10
108
(K10a
119
)
A knot diagram
1
Linearized knot diagam
8 6 7 9 2 10 1 5 4 3
Solving Sequence
3,7 4,10
1 8 6 2 5 9
c
3
c
10
c
7
c
6
c
2
c
5
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−11u
13
− 5u
12
+ ··· + 4b + 21, a − 1,
u
14
+ 3u
12
− u
11
+ 10u
10
− 2u
9
+ 12u
8
− 2u
7
+ 12u
6
− u
5
+ 5u
4
− 4u
3
− 2u + 1i
I
u
2
= h−1.14931 × 10
20
u
23
+ 8.36148 × 10
19
u
22
+ ··· + 1.67024 × 10
21
b + 1.69140 × 10
22
,
− 7.94884 × 10
24
u
23
+ 4.40832 × 10
25
u
22
+ ··· + 8.43824 × 10
25
a + 2.32671 × 10
27
,
u
24
+ 3u
23
+ ··· + 6u + 19i
I
u
3
= h−u
6
− u
5
− 3u
3
− 3u
2
+ 2b − u + 1, a + 1, u
7
+ u
5
+ u
4
+ 2u
3
− 1i
* 3 irreducible components of dim
C
= 0, with total 45 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−11u
13
− 5u
12
+ · · · + 4b + 21, a − 1, u
14
+ 3u
12
+ · · · − 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
10
=
1
11
4
u
13
+
5
4
u
12
+ ··· −
1
4
u −
21
4
a
1
=
−
11
4
u
13
−
5
4
u
12
+ ··· +
1
4
u +
25
4
11
4
u
13
+
5
4
u
12
+ ··· −
1
4
u −
21
4
a
8
=
7
2
u
13
+ u
12
+ ··· +
1
2
u − 6
−
9
4
u
13
−
1
4
u
12
+ ··· −
1
4
u +
13
4
a
6
=
−u
−
5
4
u
13
−
3
4
u
12
+ ··· +
3
4
u +
11
4
a
2
=
−
3
4
u
13
−
1
4
u
12
+ ··· +
1
4
u +
9
4
u
13
+
1
2
u
12
+ ··· + u −
7
2
a
5
=
−
7
4
u
13
−
7
4
u
12
+ ··· +
5
4
u +
15
4
−
1
4
u
13
+
1
4
u
12
+ ··· +
3
4
u +
3
4
a
9
=
−
11
4
u
13
−
5
4
u
12
+ ··· +
1
4
u +
25
4
7
2
u
13
+
3
2
u
12
+ ··· −
1
2
u −
13
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
63
4
u
13
−
39
4
u
12
− 49u
11
−
61
4
u
10
−
623
4
u
9
−
289
4
u
8
−
781
4
u
7
−
407
4
u
6
−
843
4
u
5
− 125u
4
−
479
4
u
3
−
23
4
u
2
+
41
4
u +
111
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
u
14
+ u
13
+ ··· + u + 1
c
3
, c
6
u
14
+ 3u
12
+ ··· + 2u + 1
c
4
, c
8
, c
9
u
14
− 7u
13
+ ··· − 56u + 8
c
10
u
14
− 13u
13
+ ··· − 128u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
y
14
− 15y
13
+ ··· + 7y + 1
c
3
, c
6
y
14
+ 6y
13
+ ··· − 4y + 1
c
4
, c
8
, c
9
y
14
+ 13y
13
+ ··· − 32y + 64
c
10
y
14
− 3y
13
+ ··· − 128y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.449224 + 0.834596I
a = 1.00000
b = 1.63158 + 1.27198I
−6.03431 − 1.96052I −7.98588 + 3.63018I
u = −0.449224 − 0.834596I
a = 1.00000
b = 1.63158 − 1.27198I
−6.03431 + 1.96052I −7.98588 − 3.63018I
u = −0.078710 + 0.897903I
a = 1.00000
b = 1.17059 − 1.56821I
−13.45690 − 3.91206I −10.44278 + 2.90737I
u = −0.078710 − 0.897903I
a = 1.00000
b = 1.17059 + 1.56821I
−13.45690 + 3.91206I −10.44278 − 2.90737I
u = −0.605476 + 0.511603I
a = 1.00000
b = 0.410349 + 0.397635I
1.004190 − 0.960325I 4.75919 + 2.76007I
u = −0.605476 − 0.511603I
a = 1.00000
b = 0.410349 − 0.397635I
1.004190 + 0.960325I 4.75919 − 2.76007I
u = 0.777537 + 1.051940I
a = 1.00000
b = 0.686906 − 0.276246I
−4.00326 + 3.48344I 0.15043 − 1.66516I
u = 0.777537 − 1.051940I
a = 1.00000
b = 0.686906 + 0.276246I
−4.00326 − 3.48344I 0.15043 + 1.66516I
u = 0.803725 + 1.091800I
a = 1.00000
b = 1.43309 − 0.98357I
−6.98628 + 8.54350I −5.70825 − 6.73218I
u = 0.803725 − 1.091800I
a = 1.00000
b = 1.43309 + 0.98357I
−6.98628 − 8.54350I −5.70825 + 6.73218I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.497537 + 0.019222I
a = 1.00000
b = −0.118230 + 0.827768I
−0.78382 + 1.56236I −0.68409 − 4.99180I
u = 0.497537 − 0.019222I
a = 1.00000
b = −0.118230 − 0.827768I
−0.78382 − 1.56236I −0.68409 + 4.99180I
u = −0.94539 + 1.37947I
a = 1.00000
b = 1.28571 + 0.96390I
−14.9753 − 12.9046I −7.08862 + 6.20783I
u = −0.94539 − 1.37947I
a = 1.00000
b = 1.28571 − 0.96390I
−14.9753 + 12.9046I −7.08862 − 6.20783I
6
II. I
u
2
= h−1.15 × 10
20
u
23
+ 8.36 × 10
19
u
22
+ · · · + 1.67 × 10
21
b + 1.69 ×
10
22
, −7.95 × 10
24
u
23
+ 4.41 × 10
25
u
22
+ · · · + 8.44 × 10
25
a + 2.33 ×
10
27
, u
24
+ 3u
23
+ · · · + 6u + 19i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
10
=
0.0942002u
23
− 0.522421u
22
+ ··· − 0.770681u − 27.5734
0.0688111u
23
− 0.0500614u
22
+ ··· − 0.114391u − 10.1267
a
1
=
0.0253891u
23
− 0.472360u
22
+ ··· − 0.656290u − 17.4468
0.0688111u
23
− 0.0500614u
22
+ ··· − 0.114391u − 10.1267
a
8
=
0.210439u
23
− 0.295381u
22
+ ··· + 0.388096u − 30.0893
0.142257u
23
+ 0.385777u
22
+ ··· + 6.96971u + 1.18135
a
6
=
0.412085u
23
+ 0.307609u
22
+ ··· + 8.81510u − 23.1359
0.0593893u
23
+ 0.217213u
22
+ ··· + 3.45729u + 5.77204
a
2
=
−1.13112u
23
− 2.14478u
22
+ ··· − 38.0006u + 30.5997
−0.121067u
23
− 0.349984u
22
+ ··· − 8.56109u − 3.83752
a
5
=
−1.76389u
23
− 5.24938u
22
+ ··· − 57.7044u − 1.81964
−0.503760u
23
− 1.53601u
22
+ ··· − 18.2795u − 3.57211
a
9
=
−0.0384844u
23
− 1.08724u
22
+ ··· − 3.69662u − 32.7422
−0.0389677u
23
− 0.469297u
22
+ ··· − 3.63598u − 13.2952
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
6347129273848677988188136
4441180884722885954957147
u
23
−
14886253620454834880565064
4441180884722885954957147
u
22
+
··· −
311699801293938790854796424
4441180884722885954957147
u +
98611701692235728991949218
4441180884722885954957147
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
u
24
− u
23
+ ··· + 12u + 1
c
3
, c
6
u
24
− 3u
23
+ ··· − 6u + 19
c
4
, c
8
, c
9
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
6
c
10
(u
3
+ u
2
− 1)
8
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
y
24
− 21y
23
+ ··· + 220y + 1
c
3
, c
6
y
24
+ 7y
23
+ ··· + 5436y + 361
c
4
, c
8
, c
9
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
6
c
10
(y
3
− y
2
+ 2y −1)
8
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.527689 + 0.759509I
a = −1.56824 − 0.01791I
b = −0.877439 + 0.744862I
−1.69967 + 4.24323I −2.66351 − 7.88819I
u = 0.527689 − 0.759509I
a = −1.56824 + 0.01791I
b = −0.877439 − 0.744862I
−1.69967 − 4.24323I −2.66351 + 7.88819I
u = 0.076109 + 0.834463I
a = −1.29027 + 0.93223I
b = −0.877439 − 0.744862I
−8.70142 + 0.33584I −6.31698 + 0.41465I
u = 0.076109 − 0.834463I
a = −1.29027 − 0.93223I
b = −0.877439 + 0.744862I
−8.70142 − 0.33584I −6.31698 − 0.41465I
u = 0.448386 + 0.692782I
a = −0.608916 − 0.502989I
b = −0.877439 + 0.744862I
−1.69967 + 1.41302I −2.66351 + 1.92930I
u = 0.448386 − 0.692782I
a = −0.608916 + 0.502989I
b = −0.877439 − 0.744862I
−1.69967 − 1.41302I −2.66351 − 1.92930I
u = −0.384009 + 0.725091I
a = 0.33711 − 1.83607I
b = 0.754878
−5.83725 − 1.41510I −9.19277 + 4.90874I
u = −0.384009 − 0.725091I
a = 0.33711 + 1.83607I
b = 0.754878
−5.83725 + 1.41510I −9.19277 − 4.90874I
u = −0.793266 + 0.923818I
a = −1.61039 + 0.28080I
b = −0.877439 − 0.744862I
−8.70142 − 5.99209I −6.31698 + 5.54425I
u = −0.793266 − 0.923818I
a = −1.61039 − 0.28080I
b = −0.877439 + 0.744862I
−8.70142 + 5.99209I −6.31698 − 5.54425I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.090233 + 0.756403I
a = 2.06937 + 2.25178I
b = 0.754878
−12.83900 + 3.16396I −12.84625 − 2.56480I
u = −0.090233 − 0.756403I
a = 2.06937 − 2.25178I
b = 0.754878
−12.83900 − 3.16396I −12.84625 + 2.56480I
u = −0.876115 + 1.005730I
a = −0.509213 + 0.367913I
b = −0.877439 + 0.744862I
−8.70142 − 0.33584I −6.31698 − 0.41465I
u = −0.876115 − 1.005730I
a = −0.509213 − 0.367913I
b = −0.877439 − 0.744862I
−8.70142 + 0.33584I −6.31698 + 0.41465I
u = 0.075432 + 0.647379I
a = −0.976176 − 0.806360I
b = −0.877439 − 0.744862I
−1.69967 − 1.41302I −2.66351 − 1.92930I
u = 0.075432 − 0.647379I
a = −0.976176 + 0.806360I
b = −0.877439 + 0.744862I
−1.69967 + 1.41302I −2.66351 + 1.92930I
u = −0.81394 + 1.20054I
a = −0.637576 − 0.007282I
b = −0.877439 − 0.744862I
−1.69967 − 4.24323I −2.66351 + 7.88819I
u = −0.81394 − 1.20054I
a = −0.637576 + 0.007282I
b = −0.877439 + 0.744862I
−1.69967 + 4.24323I −2.66351 − 7.88819I
u = 1.20186 + 0.94950I
a = 0.096737 + 0.526881I
b = 0.754878
−5.83725 − 1.41510I −9.19277 + 4.90874I
u = 1.20186 − 0.94950I
a = 0.096737 − 0.526881I
b = 0.754878
−5.83725 + 1.41510I −9.19277 − 4.90874I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.01806 + 1.71046I
a = −0.602643 + 0.105082I
b = −0.877439 + 0.744862I
−8.70142 + 5.99209I −6.31698 − 5.54425I
u = 1.01806 − 1.71046I
a = −0.602643 − 0.105082I
b = −0.877439 − 0.744862I
−8.70142 − 5.99209I −6.31698 + 5.54425I
u = −1.88998 + 1.36209I
a = 0.221256 − 0.240760I
b = 0.754878
−12.83900 + 3.16396I 0
u = −1.88998 − 1.36209I
a = 0.221256 + 0.240760I
b = 0.754878
−12.83900 − 3.16396I 0
12
III. I
u
3
= h−u
6
− u
5
− 3u
3
− 3u
2
+ 2b − u + 1, a + 1, u
7
+ u
5
+ u
4
+ 2u
3
− 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
10
=
−1
1
2
u
6
+
1
2
u
5
+ ··· +
1
2
u −
1
2
a
1
=
−
1
2
u
6
−
1
2
u
5
+ ··· −
1
2
u −
1
2
1
2
u
6
+
1
2
u
5
+ ··· +
1
2
u −
1
2
a
8
=
−
3
2
u
6
+
1
2
u
5
+ ··· −
1
2
u −
1
2
u
6
+ u
4
+ u
3
+ 2u
2
+ u
a
6
=
−u
1
2
u
6
−
1
2
u
5
+ ··· +
1
2
u +
1
2
a
2
=
−
1
2
u
6
+
1
2
u
5
+ ··· +
1
2
u +
3
2
1
2
u
6
−
1
2
u
5
+ ··· −
1
2
u −
3
2
a
5
=
u
6
+ u
4
+ 2u
3
+ u
2
+ u
−
1
2
u
6
−
1
2
u
5
+ ··· −
3
2
u +
1
2
a
9
=
−
1
2
u
6
−
1
2
u
5
+ ··· −
1
2
u −
1
2
u
5
− u
4
+ u
3
+ u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
6
+ 2u
5
− u
4
− 3u
3
− 3u
2
+ 3u − 6
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
7
− u
6
− 3u
5
+ 3u
4
+ 3u
3
− 2u
2
− u + 1
c
2
, c
7
u
7
+ u
6
− 3u
5
− 3u
4
+ 3u
3
+ 2u
2
− u − 1
c
3
, c
6
u
7
+ u
5
+ u
4
+ 2u
3
− 1
c
4
u
7
+ 4u
5
+ 4u
3
− u
2
− 1
c
8
, c
9
u
7
+ 4u
5
+ 4u
3
+ u
2
+ 1
c
10
u
7
− 2u
6
+ u
5
+ 2u
4
− 2u
3
+ 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
y
7
− 7y
6
+ 21y
5
− 33y
4
+ 29y
3
− 16y
2
+ 5y −1
c
3
, c
6
y
7
+ 2y
6
+ 5y
5
+ 3y
4
+ 4y
3
+ 2y
2
− 1
c
4
, c
8
, c
9
y
7
+ 8y
6
+ 24y
5
+ 32y
4
+ 16y
3
− y
2
− 2y −1
c
10
y
7
− 2y
6
+ 5y
5
− 8y
4
+ 8y
3
− 4y
2
− 1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.796153 + 0.643678I
a = −1.00000
b = 0.361823 + 0.541221I
−11.54960 + 2.86772I −5.28046 − 0.77527I
u = −0.796153 − 0.643678I
a = −1.00000
b = 0.361823 − 0.541221I
−11.54960 − 2.86772I −5.28046 + 0.77527I
u = −0.271378 + 0.816016I
a = −1.00000
b = −0.905465 − 0.998646I
−2.23879 − 2.27150I −8.12085 + 5.44639I
u = −0.271378 − 0.816016I
a = −1.00000
b = −0.905465 + 0.998646I
−2.23879 + 2.27150I −8.12085 − 5.44639I
u = 0.670242
a = −1.00000
b = 1.07355
−5.45683 −6.44350
u = 0.732410 + 1.178280I
a = −1.00000
b = −0.993133 + 0.472371I
−4.86730 + 3.93356I −8.37695 − 4.94972I
u = 0.732410 − 1.178280I
a = −1.00000
b = −0.993133 − 0.472371I
−4.86730 − 3.93356I −8.37695 + 4.94972I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
7
− u
6
+ ··· − u + 1)(u
14
+ u
13
+ ··· + u + 1)
· (u
24
− u
23
+ ··· + 12u + 1)
c
2
, c
7
(u
7
+ u
6
+ ··· − u − 1)(u
14
+ u
13
+ ··· + u + 1)
· (u
24
− u
23
+ ··· + 12u + 1)
c
3
, c
6
(u
7
+ u
5
+ u
4
+ 2u
3
− 1)(u
14
+ 3u
12
+ ··· + 2u + 1)
· (u
24
− 3u
23
+ ··· − 6u + 19)
c
4
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
6
(u
7
+ 4u
5
+ 4u
3
− u
2
− 1)
· (u
14
− 7u
13
+ ··· − 56u + 8)
c
8
, c
9
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
6
(u
7
+ 4u
5
+ 4u
3
+ u
2
+ 1)
· (u
14
− 7u
13
+ ··· − 56u + 8)
c
10
(u
3
+ u
2
− 1)
8
(u
7
− 2u
6
+ u
5
+ 2u
4
− 2u
3
+ 1)
· (u
14
− 13u
13
+ ··· − 128u + 16)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
(y
7
− 7y
6
+ 21y
5
− 33y
4
+ 29y
3
− 16y
2
+ 5y −1)
· (y
14
− 15y
13
+ ··· + 7y + 1)(y
24
− 21y
23
+ ··· + 220y + 1)
c
3
, c
6
(y
7
+ 2y
6
+ ··· + 2y
2
− 1)(y
14
+ 6y
13
+ ··· − 4y + 1)
· (y
24
+ 7y
23
+ ··· + 5436y + 361)
c
4
, c
8
, c
9
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
6
· (y
7
+ 8y
6
+ 24y
5
+ 32y
4
+ 16y
3
− y
2
− 2y −1)
· (y
14
+ 13y
13
+ ··· − 32y + 64)
c
10
(y
3
− y
2
+ 2y −1)
8
(y
7
− 2y
6
+ 5y
5
− 8y
4
+ 8y
3
− 4y
2
− 1)
· (y
14
− 3y
13
+ ··· − 128y + 256)
18
