9
49
(K9n
8
)
A knot diagram
1
Linearized knot diagam
8 6 8 2 9 2 5 3 5
Solving Sequence
2,8 1,5
4 3 7 6 9
c
1
c
4
c
3
c
7
c
6
c
9
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, a − 1, u
3
− 3u
2
+ 2u + 1i
I
u
2
= hb + u, a
2
− au + 2u + 4, u
2
+ u − 1i
I
u
3
= hu
3
− 3u
2
+ 2b + 3u − 4, −u
3
+ 2u
2
+ 2a − 2u + 3, u
4
− 3u
3
+ 5u
2
− 6u + 4i
I
u
4
= hb
2
− bu + b + 2, a − 1, u
2
+ u − 1i
I
u
5
= hb + u, a + 1, u
3
+ u
2
+ 1i
* 5 irreducible components of dim
C
= 0, with total 18 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
3
− 3u
2
+ 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
1
=
1
−u
2
a
5
=
1
−u
a
4
=
−u + 1
−u
a
3
=
−u + 1
2u
2
− 3u − 1
a
7
=
u
−u
2
+ u
a
6
=
−u
2
+ 2u
−u
2
+ u
a
9
=
−u
2
+ u + 1
u
2
− 2u − 1
a
9
=
−u
2
+ u + 1
u
2
− 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
2
− 15
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
u
3
− 3u
2
+ 2u + 1
c
2
, c
3
, c
5
c
6
, c
8
, c
9
u
3
+ 2u
2
+ 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
y
3
− 5y
2
+ 10y −1
c
2
, c
3
, c
5
c
6
, c
8
, c
9
y
3
+ 2y
2
+ 5y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.324718
a = 1.00000
b = 0.324718
−0.674976 −14.6840
u = 1.66236 + 0.56228I
a = 1.00000
b = −1.66236 − 0.56228I
−1.30745 − 9.42707I −7.65816 + 5.60826I
u = 1.66236 − 0.56228I
a = 1.00000
b = −1.66236 + 0.56228I
−1.30745 + 9.42707I −7.65816 − 5.60826I
5
II. I
u
2
= hb + u, a
2
− au + 2u + 4, u
2
+ u − 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
1
=
1
u − 1
a
5
=
a
−u
a
4
=
a − u
−u
a
3
=
a − u
au − a + u − 1
a
7
=
−au + a − 2u − 2
au − a + u
a
6
=
−u − 2
au − a + u
a
9
=
−au + a + 3
2au − a + 2u − 2
a
9
=
−au + a + 3
2au − a + 2u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4au − 4a + 4u − 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
2
+ u − 1)
2
c
2
, c
5
, c
6
c
9
(u
2
− u + 1)
2
c
3
, c
8
u
4
+ 3u
3
+ 5u
2
+ 6u + 4
c
7
u
4
− 3u
3
+ 5u
2
− 6u + 4
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
2
− 3y + 1)
2
c
2
, c
5
, c
6
c
9
(y
2
+ y + 1)
2
c
3
, c
7
, c
8
y
4
+ y
3
− 3y
2
+ 4y + 16
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0.30902 + 2.26728I
b = −0.618034
3.94784 + 2.02988I −8.00000 − 3.46410I
u = 0.618034
a = 0.30902 − 2.26728I
b = −0.618034
3.94784 − 2.02988I −8.00000 + 3.46410I
u = −1.61803
a = −0.809017 + 0.330792I
b = 1.61803
−3.94784 + 2.02988I −8.00000 − 3.46410I
u = −1.61803
a = −0.809017 − 0.330792I
b = 1.61803
−3.94784 − 2.02988I −8.00000 + 3.46410I
9
III.
I
u
3
= hu
3
− 3u
2
+ 2b + 3u − 4, −u
3
+ 2u
2
+ 2a − 2u + 3, u
4
− 3u
3
+ 5u
2
− 6u + 4i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
1
=
1
−u
2
a
5
=
1
2
u
3
− u
2
+ u −
3
2
−
1
2
u
3
+
3
2
u
2
−
3
2
u + 2
a
4
=
1
2
u
2
−
1
2
u +
1
2
−
1
2
u
3
+
3
2
u
2
−
3
2
u + 2
a
3
=
1
2
u
2
−
1
2
u +
1
2
−
3
2
u
3
+
7
2
u
2
−
9
2
u + 4
a
7
=
−
3
4
u
3
+
7
4
u
2
−
9
4
u + 3
1
2
u
3
−
3
2
u
2
+
5
2
u − 3
a
6
=
−
1
4
u
3
+
1
4
u
2
+
1
4
u
1
2
u
3
−
3
2
u
2
+
5
2
u − 3
a
9
=
−
1
4
u
3
+
1
4
u
2
−
3
4
u + 1
1
2
u
3
−
3
2
u
2
+
3
2
u − 1
a
9
=
−
1
4
u
3
+
1
4
u
2
−
3
4
u + 1
1
2
u
3
−
3
2
u
2
+
3
2
u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
3
− 2u
2
+ 2u − 10
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− 3u
3
+ 5u
2
− 6u + 4
c
2
, c
6
u
4
+ 3u
3
+ 5u
2
+ 6u + 4
c
3
, c
5
, c
8
c
9
(u
2
− u + 1)
2
c
4
, c
7
(u
2
+ u − 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
4
+ y
3
− 3y
2
+ 4y + 16
c
3
, c
5
, c
8
c
9
(y
2
+ y + 1)
2
c
4
, c
7
(y
2
− 3y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.30902 + 0.53523I
a = −1.059020 + 0.433013I
b = 1.61803
−3.94784 − 2.02988I −8.00000 + 3.46410I
u = 1.30902 − 0.53523I
a = −1.059020 − 0.433013I
b = 1.61803
−3.94784 + 2.02988I −8.00000 − 3.46410I
u = 0.19098 + 1.40126I
a = 0.059017 − 0.433013I
b = −0.618034
3.94784 + 2.02988I −8.00000 − 3.46410I
u = 0.19098 − 1.40126I
a = 0.059017 + 0.433013I
b = −0.618034
3.94784 − 2.02988I −8.00000 + 3.46410I
13
IV. I
u
4
= hb
2
− bu + b + 2, a − 1, u
2
+ u − 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
1
=
1
u − 1
a
5
=
1
b
a
4
=
b + 1
b
a
3
=
b + 1
bu + u − 1
a
7
=
u
bu + u
a
6
=
bu + 2u
bu + u
a
9
=
−b + u
u + 1
a
9
=
−b + u
u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4bu + 4u − 10
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
2
+ u − 1)
2
c
2
, c
3
, c
6
c
8
(u
2
− u + 1)
2
c
4
u
4
− 3u
3
+ 5u
2
− 6u + 4
c
5
, c
9
u
4
+ 3u
3
+ 5u
2
+ 6u + 4
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
2
− 3y + 1)
2
c
2
, c
3
, c
6
c
8
(y
2
+ y + 1)
2
c
4
, c
5
, c
9
y
4
+ y
3
− 3y
2
+ 4y + 16
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.00000
b = −0.19098 + 1.40126I
3.94784 − 2.02988I −8.00000 + 3.46410I
u = 0.618034
a = 1.00000
b = −0.19098 − 1.40126I
3.94784 + 2.02988I −8.00000 − 3.46410I
u = −1.61803
a = 1.00000
b = −1.30902 + 0.53523I
−3.94784 + 2.02988I −8.00000 − 3.46410I
u = −1.61803
a = 1.00000
b = −1.30902 − 0.53523I
−3.94784 − 2.02988I −8.00000 + 3.46410I
17
V. I
u
5
= hb + u, a + 1, u
3
+ u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
1
=
1
−u
2
a
5
=
−1
−u
a
4
=
−u − 1
−u
a
3
=
−u − 1
−u − 1
a
7
=
u
u
2
+ u
a
6
=
u
2
+ 2u
u
2
+ u
a
9
=
−u
2
− u + 1
−u
2
+ 1
a
9
=
−u
2
− u + 1
−u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
2
− 3
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
u
3
+ u
2
+ 1
c
2
, c
5
, c
8
u
3
+ u − 1
c
3
, c
6
, c
9
u
3
+ u + 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
y
3
− y
2
− 2y −1
c
2
, c
3
, c
5
c
6
, c
8
, c
9
y
3
+ 2y
2
+ y −1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.232786 + 0.792552I
a = −1.00000
b = −0.232786 − 0.792552I
5.50124 + 1.58317I −1.27815 − 1.10697I
u = 0.232786 − 0.792552I
a = −1.00000
b = −0.232786 + 0.792552I
5.50124 − 1.58317I −1.27815 + 1.10697I
u = −1.46557
a = −1.00000
b = 1.46557
−4.42273 −9.44370
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
7
((u
2
+ u − 1)
4
)(u
3
− 3u
2
+ 2u + 1)(u
3
+ u
2
+ 1)(u
4
− 3u
3
+ ··· − 6u + 4)
c
2
, c
5
, c
8
((u
2
− u + 1)
4
)(u
3
+ u − 1)(u
3
+ 2u
2
+ 3u + 1)(u
4
+ 3u
3
+ ··· + 6u + 4)
c
3
, c
6
, c
9
((u
2
− u + 1)
4
)(u
3
+ u + 1)(u
3
+ 2u
2
+ 3u + 1)(u
4
+ 3u
3
+ ··· + 6u + 4)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
(y
2
− 3y + 1)
4
(y
3
− 5y
2
+ 10y −1)(y
3
− y
2
− 2y −1)
· (y
4
+ y
3
− 3y
2
+ 4y + 16)
c
2
, c
3
, c
5
c
6
, c
8
, c
9
(y
2
+ y + 1)
4
(y
3
+ 2y
2
+ y −1)(y
3
+ 2y
2
+ 5y −1)
· (y
4
+ y
3
− 3y
2
+ 4y + 16)
23
