9
42
(K9n
4
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 9 5 6 7
Solving Sequence
2,6
5 3
1,9
8 4 7
c
5
c
2
c
1
c
8
c
4
c
7
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
4
+ u
3
+ u
2
+ b + 1, u
4
+ u
3
+ u
2
+ a − u + 1, u
5
+ 2u
4
+ 2u
3
+ u + 1i
I
u
2
= hb + 1, a − u + 1, u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 7 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
4
+ u
3
+ u
2
+ b + 1, u
4
+ u
3
+ u
2
+ a − u + 1, u
5
+ 2u
4
+ 2u
3
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
−2u
4
− u
3
− 1
a
9
=
−u
4
− u
3
− u
2
+ u − 1
−u
4
− u
3
− u
2
− 1
a
8
=
u
−u
4
− u
3
− u
2
− 1
a
4
=
−u
3
u
3
+ u
a
7
=
−u
4
− 2u
3
− u
2
+ u − 1
−u
3
− u
2
+ u − 1
a
7
=
−u
4
− 2u
3
− u
2
+ u − 1
−u
3
− u
2
+ u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
4
+ u
3
− 2u
2
− 5u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ 6u
3
+ u − 1
c
2
, c
5
u
5
+ 2u
4
+ 2u
3
+ u + 1
c
3
u
5
− 2u
4
+ 14u
3
+ 16u
2
+ 9u + 9
c
4
, c
7
u
5
− u
4
+ 8u
3
− u
2
− 4u − 4
c
6
, c
8
, c
9
u
5
+ 3u
4
− u
3
− 6u
2
− 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y −1
c
2
, c
5
y
5
+ 6y
3
+ y −1
c
3
y
5
+ 24y
4
+ 278y
3
+ 32y
2
− 207y −81
c
4
, c
7
y
5
+ 15y
4
+ 54y
3
− 73y
2
+ 8y −16
c
6
, c
8
, c
9
y
5
− 11y
4
+ 37y
3
− 30y
2
− 12y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.436447 + 0.655029I
a = 0.423679 + 0.262806I
b = −0.012768 − 0.392223I
−0.057511 + 1.373620I −0.45374 − 4.59823I
u = 0.436447 − 0.655029I
a = 0.423679 − 0.262806I
b = −0.012768 + 0.392223I
−0.057511 − 1.373620I −0.45374 + 4.59823I
u = −0.668466
a = −2.01628
b = −1.34782
2.55277 4.34960
u = −1.10221 + 1.09532I
a = 1.084460 + 0.905094I
b = 2.18668 − 0.19022I
17.6979 − 4.0569I 4.27894 + 1.95729I
u = −1.10221 − 1.09532I
a = 1.084460 − 0.905094I
b = 2.18668 + 0.19022I
17.6979 + 4.0569I 4.27894 − 1.95729I
5
II. I
u
2
= hb + 1, a − u + 1, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u − 1
a
3
=
u
u − 1
a
1
=
−1
0
a
9
=
u − 1
−1
a
8
=
u
−1
a
4
=
1
u − 1
a
7
=
u
−1
a
7
=
u
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u + 5
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
u
2
− u + 1
c
2
u
2
+ u + 1
c
4
, c
7
u
2
c
6
(u + 1)
2
c
8
, c
9
(u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
y
2
+ y + 1
c
4
, c
7
y
2
c
6
, c
8
, c
9
(y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = −1.00000
1.64493 + 2.02988I 3.00000 − 3.46410I
u = 0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = −1.00000
1.64493 − 2.02988I 3.00000 + 3.46410I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
5
+ 6u
3
+ u − 1)
c
2
(u
2
+ u + 1)(u
5
+ 2u
4
+ 2u
3
+ u + 1)
c
3
(u
2
− u + 1)(u
5
− 2u
4
+ 14u
3
+ 16u
2
+ 9u + 9)
c
4
, c
7
u
2
(u
5
− u
4
+ 8u
3
− u
2
− 4u − 4)
c
5
(u
2
− u + 1)(u
5
+ 2u
4
+ 2u
3
+ u + 1)
c
6
(u + 1)
2
(u
5
+ 3u
4
− u
3
− 6u
2
− 1)
c
8
, c
9
(u − 1)
2
(u
5
+ 3u
4
− u
3
− 6u
2
− 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y −1)
c
2
, c
5
(y
2
+ y + 1)(y
5
+ 6y
3
+ y −1)
c
3
(y
2
+ y + 1)(y
5
+ 24y
4
+ 278y
3
+ 32y
2
− 207y −81)
c
4
, c
7
y
2
(y
5
+ 15y
4
+ 54y
3
− 73y
2
+ 8y −16)
c
6
, c
8
, c
9
(y −1)
2
(y
5
− 11y
4
+ 37y
3
− 30y
2
− 12y −1)
11
