10
99
(K10a
103
)
A knot diagram
1
Linearized knot diagam
5 9 1 8 2 10 4 6 3 7
Solving Sequence
2,9
3
6,10
7 5 1 8 4
c
2
c
9
c
6
c
5
c
1
c
8
c
4
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= ⟨b + u, −2u
7
− u
6
+ 5u
5
+ 3u
4
− 4u
3
+ u
2
+ 2a − 4u − 4, u
8
− 3u
6
+ 3u
4
− 2u
3
+ 2u
2
+ 2u − 1⟩
I
u
2
= ⟨246u
11
− 474u
10
+ ··· + 72b − 283, −1686u
11
+ 3984u
10
+ ··· + 552a + 4645,
3u
12
− 12u
11
+ 14u
10
+ 4u
9
− 20u
8
+ 10u
7
+ 32u
6
− 108u
5
+ 163u
4
− 142u
3
+ 96u
2
− 62u + 23⟩
I
u
3
= ⟨b, a − 1, u
3
− u + 1⟩
I
u
4
= ⟨b − 1, a − u, u
3
− u − 1⟩
I
u
5
= ⟨a
2
+ b + a, a
3
+ 2a
2
+ a + 1, u + 1⟩
I
u
6
= ⟨ba + a − 1, u + 1⟩
I
u
7
= ⟨b − 1, u
2
a − au − 1⟩
I
u
8
= ⟨b − 1, u + 1⟩
I
v
1
= ⟨a, b
3
− b − 1, v − 1⟩
* 6 irreducible components of dim
C
= 0, with total 32 representations.
* 3 irreducible components of dim
C
= 1
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
= ⟨b + u, −2u
7
− u
6
+ · · · + 2a − 4, u
8
− 3u
6
+ 3u
4
− 2u
3
+ 2u
2
+ 2u − 1⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
6
=
u
7
+
1
2
u
6
+ ··· + 2u + 2
−u
a
10
=
−u
−u
3
+ u
a
7
=
u
7
+
1
2
u
6
+ ··· +
3
2
u +
5
2
−
1
2
u
7
+
3
2
u
5
+ ··· −
1
2
u −
1
2
a
5
=
u
7
+
1
2
u
6
+ ··· + u + 2
−u
a
1
=
1
2
u
7
+
1
2
u
6
+ ··· − u
2
+ 2
−u
2
a
8
=
−u
7
+ 3u
5
−
5
2
u
3
+ 2u
2
−
3
2
u −
5
2
1
2
u
7
− u
5
+ u
3
− u
2
+ u +
1
2
a
4
=
1
2
u
3
− u
2
−
1
2
u +
1
2
−
1
2
u
4
+
1
2
u
3
+
1
2
u
2
− u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
7
− 10u
5
+ 2u
4
+ 6u
3
− 12u
2
+ 10u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
9
u
8
− 3u
6
+ 3u
4
− 2u
3
+ 2u
2
+ 2u − 1
c
3
2(2u
8
− 10u
7
+ 23u
6
− 22u
5
− 7u
4
+ 37u
3
− 30u
2
+ 8)
c
4
, c
6
, c
7
c
10
u
8
− 3u
6
+ 3u
4
+ 2u
3
+ 2u
2
− 2u − 1
c
8
2(2u
8
+ 10u
7
+ 23u
6
+ 22u
5
− 7u
4
− 37u
3
− 30u
2
+ 8)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
y
8
− 6y
7
+ 15y
6
− 14y
5
− 5y
4
+ 14y
3
+ 6y
2
− 8y + 1
c
3
, c
8
4(4y
8
− 8y
7
+ ··· − 480y + 64)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.221678 + 0.868597I
a = −0.558946 + 1.189130I
b = −0.221678 − 0.868597I
7.42191 + 3.34562I 7.11001 − 1.68383I
u = 0.221678 − 0.868597I
a = −0.558946 − 1.189130I
b = −0.221678 + 0.868597I
7.42191 − 3.34562I 7.11001 + 1.68383I
u = −0.752536
a = −0.564130
b = 0.752536
−1.28346 −8.36990
u = 1.352820 + 0.318023I
a = −0.432640 + 0.858986I
b = −1.352820 − 0.318023I
−7.42191 − 3.34562I −7.11001 + 1.68383I
u = 1.352820 − 0.318023I
a = −0.432640 − 0.858986I
b = −1.352820 + 0.318023I
−7.42191 + 3.34562I −7.11001 − 1.68383I
u = −1.38933 + 0.55684I
a = 0.396967 + 1.206200I
b = 1.38933 − 0.55684I
14.3343I 0. − 7.84155I
u = −1.38933 − 0.55684I
a = 0.396967 − 1.206200I
b = 1.38933 + 0.55684I
− 14.3343I 0. + 7.84155I
u = 0.382196
a = 2.75337
b = −0.382196
1.28346 8.36990
5
II. I
u
2
= ⟨246u
11
− 474u
10
+ · · · + 72b − 283, −1686u
11
+ 3984u
10
+ · · · +
552a + 4645, 3u
12
− 12u
11
+ · · · − 62u + 23⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
6
=
3.05435u
11
− 7.21739u
10
+ ··· + 25.7808u − 8.41486
−3.41667u
11
+ 6.58333u
10
+ ··· − 21.3194u + 3.93056
a
10
=
−u
−u
3
+ u
a
7
=
−7.48732u
11
+ 25.2409u
10
+ ··· − 113.539u + 63.1407
1.08333u
11
− 0.541667u
10
+ ··· − 1.81944u + 6.80556
a
5
=
−0.362319u
11
− 0.634058u
10
+ ··· + 4.46135u − 4.48430
−3.41667u
11
+ 6.58333u
10
+ ··· − 21.3194u + 3.93056
a
1
=
0.331522u
11
+ 0.423913u
10
+ ··· − 3.47464u + 3.81522
5
4
u
11
−
13
8
u
10
+ ··· +
131
24
u −
1
3
a
8
=
1.41848u
11
− 3.79891u
10
+ ··· + 14.0163u − 6.06522
−1.62500u
11
+ 4.37500u
10
+ ··· − 14.8333u + 7.04167
a
4
=
3.68116u
11
− 12.3080u
10
+ ··· + 55.6027u − 29.5495
1.70833u
11
− 7.79167u
10
+ ··· + 35.5972u − 23.5278
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
1
18
u
11
+
38
9
u
10
−
295
54
u
9
−
187
54
u
8
+
275
54
u
7
−
65
27
u
6
−2u
5
+
71
2
u
4
−
2645
54
u
3
+
1855
54
u
2
−
785
27
u+
484
27
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
9
3(3u
12
− 12u
11
+ ··· − 62u + 23)
c
3
(u
6
+ u
5
+ 2u
4
− u
3
+ 2u
2
+ 3)
2
c
4
, c
6
, c
7
c
10
3(3u
12
+ 12u
11
+ ··· + 62u + 23)
c
8
(u
6
− u
5
+ 2u
4
+ u
3
+ 2u
2
+ 3)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
9(9y
12
− 60y
11
+ ··· + 572y + 529)
c
3
, c
8
(y
6
+ 3y
5
+ 10y
4
+ 13y
3
+ 16y
2
+ 12y + 9)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.079480 + 0.450431I
a = −0.019332 − 0.915767I
b = 0.187861 + 0.726416I
− 4.33667I 0. + 5.70400I
u = 1.079480 − 0.450431I
a = −0.019332 + 0.915767I
b = 0.187861 − 0.726416I
4.33667I 0. − 5.70400I
u = 0.052828 + 1.195260I
a = 0.550361 + 0.680226I
b = −1.171280 − 0.484667I
4.49149 − 8.24229I 3.01193 + 6.51979I
u = 0.052828 − 1.195260I
a = 0.550361 − 0.680226I
b = −1.171280 + 0.484667I
4.49149 + 8.24229I 3.01193 − 6.51979I
u = −0.187861 + 0.726416I
a = 1.15611 − 0.83810I
b = −1.079480 + 0.450431I
4.33667I 0. − 5.70400I
u = −0.187861 − 0.726416I
a = 1.15611 + 0.83810I
b = −1.079480 − 0.450431I
− 4.33667I 0. + 5.70400I
u = 1.171280 + 0.484667I
a = −0.362217 + 0.742191I
b = −0.052828 − 1.195260I
4.49149 − 8.24229I 3.01193 + 6.51979I
u = 1.171280 − 0.484667I
a = −0.362217 − 0.742191I
b = −0.052828 + 1.195260I
4.49149 + 8.24229I 3.01193 − 6.51979I
u = 1.296770 + 0.356378I
a = 0.391471 − 1.079490I
b = 1.41250 + 0.63054I
−4.49149 − 8.24229I −3.01193 + 6.51979I
u = 1.296770 − 0.356378I
a = 0.391471 + 1.079490I
b = 1.41250 − 0.63054I
−4.49149 + 8.24229I −3.01193 − 6.51979I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.41250 + 0.63054I
a = −0.194653 − 0.979163I
b = −1.296770 + 0.356378I
−4.49149 + 8.24229I −3.01193 − 6.51979I
u = −1.41250 − 0.63054I
a = −0.194653 + 0.979163I
b = −1.296770 − 0.356378I
−4.49149 − 8.24229I −3.01193 + 6.51979I
10
III. I
u
3
= ⟨b, a − 1, u
3
− u + 1⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
6
=
1
0
a
10
=
−u
1
a
7
=
u + 1
−1
a
5
=
1
0
a
1
=
1
0
a
8
=
−u
u
a
4
=
−u
2
+ 1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
3
c
2
, c
4
, c
7
c
8
, c
9
u
3
− u + 1
c
3
u
3
+ 2u
2
+ u + 1
c
6
, c
10
(u − 1)
3
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
3
c
2
, c
4
, c
7
c
8
, c
9
y
3
− 2y
2
+ y − 1
c
3
y
3
− 2y
2
− 3y − 1
c
6
, c
10
(y − 1)
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.662359 + 0.562280I
a = 1.00000
b = 0
1.64493 6.00000
u = 0.662359 − 0.562280I
a = 1.00000
b = 0
1.64493 6.00000
u = −1.32472
a = 1.00000
b = 0
1.64493 6.00000
14
IV. I
u
4
= ⟨b − 1, a − u, u
3
− u − 1⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
6
=
u
1
a
10
=
−u
−1
a
7
=
u
1
a
5
=
u + 1
1
a
1
=
−u
−1
a
8
=
−u − 1
−u
2
+ u
a
4
=
−u
2
+ 1
u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u + 1)
3
c
2
, c
3
, c
4
c
7
, c
9
u
3
− u − 1
c
6
, c
10
u
3
c
8
u
3
− 2u
2
+ u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y − 1)
3
c
2
, c
3
, c
4
c
7
, c
9
y
3
− 2y
2
+ y − 1
c
6
, c
10
y
3
c
8
y
3
− 2y
2
− 3y − 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.662359 + 0.562280I
a = −0.662359 + 0.562280I
b = 1.00000
−1.64493 −6.00000
u = −0.662359 − 0.562280I
a = −0.662359 − 0.562280I
b = 1.00000
−1.64493 −6.00000
u = 1.32472
a = 1.32472
b = 1.00000
−1.64493 −6.00000
18
V. I
u
5
= ⟨a
2
+ b + a, a
3
+ 2a
2
+ a + 1, u + 1⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
0
−1
a
3
=
1
1
a
6
=
a
−a
2
− a
a
10
=
1
0
a
7
=
−a
2
−a
2
− a
a
5
=
−a
2
−a
2
− a
a
1
=
−a
2
a
a
8
=
a
2
a
2
+ a
a
4
=
−a
2
−a
2
− a
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
6
, c
10
u
3
− u − 1
c
2
, c
9
(u + 1)
3
c
4
, c
7
u
3
c
8
u
3
− 2u
2
+ u − 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
6
, c
10
y
3
− 2y
2
+ y − 1
c
2
, c
9
(y − 1)
3
c
4
, c
7
y
3
c
8
y
3
− 2y
2
− 3y − 1
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.122561 + 0.744862I
b = 0.662359 − 0.562280I
−1.64493 −6.00000
u = −1.00000
a = −0.122561 − 0.744862I
b = 0.662359 + 0.562280I
−1.64493 −6.00000
u = −1.00000
a = −1.75488
b = −1.32472
−1.64493 −6.00000
22
VI. I
u
6
= ⟨ba + a − 1, u + 1⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
0
−1
a
3
=
1
1
a
6
=
a
b
a
10
=
1
0
a
7
=
b + a
b
a
5
=
b + a
b
a
1
=
−b
2
+ a
−b
2
a
8
=
a
2
−a
a
4
=
−a
2
+ b + a
b + a
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
23
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
24
VII. I
u
7
= ⟨b − 1, u
2
a − au − 1⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
6
=
a
1
a
10
=
−u
−u
3
+ u
a
7
=
a − u
−u
3
+ u + 1
a
5
=
a + 1
1
a
1
=
−a
−1
a
8
=
−a
2
u
−au + u
a
4
=
−a
2
u + 1
−au + u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
25
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
26
VIII. I
u
8
= ⟨b − 1, u + 1⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
0
−1
a
3
=
1
1
a
6
=
a
1
a
10
=
1
0
a
7
=
a + 1
1
a
5
=
a + 1
1
a
1
=
−a
−1
a
8
=
a
2
a − 1
a
4
=
−a
2
+ a + 1
−a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
27
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
28
IX. I
v
1
= ⟨a, b
3
− b − 1, v − 1⟩
(i) Arc colorings
a
2
=
1
0
a
9
=
1
0
a
3
=
1
0
a
6
=
0
b
a
10
=
1
0
a
7
=
b
b
a
5
=
b
b
a
1
=
−b
2
+ 1
−b
2
a
8
=
1
b
2
a
4
=
b + 1
b
2
+ b
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
29
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
8
, c
10
u
3
− u + 1
c
2
, c
9
u
3
c
3
u
3
+ 2u
2
+ u + 1
c
4
, c
7
(u − 1)
3
30
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
8
, c
10
y
3
− 2y
2
+ y − 1
c
2
, c
9
y
3
c
3
y
3
− 2y
2
− 3y − 1
c
4
, c
7
(y − 1)
3
31
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −0.662359 + 0.562280I
1.64493 6.00000
v = 1.00000
a = 0
b = −0.662359 − 0.562280I
1.64493 6.00000
v = 1.00000
a = 0
b = 1.32472
1.64493 6.00000
32
X. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
9
3u
3
(u + 1)
3
(u
3
− u − 1)(u
3
− u + 1)(u
8
− 3u
6
+ ··· + 2u − 1)
· (3u
12
− 12u
11
+ ··· − 62u + 23)
c
3
2(u
3
− u − 1)
2
(u
3
+ 2u
2
+ u + 1)
2
(u
6
+ u
5
+ 2u
4
− u
3
+ 2u
2
+ 3)
2
· (2u
8
− 10u
7
+ 23u
6
− 22u
5
− 7u
4
+ 37u
3
− 30u
2
+ 8)
c
4
, c
6
, c
7
c
10
3u
3
(u − 1)
3
(u
3
− u − 1)(u
3
− u + 1)(u
8
− 3u
6
+ ··· − 2u − 1)
· (3u
12
+ 12u
11
+ ··· + 62u + 23)
c
8
2(u
3
− u + 1)
2
(u
3
− 2u
2
+ u − 1)
2
(u
6
− u
5
+ 2u
4
+ u
3
+ 2u
2
+ 3)
2
· (2u
8
+ 10u
7
+ 23u
6
+ 22u
5
− 7u
4
− 37u
3
− 30u
2
+ 8)
33
XI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
9y
3
(y − 1)
3
(y
3
− 2y
2
+ y − 1)
2
· (y
8
− 6y
7
+ 15y
6
− 14y
5
− 5y
4
+ 14y
3
+ 6y
2
− 8y + 1)
· (9y
12
− 60y
11
+ ··· + 572y + 529)
c
3
, c
8
4(y
3
− 2y
2
− 3y − 1)
2
(y
3
− 2y
2
+ y − 1)
2
· (y
6
+ 3y
5
+ 10y
4
+ 13y
3
+ 16y
2
+ 12y + 9)
2
· (4y
8
− 8y
7
+ 61y
6
− 186y
5
+ 329y
4
− 581y
3
+ 788y
2
− 480y + 64)
34
