
9
(y + 1)
2
c
4
, c
7
(y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −1.00000
b = 1.000000I
−1.64493 0
u = − 1.000000I
a = −1.00000
b = − 1.000000I
−1.64493 0
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
2
(u
2
+ 1)(u
4
+ 3u
3
+ 5u
2
+ 3u + 2)
c
2
, c
3
, c
6
c
8
, c
9
(u
2
+ 1)(u
2
+ u + 2)(u
4
+ 4u
2
− 2u + 1)
c
4
(u + 1)
4
(u
4
− u
3
+ 11u
2
− 11u + 4)
c
7
(u + 1)
2
(u
2
+ 3u + 4)(u
4
+ 8u
3
+ 18u
2
+ 4u + 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y −1)
2
(y + 1)
2
(y
4
+ y
3
+ 11y
2
+ 11y + 4)
c
2
, c
3
, c
6
c
8
, c
9
(y + 1)
2
(y
2
+ 3y + 4)(y
4
+ 8y
3
+ 18y
2
+ 4y + 1)
c
4
(y −1)
4
(y
4
+ 21y
3
+ 107y
2
− 33y + 16)
c
7
(y −1)
2
(y
2
− y + 16)(y
4
− 28y
3
+ 262y
2
+ 20y + 1)
15
