8
20
(K8n
1
)
A knot diagram
1
Linearized knot diagam
4 7 5 2 8 5 2 6
Solving Sequence
2,5
4 1
3,8
7 6
c
4
c
1
c
3
c
7
c
6
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
4
− u
3
− 2u
2
+ b + 1, −u
4
+ u
3
+ 2u
2
+ a − 2, u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 1i
I
u
2
= hb + 1, a, u + 1i
* 2 irreducible components of dim
C
= 0, with total 6 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
−2u
2
+b+1, −u
4
+u
3
+2u
2
+a−2, u
5
−2u
4
−2u
3
+3u
2
+3u−1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
4
=
1
−u
2
a
1
=
u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
2
a
8
=
u
4
− u
3
− 2u
2
+ 2
−u
4
+ u
3
+ 2u
2
− 1
a
7
=
u
4
− u
3
− 2u
2
+ 2
u
4
− 2u
2
− 2u
a
6
=
−u
3
+ 2u + 2
u
4
− 2u
2
− 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
− 6u
3
− 8u
2
+ 6u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 1
c
2
, c
7
u
5
+ u
4
+ 5u
3
+ u
2
+ 2u − 2
c
3
u
5
+ 8u
4
+ 22u
3
+ 25u
2
+ 15u + 1
c
5
, c
8
u
5
+ 2u
4
+ 2u
3
− u
2
− u − 1
c
6
u
5
+ 6u
3
− u
2
− u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
5
− 8y
4
+ 22y
3
− 25y
2
+ 15y −1
c
2
, c
7
y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y −4
c
3
y
5
− 20y
4
+ 114y
3
+ 19y
2
+ 175y −1
c
5
, c
8
y
5
+ 6y
3
− y
2
− y −1
c
6
y
5
+ 12y
4
+ 34y
3
− 13y
2
− y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.949895 + 0.441667I
a = 0.682871 − 0.618084I
b = 0.317129 + 0.618084I
−1.85138 + 1.10891I −2.36548 − 2.04112I
u = −0.949895 − 0.441667I
a = 0.682871 + 0.618084I
b = 0.317129 − 0.618084I
−1.85138 − 1.10891I −2.36548 + 2.04112I
u = 0.274898
a = 1.83380
b = −0.833800
1.20365 8.94300
u = 1.81245 + 0.17314I
a = −0.099771 + 1.129450I
b = 1.09977 − 1.12945I
−11.90990 − 4.12490I −1.10604 + 2.15443I
u = 1.81245 − 0.17314I
a = −0.099771 − 1.129450I
b = 1.09977 + 1.12945I
−11.90990 + 4.12490I −1.10604 − 2.15443I
5
II. I
u
2
= hb + 1, a, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
4
=
1
−1
a
1
=
−1
0
a
3
=
0
−1
a
8
=
0
−1
a
7
=
0
−1
a
6
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
8
u − 1
c
2
, c
7
u
c
4
, c
5
u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
8
y −1
c
2
, c
7
y
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = −1.00000
0 0
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 1)
c
2
, c
7
u(u
5
+ u
4
+ 5u
3
+ u
2
+ 2u − 2)
c
3
(u − 1)(u
5
+ 8u
4
+ 22u
3
+ 25u
2
+ 15u + 1)
c
4
(u + 1)(u
5
− 2u
4
− 2u
3
+ 3u
2
+ 3u − 1)
c
5
(u + 1)(u
5
+ 2u
4
+ 2u
3
− u
2
− u − 1)
c
6
(u − 1)(u
5
+ 6u
3
− u
2
− u − 1)
c
8
(u − 1)(u
5
+ 2u
4
+ 2u
3
− u
2
− u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y −1)(y
5
− 8y
4
+ 22y
3
− 25y
2
+ 15y −1)
c
2
, c
7
y(y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y −4)
c
3
(y −1)(y
5
− 20y
4
+ 114y
3
+ 19y
2
+ 175y −1)
c
5
, c
8
(y −1)(y
5
+ 6y
3
− y
2
− y −1)
c
6
(y −1)(y
5
+ 12y
4
+ 34y
3
− 13y
2
− y −1)
11
