9
36
(K9a
9
)
A knot diagram
1
Linearized knot diagam
8 6 2 9 3 5 1 4 7
Solving Sequence
2,6
3 4 5
7,9
8 1
c
2
c
3
c
5
c
6
c
8
c
1
c
4
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
19
− u
18
+ ··· + b − 1, −u
19
+ 3u
18
+ ··· + a + 2u, u
20
− 2u
19
+ ··· + 2u + 1i
I
u
2
= hb − 1, a + u + 1, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 22 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−u
19
−u
18
+· · ·+b−1, −u
19
+3u
18
+· · ·+a+2u, u
20
−2u
19
+· · ·+2u+1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
−u
2
a
4
=
u
2
+ 1
−u
2
a
5
=
−u
u
3
+ u
a
7
=
−u
3
u
5
+ u
3
+ u
a
9
=
u
19
− 3u
18
+ ··· + u
2
− 2u
u
19
+ u
18
+ ··· + 2u + 1
a
8
=
u
19
− u
18
+ ··· + 3u + 1
u
18
− u
17
+ ··· + u + 1
a
1
=
u
19
− 2u
18
+ ··· + u + 1
u
18
− u
17
+ ··· + 2u + 1
a
1
=
u
19
− 2u
18
+ ··· + u + 1
u
18
− u
17
+ ··· + 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
19
− 5u
18
+ 9u
17
− 18u
16
+ 24u
15
− 43u
14
+ 43u
13
− 64u
12
+
51u
11
− 69u
10
+ 27u
9
− 40u
8
− 10u
7
− 4u
6
− 38u
5
+ 14u
4
− 37u
3
+ 11u
2
− 9u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
9
u
20
+ 3u
19
+ ··· − u − 1
c
2
, c
5
u
20
+ 2u
19
+ ··· − 2u + 1
c
3
, c
6
u
20
+ 6u
19
+ ··· − 2u + 1
c
4
, c
8
u
20
− u
19
+ ··· + 8u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
y
20
− 21y
19
+ ··· − 13y + 1
c
2
, c
5
y
20
+ 6y
19
+ ··· − 2y + 1
c
3
, c
6
y
20
+ 18y
19
+ ··· − 86y + 1
c
4
, c
8
y
20
− 15y
19
+ ··· − 24y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.584423 + 0.858889I
a = 0.243370 + 0.067189I
b = 0.199938 − 0.169761I
0.46628 − 2.30782I 1.88733 + 3.58910I
u = −0.584423 − 0.858889I
a = 0.243370 − 0.067189I
b = 0.199938 + 0.169761I
0.46628 + 2.30782I 1.88733 − 3.58910I
u = −0.178424 + 0.888583I
a = 0.314733 − 0.630728I
b = −0.504299 − 0.392204I
−1.44173 − 1.82256I 0.87459 + 5.12436I
u = −0.178424 − 0.888583I
a = 0.314733 + 0.630728I
b = −0.504299 + 0.392204I
−1.44173 + 1.82256I 0.87459 − 5.12436I
u = 0.792511 + 0.823295I
a = 1.20713 + 1.81447I
b = 0.53718 − 2.43181I
4.53977 + 0.19167I 9.73570 + 0.22109I
u = 0.792511 − 0.823295I
a = 1.20713 − 1.81447I
b = 0.53718 + 2.43181I
4.53977 − 0.19167I 9.73570 − 0.22109I
u = −0.840464
a = −0.636029
b = −0.534560
7.40368 12.6680
u = −0.303359 + 1.135910I
a = −0.484298 + 0.279243I
b = 0.170280 + 0.634831I
3.57238 − 3.88098I 8.06498 + 4.02252I
u = −0.303359 − 1.135910I
a = −0.484298 − 0.279243I
b = 0.170280 − 0.634831I
3.57238 + 3.88098I 8.06498 − 4.02252I
u = 0.914869 + 0.748366I
a = −0.87489 − 1.67983I
b = −0.45672 + 2.19157I
11.87210 − 3.56941I 11.71587 + 1.00735I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.914869 − 0.748366I
a = −0.87489 + 1.67983I
b = −0.45672 − 2.19157I
11.87210 + 3.56941I 11.71587 − 1.00735I
u = −0.791805 + 0.888234I
a = −0.389342 − 0.061647I
b = −0.363039 + 0.297014I
6.53428 − 2.97363I 9.92336 + 2.68538I
u = −0.791805 − 0.888234I
a = −0.389342 + 0.061647I
b = −0.363039 − 0.297014I
6.53428 + 2.97363I 9.92336 − 2.68538I
u = 0.764902 + 0.939137I
a = −1.51148 − 1.52126I
b = −0.27254 + 2.58310I
4.18332 + 5.67427I 8.59597 − 5.66395I
u = 0.764902 − 0.939137I
a = −1.51148 + 1.52126I
b = −0.27254 − 2.58310I
4.18332 − 5.67427I 8.59597 + 5.66395I
u = 0.795971 + 1.032250I
a = 1.43808 + 1.21025I
b = 0.10460 − 2.44777I
10.9814 + 9.8846I 10.38252 − 5.77638I
u = 0.795971 − 1.032250I
a = 1.43808 − 1.21025I
b = 0.10460 + 2.44777I
10.9814 − 9.8846I 10.38252 + 5.77638I
u = 0.175936 + 0.650679I
a = −0.26288 + 1.68135I
b = 1.140270 − 0.124755I
1.21872 + 0.86143I 5.55325 + 0.99952I
u = 0.175936 − 0.650679I
a = −0.26288 − 1.68135I
b = 1.140270 + 0.124755I
1.21872 − 0.86143I 5.55325 − 0.99952I
u = −0.331892
a = 1.27519
b = 0.423225
0.859562 11.8650
6
II. I
u
2
= hb − 1, a + u + 1, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u + 1
a
4
=
−u
u + 1
a
5
=
−u
u + 1
a
7
=
−1
0
a
9
=
−u − 1
1
a
8
=
−u − 1
1
a
1
=
−u
1
a
1
=
−u
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 11
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
9
(u − 1)
2
c
2
, c
3
, c
6
u
2
+ u + 1
c
4
, c
8
u
2
c
5
u
2
− u + 1
c
7
(u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
(y − 1)
2
c
2
, c
3
, c
5
c
6
y
2
+ y + 1
c
4
, c
8
y
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = 1.00000
1.64493 − 2.02988I 9.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = 1.00000
1.64493 + 2.02988I 9.00000 − 3.46410I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
9
((u − 1)
2
)(u
20
+ 3u
19
+ ··· − u − 1)
c
2
(u
2
+ u + 1)(u
20
+ 2u
19
+ ··· − 2u + 1)
c
3
, c
6
(u
2
+ u + 1)(u
20
+ 6u
19
+ ··· − 2u + 1)
c
4
, c
8
u
2
(u
20
− u
19
+ ··· + 8u − 4)
c
5
(u
2
− u + 1)(u
20
+ 2u
19
+ ··· − 2u + 1)
c
7
((u + 1)
2
)(u
20
+ 3u
19
+ ··· − u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
((y − 1)
2
)(y
20
− 21y
19
+ ··· − 13y + 1)
c
2
, c
5
(y
2
+ y + 1)(y
20
+ 6y
19
+ ··· − 2y + 1)
c
3
, c
6
(y
2
+ y + 1)(y
20
+ 18y
19
+ ··· − 86y + 1)
c
4
, c
8
y
2
(y
20
− 15y
19
+ ··· − 24y + 16)
12
