8
10
(K8a
3
)
A knot diagram
1
Linearized knot diagam
7 5 6 8 3 1 4 2
Solving Sequence
4,7
8
2,5
1 6 3
c
7
c
4
c
1
c
6
c
3
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
10
+ 2u
9
+ 5u
8
− 2u
7
− 6u
6
− 7u
4
+ 14u
3
+ u
2
+ 4b − 4u + 2,
2u
10
− u
9
− 5u
8
+ 4u
6
+ u
5
+ 9u
4
− 9u
3
− u
2
+ 4a + 4u − 6,
u
11
− 2u
10
− u
9
+ 3u
8
+ u
7
− 2u
6
+ 4u
5
− 11u
4
+ 9u
3
− u
2
− 2u + 2i
I
u
2
= h−a
2
+ b + 2a − 2, a
3
− 2a
2
+ 3a − 1, u + 1i
I
v
1
= ha, b − 1, v −1i
* 3 irreducible components of dim
C
= 0, with total 15 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−2u
10
+2u
9
+· · ·+4b+2, 2u
10
−u
9
+· · ·+4a−6, u
11
−2u
10
+· · ·−2u+2i
(i) Arc colorings
a
4
=
0
u
a
7
=
1
0
a
8
=
1
−u
2
a
2
=
−
1
2
u
10
+
1
4
u
9
+ ··· − u +
3
2
1
2
u
10
−
1
2
u
9
+ ··· + u −
1
2
a
5
=
u
−u
3
+ u
a
1
=
−
1
4
u
9
+
1
2
u
7
+ ··· −
5
4
u
3
+ 1
1
2
u
10
−
1
2
u
9
+ ··· + u −
1
2
a
6
=
−
1
4
u
10
+
3
4
u
8
+ ··· −
3
2
u +
1
2
1
4
u
10
−
1
4
u
9
+ ··· −
1
2
u − 1
a
3
=
1
4
u
8
−
1
2
u
6
+ ··· −
1
2
u +
1
2
1
2
u
5
−
1
2
u
3
−
1
2
u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
10
+ 6u
8
+ 2u
7
− 6u
6
− 4u
5
− 8u
4
+ 8u
3
+ 10u
2
− 8u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1
c
2
, c
3
, c
5
u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 7u
5
+ 2u
4
+ 7u
3
− 3u
2
− u − 1
c
4
, c
7
u
11
+ 2u
10
− u
9
− 3u
8
+ u
7
+ 2u
6
+ 4u
5
+ 11u
4
+ 9u
3
+ u
2
− 2u − 2
c
8
u
11
+ 4u
10
+ ··· + 11u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
11
− 4y
10
+ ··· + 11y −1
c
2
, c
3
, c
5
y
11
− 12y
10
+ ··· − 5y −1
c
4
, c
7
y
11
− 6y
10
+ ··· + 8y −4
c
8
y
11
+ 8y
10
+ ··· + 67y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.217339 + 1.116860I
a = 0.486755 + 0.161793I
b = 0.850023 − 0.614930I
3.20561 + 2.41892I 4.92816 − 2.88947I
u = −0.217339 − 1.116860I
a = 0.486755 − 0.161793I
b = 0.850023 + 0.614930I
3.20561 − 2.41892I 4.92816 + 2.88947I
u = 1.116820 + 0.404951I
a = 0.06010 − 1.67645I
b = −0.978643 + 0.595733I
0.67123 + 4.69742I 2.91876 − 5.88322I
u = 1.116820 − 0.404951I
a = 0.06010 + 1.67645I
b = −0.978643 − 0.595733I
0.67123 − 4.69742I 2.91876 + 5.88322I
u = 0.323694 + 0.583510I
a = 0.505484 − 0.058656I
b = 0.952018 + 0.226513I
−1.73094 − 0.74196I −3.53927 + 1.11909I
u = 0.323694 − 0.583510I
a = 0.505484 + 0.058656I
b = 0.952018 − 0.226513I
−1.73094 + 0.74196I −3.53927 − 1.11909I
u = 1.38823 + 0.36743I
a = 0.423130 + 0.842208I
b = −0.523691 − 0.948055I
8.61577 + 2.58451I 8.19194 − 1.01660I
u = 1.38823 − 0.36743I
a = 0.423130 − 0.842208I
b = −0.523691 + 0.948055I
8.61577 − 2.58451I 8.19194 + 1.01660I
u = −0.552641
a = 1.53210
b = −0.347303
1.12618 9.42940
u = −1.33508 + 0.61220I
a = −0.241523 + 1.362970I
b = −1.126060 − 0.711355I
6.76952 − 8.65115I 5.78570 + 5.57892I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.33508 − 0.61220I
a = −0.241523 − 1.362970I
b = −1.126060 + 0.711355I
6.76952 + 8.65115I 5.78570 − 5.57892I
6
II. I
u
2
= h−a
2
+ b + 2a − 2, a
3
− 2a
2
+ 3a − 1, u + 1i
(i) Arc colorings
a
4
=
0
−1
a
7
=
1
0
a
8
=
1
−1
a
2
=
a
a
2
− 2a + 2
a
5
=
−1
0
a
1
=
a
2
− a + 2
a
2
− 2a + 2
a
6
=
−a
2
+ 2a − 2
−a
2
+ a − 2
a
3
=
a
2
− a + 2
a
2
− 2a + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
u
3
− u + 1
c
4
, c
7
(u − 1)
3
c
8
u
3
+ 2u
2
+ u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
y
3
− 2y
2
+ y − 1
c
4
, c
7
(y − 1)
3
c
8
y
3
− 2y
2
− 3y − 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.78492 + 1.30714I
b = −0.662359 − 0.562280I
1.64493 6.00000
u = −1.00000
a = 0.78492 − 1.30714I
b = −0.662359 + 0.562280I
1.64493 6.00000
u = −1.00000
a = 0.430160
b = 1.32472
1.64493 6.00000
10
III. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
1
0
a
8
=
1
0
a
2
=
0
1
a
5
=
1
0
a
1
=
1
1
a
6
=
0
−1
a
3
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
u + 1
c
4
, c
7
u
c
5
, c
6
, c
8
u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
8
y − 1
c
4
, c
7
y
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
3
− u + 1)
· (u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1)
c
2
, c
3
(u + 1)(u
3
− u + 1)
· (u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 7u
5
+ 2u
4
+ 7u
3
− 3u
2
− u − 1)
c
4
, c
7
u(u − 1)
3
· (u
11
+ 2u
10
− u
9
− 3u
8
+ u
7
+ 2u
6
+ 4u
5
+ 11u
4
+ 9u
3
+ u
2
− 2u − 2)
c
5
(u − 1)(u
3
− u + 1)
· (u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 7u
5
+ 2u
4
+ 7u
3
− 3u
2
− u − 1)
c
6
(u − 1)(u
3
− u + 1)
· (u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1)
c
8
(u − 1)(u
3
+ 2u
2
+ u + 1)(u
11
+ 4u
10
+ ··· + 11u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y − 1)(y
3
− 2y
2
+ y − 1)(y
11
− 4y
10
+ ··· + 11y − 1)
c
2
, c
3
, c
5
(y − 1)(y
3
− 2y
2
+ y − 1)(y
11
− 12y
10
+ ··· − 5y − 1)
c
4
, c
7
y(y − 1)
3
(y
11
− 6y
10
+ ··· + 8y − 4)
c
8
(y − 1)(y
3
− 2y
2
− 3y − 1)(y
11
+ 8y
10
+ ··· + 67y − 1)
16
