8
15
(K8a
2
)
A knot diagram
1
Linearized knot diagam
4 8 5 2 7 1 3 6
Solving Sequence
2,8 3,5
4 1 7 6
c
2
c
4
c
1
c
7
c
5
c
3
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
− 2u
5
+ 3u
4
− 2u
3
+ b + u − 1, −u
6
+ 3u
5
− 4u
4
+ 3u
3
− u
2
+ 2a − u,
u
7
− 3u
6
+ 6u
5
− 7u
4
+ 5u
3
− u
2
− 2u + 2i
I
u
2
= hu
4
a + u
2
a + u
3
− au + b + a + u − 1, −u
3
a − 2u
2
a + u
3
+ a
2
− 2au + u
2
− 2a + u + 1,
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
I
v
1
= ha, b − 1, v + 1i
* 3 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
= hu
6
− 2u
5
+ 3u
4
− 2u
3
+ b + u − 1, −u
6
+ 3u
5
− 4u
4
+ 3u
3
− u
2
+
2a − u, u
7
− 3u
6
+ 6u
5
− 7u
4
+ 5u
3
− u
2
− 2u + 2i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
5
=
1
2
u
6
−
3
2
u
5
+ ··· +
1
2
u
2
+
1
2
u
−u
6
+ 2u
5
− 3u
4
+ 2u
3
− u + 1
a
4
=
−
1
2
u
6
+
1
2
u
5
+ ··· −
1
2
u + 1
−u
6
+ 2u
5
− 3u
4
+ 2u
3
− u + 1
a
1
=
1
2
u
6
−
3
2
u
5
+ ··· +
1
2
u − 1
u
4
− u
3
+ u
2
− 1
a
7
=
u
u
3
+ u
a
6
=
1
2
u
6
−
1
2
u
5
+ ··· +
1
2
u
2
−
1
2
u
u
6
− 2u
5
+ 3u
4
− 3u
3
+ u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
6
− 8u
5
+ 10u
4
− 10u
3
+ 4u − 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
+ u + 1
c
2
, c
7
u
7
− 3u
6
+ 6u
5
− 7u
4
+ 5u
3
− u
2
− 2u + 2
c
3
, c
5
u
7
+ 3u
6
+ 7u
5
+ 8u
4
+ 9u
3
+ 6u
2
+ 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
y
7
− 3y
6
+ 7y
5
− 8y
4
+ 9y
3
− 6y
2
+ 5y −1
c
2
, c
7
y
7
+ 3y
6
+ 4y
5
+ y
4
− y
3
+ 7y
2
+ 8y −4
c
3
, c
5
y
7
+ 5y
6
+ 19y
5
+ 36y
4
+ 49y
3
+ 38y
2
+ 13y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.984140 + 0.426152I
a = 0.472917 + 0.120643I
b = 0.985336 − 0.506466I
−2.09542 + 3.93070I −10.25941 − 4.87230I
u = 0.984140 − 0.426152I
a = 0.472917 − 0.120643I
b = 0.985336 + 0.506466I
−2.09542 − 3.93070I −10.25941 + 4.87230I
u = 0.167785 + 1.218780I
a = 0.529166 − 1.016880I
b = −0.597306 + 0.773845I
3.85236 + 0.95540I −3.31071 − 2.37083I
u = 0.167785 − 1.218780I
a = 0.529166 + 1.016880I
b = −0.597306 − 0.773845I
3.85236 − 0.95540I −3.31071 + 2.37083I
u = 0.654547 + 1.202470I
a = −0.33478 + 1.51279I
b = −1.139460 − 0.630170I
0.36369 − 9.93065I −8.46028 + 7.33664I
u = 0.654547 − 1.202470I
a = −0.33478 − 1.51279I
b = −1.139460 + 0.630170I
0.36369 + 9.93065I −8.46028 − 7.33664I
u = −0.612945
a = 0.665400
b = 0.502855
−0.951399 −9.93920
5
II. I
u
2
= hu
4
a + u
2
a + u
3
− au + b + a + u − 1, −u
3
a + u
3
+ · · · − 2a +
1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
5
=
a
−u
4
a − u
2
a − u
3
+ au − a − u + 1
a
4
=
−u
4
a − u
2
a − u
3
+ au − u + 1
−u
4
a − u
2
a − u
3
+ au − a − u + 1
a
1
=
u
4
a + 2u
2
a + u
3
− au + 2a + u − 1
u
4
a + 2u
2
a + u
3
− au + a + u − 1
a
7
=
u
u
3
+ u
a
6
=
u
4
a − u
4
+ 2u
2
a − u
3
− 2u
2
+ 2a − u − 1
u
4
a + u
3
a − u
4
+ 2u
2
a − 2u
3
+ au − 2u
2
+ a − u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
+ 4u
2
+ 4u − 6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
u
10
− u
9
− 2u
8
+ 4u
7
− 4u
5
+ 3u
4
+ u
3
− 2u
2
+ 1
c
2
, c
7
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
3
, c
5
u
10
+ 5u
9
+ ··· + 4u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
y
10
− 5y
9
+ ··· − 4y + 1
c
2
, c
7
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
3
, c
5
y
10
− y
9
− 6y
7
+ 22y
6
+ 6y
5
+ 45y
4
+ 15y
3
+ 22y
2
+ 4y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 0.445032 + 0.031192I
b = 1.236040 − 0.156723I
−2.96077 − 1.53058I −9.48489 + 4.43065I
u = 0.339110 + 0.822375I
a = 0.46155 + 2.45660I
b = −0.926127 − 0.393188I
−2.96077 − 1.53058I −9.48489 + 4.43065I
u = 0.339110 − 0.822375I
a = 0.445032 − 0.031192I
b = 1.236040 + 0.156723I
−2.96077 + 1.53058I −9.48489 − 4.43065I
u = 0.339110 − 0.822375I
a = 0.46155 − 2.45660I
b = −0.926127 + 0.393188I
−2.96077 + 1.53058I −9.48489 − 4.43065I
u = −0.766826
a = 0.595741 + 0.124010I
b = 0.608868 − 0.334904I
−0.888787 −8.51890
u = −0.766826
a = 0.595741 − 0.124010I
b = 0.608868 + 0.334904I
−0.888787 −8.51890
u = −0.455697 + 1.200150I
a = 0.542114 + 0.781069I
b = −0.400287 − 0.864056I
2.58269 + 4.40083I −5.25569 − 3.49859I
u = −0.455697 + 1.200150I
a = −0.04444 − 1.54938I
b = −1.018500 + 0.644891I
2.58269 + 4.40083I −5.25569 − 3.49859I
u = −0.455697 − 1.200150I
a = 0.542114 − 0.781069I
b = −0.400287 + 0.864056I
2.58269 − 4.40083I −5.25569 + 3.49859I
u = −0.455697 − 1.200150I
a = −0.04444 + 1.54938I
b = −1.018500 − 0.644891I
2.58269 − 4.40083I −5.25569 + 3.49859I
9
III. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
−1
0
a
3
=
1
0
a
5
=
0
1
a
4
=
1
1
a
1
=
0
−1
a
7
=
−1
0
a
6
=
−1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
6
u − 1
c
2
, c
7
u
c
4
, c
8
u + 1
11
(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
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u − 1)(u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
+ u + 1)
· (u
10
− u
9
− 2u
8
+ 4u
7
− 4u
5
+ 3u
4
+ u
3
− 2u
2
+ 1)
c
2
, c
7
u(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
· (u
7
− 3u
6
+ 6u
5
− 7u
4
+ 5u
3
− u
2
− 2u + 2)
c
3
, c
5
(u − 1)(u
7
+ 3u
6
+ 7u
5
+ 8u
4
+ 9u
3
+ 6u
2
+ 5u + 1)
· (u
10
+ 5u
9
+ ··· + 4u + 1)
c
4
, c
8
(u + 1)(u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
+ u + 1)
· (u
10
− u
9
− 2u
8
+ 4u
7
− 4u
5
+ 3u
4
+ u
3
− 2u
2
+ 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
(y −1)(y
7
− 3y
6
+ 7y
5
− 8y
4
+ 9y
3
− 6y
2
+ 5y −1)
· (y
10
− 5y
9
+ ··· − 4y + 1)
c
2
, c
7
y(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
· (y
7
+ 3y
6
+ 4y
5
+ y
4
− y
3
+ 7y
2
+ 8y −4)
c
3
, c
5
(y −1)(y
7
+ 5y
6
+ 19y
5
+ 36y
4
+ 49y
3
+ 38y
2
+ 13y −1)
· (y
10
− y
9
− 6y
7
+ 22y
6
+ 6y
5
+ 45y
4
+ 15y
3
+ 22y
2
+ 4y + 1)
15
