10
65
(K10a
42
)
A knot diagram
1
Linearized knot diagam
5 9 8 6 2 10 1 3 4 7
Solving Sequence
2,9 3,6
5 1 8 4 10 7
c
2
c
5
c
1
c
8
c
3
c
9
c
7
c
4
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
24
+ 2u
23
+ ··· + 4b + 2, 2u
24
− u
23
+ ··· + 4a − 6, u
25
− 2u
24
+ ··· − 4u + 2i
I
u
2
= h−a
2
u
2
+ u
2
a + 2au + b + 2a + 2u, −2a
2
u
2
+ a
3
+ 2u
2
a − 2a
2
+ 3au + 5a + u + 1, u
3
+ u
2
+ 2u + 1i
I
u
3
= hb + 1, 2a + u + 2, u
2
+ 2i
I
v
1
= ha, b − 1, v −1i
* 4 irreducible components of dim
C
= 0, with total 37 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
24
+2u
23
+· · ·+4b+2, 2u
24
−u
23
+· · ·+4a−6, u
25
−2u
24
+· · ·−4u+2i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
6
=
−
1
2
u
24
+
1
4
u
23
+ ··· − 2u +
3
2
1
2
u
24
−
1
2
u
23
+ ··· +
3
2
u −
1
2
a
5
=
−
1
4
u
23
−
5
2
u
21
+ ··· −
1
2
u + 1
1
2
u
24
−
1
2
u
23
+ ··· +
3
2
u −
1
2
a
1
=
−
1
2
u
24
+ u
23
+ ··· − 3u +
3
2
1
4
u
22
−
1
4
u
21
+ ··· −
1
2
u − 1
a
8
=
−u
u
3
+ u
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
10
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
7
=
1
4
u
18
+ 2u
16
+ ··· −
3
2
u +
1
2
1
2
u
11
+
5
2
u
9
+ ··· − u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
24
+ 4u
23
− 26u
22
+ 40u
21
− 138u
20
+ 164u
19
− 384u
18
+
346u
17
− 584u
16
+ 380u
15
− 430u
14
+ 186u
13
− 50u
12
+ 32u
11
+ 80u
10
+ 56u
9
− 74u
8
+
110u
7
− 178u
6
+ 100u
5
− 82u
4
+ 18u
3
+ 24u
2
− 14u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
25
+ 2u
24
+ ··· − u − 3
c
2
, c
3
, c
8
u
25
+ 2u
24
+ ··· − 4u − 2
c
4
u
25
+ 10u
24
+ ··· + 97u + 9
c
6
, c
7
, c
10
u
25
− 2u
24
+ ··· − 5u − 3
c
9
u
25
− 2u
24
+ ··· + 56u − 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
25
− 10y
24
+ ··· + 97y −9
c
2
, c
3
, c
8
y
25
+ 22y
24
+ ··· + 8y −4
c
4
y
25
+ 14y
24
+ ··· + 1561y −81
c
6
, c
7
, c
10
y
25
− 26y
24
+ ··· − 47y −9
c
9
y
25
− 2y
24
+ ··· − 2624y −256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498082 + 0.831864I
a = −0.210637 + 0.234020I
b = 0.969881 − 0.673526I
4.11705 + 3.30443I 6.15585 − 1.80924I
u = −0.498082 − 0.831864I
a = −0.210637 − 0.234020I
b = 0.969881 + 0.673526I
4.11705 − 3.30443I 6.15585 + 1.80924I
u = 0.404191 + 1.026880I
a = 0.433491 + 0.988124I
b = 0.686093 − 0.799024I
4.98459 + 2.21818I 7.23817 − 3.39990I
u = 0.404191 − 1.026880I
a = 0.433491 − 0.988124I
b = 0.686093 + 0.799024I
4.98459 − 2.21818I 7.23817 + 3.39990I
u = −0.814894 + 0.282583I
a = 0.16059 + 1.77022I
b = −1.096790 − 0.679709I
5.88761 − 7.92352I 7.71863 + 6.25521I
u = −0.814894 − 0.282583I
a = 0.16059 − 1.77022I
b = −1.096790 + 0.679709I
5.88761 + 7.92352I 7.71863 − 6.25521I
u = 0.045104 + 1.169880I
a = −0.509198 − 0.822038I
b = 0.611097 + 0.519026I
−2.09053 − 1.42730I 3.69318 + 4.01748I
u = 0.045104 − 1.169880I
a = −0.509198 + 0.822038I
b = 0.611097 − 0.519026I
−2.09053 + 1.42730I 3.69318 − 4.01748I
u = 0.809668 + 0.163514I
a = 0.706041 + 1.184160I
b = −0.516228 − 0.881834I
7.64625 + 2.15851I 10.42476 − 1.29245I
u = 0.809668 − 0.163514I
a = 0.706041 − 1.184160I
b = −0.516228 + 0.881834I
7.64625 − 2.15851I 10.42476 + 1.29245I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.678633 + 0.221561I
a = 0.45061 − 2.11636I
b = −0.976768 + 0.540770I
0.03499 + 4.24383I 4.60496 − 6.78537I
u = 0.678633 − 0.221561I
a = 0.45061 + 2.11636I
b = −0.976768 − 0.540770I
0.03499 − 4.24383I 4.60496 + 6.78537I
u = 0.339400 + 1.358960I
a = −0.547153 − 0.230670I
b = 0.378354 + 0.934639I
2.85055 + 6.29490I 6.20266 − 3.49250I
u = 0.339400 − 1.358960I
a = −0.547153 + 0.230670I
b = 0.378354 − 0.934639I
2.85055 − 6.29490I 6.20266 + 3.49250I
u = 0.276880 + 1.384380I
a = 0.88077 + 1.63584I
b = 1.090160 − 0.576724I
−5.06500 + 7.73599I −0.26723 − 6.67404I
u = 0.276880 − 1.384380I
a = 0.88077 − 1.63584I
b = 1.090160 + 0.576724I
−5.06500 − 7.73599I −0.26723 + 6.67404I
u = 0.11000 + 1.41509I
a = −0.998644 − 0.147362I
b = −1.121400 − 0.226598I
−7.43417 + 0.37131I −4.72924 + 0.01538I
u = 0.11000 − 1.41509I
a = −0.998644 + 0.147362I
b = −1.121400 + 0.226598I
−7.43417 − 0.37131I −4.72924 − 0.01538I
u = 0.245363 + 0.498558I
a = 0.003216 − 0.172185I
b = 0.901860 + 0.293308I
−1.52585 − 1.04428I −1.27127 + 1.42914I
u = 0.245363 − 0.498558I
a = 0.003216 + 0.172185I
b = 0.901860 − 0.293308I
−1.52585 + 1.04428I −1.27127 − 1.42914I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.33191 + 1.42709I
a = 1.00745 − 1.40966I
b = 1.172920 + 0.648513I
0.44144 − 12.07650I 3.42339 + 7.22441I
u = −0.33191 − 1.42709I
a = 1.00745 + 1.40966I
b = 1.172920 − 0.648513I
0.44144 + 12.07650I 3.42339 − 7.22441I
u = −0.06236 + 1.52702I
a = −0.863474 + 0.102113I
b = −0.881284 + 0.447818I
−3.72172 + 1.83282I 3.75932 − 4.01286I
u = −0.06236 − 1.52702I
a = −0.863474 − 0.102113I
b = −0.881284 − 0.447818I
−3.72172 − 1.83282I 3.75932 + 4.01286I
u = −0.403977
a = 1.97386
b = −0.435793
0.909052 12.0940
7
II. I
u
2
=
h−a
2
u
2
+u
2
a+2au+b+2a+2u, −2a
2
u
2
+2u
2
a+· · ·+5a+1, u
3
+u
2
+2u+1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
6
=
a
a
2
u
2
− u
2
a − 2au − 2a − 2u
a
5
=
a
2
u
2
− u
2
a − 2au − a − 2u
a
2
u
2
− u
2
a − 2au − 2a − 2u
a
1
=
a
2
u
2
− a
2
u − u
2
a − a
2
− 4au − 2u
2
− a − 4u − 2
−a
2
u − u
2
a − a
2
− 2au − 2u
2
− 2u − 2
a
8
=
−u
−u
2
− u − 1
a
4
=
u
2
+ 1
−u
2
− u − 1
a
10
=
−1
0
a
7
=
a
2
u
2
− u
2
a − 2au − a − 2u
a
2
u
2
− u
2
a − 2au − 2a − 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
2
+ 4u + 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
10
u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1
c
2
, c
3
, c
8
(u
3
− u
2
+ 2u − 1)
3
c
4
u
9
+ 6u
8
+ 15u
7
+ 21u
6
+ 19u
5
+ 12u
4
+ 7u
3
+ 5u
2
+ 2u + 1
c
9
(u
3
+ u
2
− 1)
3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
10
y
9
− 6y
8
+ 15y
7
− 21y
6
+ 19y
5
− 12y
4
+ 7y
3
− 5y
2
+ 2y −1
c
2
, c
3
, c
8
(y
3
+ 3y
2
+ 2y −1)
3
c
4
y
9
− 6y
8
+ 11y
7
− y
6
+ 11y
5
− 40y
4
− 37y
3
− 21y
2
− 6y −1
c
9
(y
3
− y
2
+ 2y −1)
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −1.110710 + 0.304480I
b = −1.324820 + 0.175904I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = −0.215080 + 1.307140I
a = −0.633796 + 0.350292I
b = 0.376870 − 0.700062I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = −0.215080 + 1.307140I
a = 0.41979 − 1.77933I
b = 0.947946 + 0.524157I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = −0.215080 − 1.307140I
a = −1.110710 − 0.304480I
b = −1.324820 − 0.175904I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = −0.215080 − 1.307140I
a = −0.633796 − 0.350292I
b = 0.376870 + 0.700062I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = −0.215080 − 1.307140I
a = 0.41979 + 1.77933I
b = 0.947946 − 0.524157I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = −0.569840
a = −0.101925
b = 1.26384
1.11345 9.01950
u = −0.569840
a = 1.37568 + 1.52573I
b = −0.631920 − 0.444935I
1.11345 9.01950
u = −0.569840
a = 1.37568 − 1.52573I
b = −0.631920 + 0.444935I
1.11345 9.01950
11
III. I
u
3
= hb + 1, 2a + u + 2, u
2
+ 2i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
2
a
6
=
−
1
2
u − 1
−1
a
5
=
−
1
2
u − 2
−1
a
1
=
−
1
2
u − 1
−1
a
8
=
−u
−u
a
4
=
−1
0
a
10
=
u
u
a
7
=
−
3
2
u − 1
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
(u + 1)
2
c
2
, c
3
, c
8
c
9
u
2
+ 2
c
4
, c
5
, c
6
c
7
(u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
7
, c
10
(y −1)
2
c
2
, c
3
, c
8
c
9
(y + 2)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = −1.000000 − 0.707107I
b = −1.00000
−4.93480 0
u = − 1.414210I
a = −1.000000 + 0.707107I
b = −1.00000
−4.93480 0
15
IV. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
0
a
3
=
1
0
a
6
=
0
1
a
5
=
1
1
a
1
=
0
−1
a
8
=
1
0
a
4
=
1
0
a
10
=
1
0
a
7
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
10
u − 1
c
2
, c
3
, c
8
c
9
u
c
5
, c
6
, c
7
u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
7
, c
10
y −1
c
2
, c
3
, c
8
c
9
y
18
(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
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u + 1)
2
(u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1)
· (u
25
+ 2u
24
+ ··· − u − 3)
c
2
, c
3
, c
8
u(u
2
+ 2)(u
3
− u
2
+ 2u − 1)
3
(u
25
+ 2u
24
+ ··· − 4u − 2)
c
4
((u − 1)
3
)(u
9
+ 6u
8
+ ··· + 2u + 1)
· (u
25
+ 10u
24
+ ··· + 97u + 9)
c
5
(u − 1)
2
(u + 1)(u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1)
· (u
25
+ 2u
24
+ ··· − u − 3)
c
6
, c
7
(u − 1)
2
(u + 1)(u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1)
· (u
25
− 2u
24
+ ··· − 5u − 3)
c
9
u(u
2
+ 2)(u
3
+ u
2
− 1)
3
(u
25
− 2u
24
+ ··· + 56u − 16)
c
10
(u − 1)(u + 1)
2
(u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1)
· (u
25
− 2u
24
+ ··· − 5u − 3)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y −1)
3
)(y
9
− 6y
8
+ ··· + 2y −1)
· (y
25
− 10y
24
+ ··· + 97y −9)
c
2
, c
3
, c
8
y(y + 2)
2
(y
3
+ 3y
2
+ 2y −1)
3
(y
25
+ 22y
24
+ ··· + 8y −4)
c
4
((y −1)
3
)(y
9
− 6y
8
+ ··· − 6y −1)
· (y
25
+ 14y
24
+ ··· + 1561y −81)
c
6
, c
7
, c
10
((y −1)
3
)(y
9
− 6y
8
+ ··· + 2y −1)
· (y
25
− 26y
24
+ ··· − 47y −9)
c
9
y(y + 2)
2
(y
3
− y
2
+ 2y −1)
3
(y
25
− 2y
24
+ ··· − 2624y −256)
21
