10
42
(K10a
31
)
A knot diagram
1
Linearized knot diagam
5 10 8 6 2 1 9 3 7 4
Solving Sequence
1,5
2 6 7 4 10 3 9 8
c
1
c
5
c
6
c
4
c
10
c
2
c
9
c
8
c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
40
+ u
39
+ ··· + 2u + 1i
* 1 irreducible components of dim
C
= 0, with total 40 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
= hu
40
+ u
39
+ · · · + 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
−u
−u
3
+ u
a
7
=
−u
3
−u
3
+ u
a
4
=
u
3
u
5
− u
3
+ u
a
10
=
u
8
− u
6
+ u
4
+ 1
u
10
− 2u
8
+ 3u
6
− 2u
4
+ u
2
a
3
=
u
16
− 3u
14
+ 5u
12
− 4u
10
+ 3u
8
− 2u
6
+ 2u
4
+ 1
u
18
− 4u
16
+ 9u
14
− 12u
12
+ 11u
10
− 6u
8
+ 2u
6
+ u
2
a
9
=
u
16
− 3u
14
+ 5u
12
− 4u
10
+ 3u
8
− 2u
6
+ 2u
4
+ 1
u
16
− 4u
14
+ 8u
12
− 8u
10
+ 4u
8
a
8
=
−u
29
+ 6u
27
+ ··· − 4u
5
− u
−u
29
+ 7u
27
+ ··· − u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
39
+ 40u
37
+ 4u
36
− 200u
35
− 36u
34
+ 644u
33
+ 164u
32
− 1472u
31
− 484u
30
+
2500u
29
+ 1020u
28
− 3236u
27
− 1616u
26
+ 3252u
25
+ 2004u
24
− 2608u
23
− 2040u
22
+
1752u
21
+ 1812u
20
−1036u
19
−1468u
18
+ 512u
17
+ 1064u
16
−160u
15
−652u
14
−16u
13
+
340u
12
+ 72u
11
−168u
10
−96u
9
+ 68u
8
+ 76u
7
−8u
6
−24u
5
−4u
4
−4u
3
−4u
2
−8u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
40
+ u
39
+ ··· + 2u + 1
c
2
u
40
+ 5u
39
+ ··· + 12u + 1
c
3
, c
8
u
40
− u
39
+ ··· − 2u
3
+ 1
c
4
u
40
+ 19u
39
+ ··· + 2u
2
+ 1
c
6
u
40
+ 3u
39
+ ··· + 61u + 16
c
7
, c
9
u
40
+ 13u
39
+ ··· − 2u
2
+ 1
c
10
u
40
− u
39
+ ··· + 70u + 25
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
40
− 19y
39
+ ··· + 2y
2
+ 1
c
2
y
40
+ y
39
+ ··· + 12y + 1
c
3
, c
8
y
40
+ 13y
39
+ ··· − 2y
2
+ 1
c
4
y
40
+ 5y
39
+ ··· + 4y + 1
c
6
y
40
+ 9y
39
+ ··· + 4695y + 256
c
7
, c
9
y
40
+ 29y
39
+ ··· − 4y + 1
c
10
y
40
− 11y
39
+ ··· − 11300y + 625
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.028980 + 0.356861I
−1.87833 + 1.42866I −1.75477 − 0.64534I
u = −1.028980 − 0.356861I
−1.87833 − 1.42866I −1.75477 + 0.64534I
u = −0.605776 + 0.668794I
4.58827 + 5.88166I 4.65065 − 6.09482I
u = −0.605776 − 0.668794I
4.58827 − 5.88166I 4.65065 + 6.09482I
u = −1.085730 + 0.202553I
−0.384043 − 0.065531I −1.65195 − 0.65182I
u = −1.085730 − 0.202553I
−0.384043 + 0.065531I −1.65195 + 0.65182I
u = 0.571687 + 0.673264I
5.18330 − 0.15085I 6.02823 + 0.49618I
u = 0.571687 − 0.673264I
5.18330 + 0.15085I 6.02823 − 0.49618I
u = −0.964797 + 0.581212I
3.52924 − 1.02826I 3.02738 + 0.15735I
u = −0.964797 − 0.581212I
3.52924 + 1.02826I 3.02738 − 0.15735I
u = 1.120660 + 0.212549I
−1.32786 + 5.57768I −3.43862 − 4.39035I
u = 1.120660 − 0.212549I
−1.32786 − 5.57768I −3.43862 + 4.39035I
u = 1.110950 + 0.292389I
−6.07985 − 0.03674I −9.04849 − 0.16943I
u = 1.110950 − 0.292389I
−6.07985 + 0.03674I −9.04849 + 0.16943I
u = 0.991959 + 0.580881I
3.94345 − 4.71182I 3.76114 + 5.41408I
u = 0.991959 − 0.580881I
3.94345 + 4.71182I 3.76114 − 5.41408I
u = −0.355458 + 0.766083I
3.32299 − 8.17729I 3.05192 + 5.82128I
u = −0.355458 − 0.766083I
3.32299 + 8.17729I 3.05192 − 5.82128I
u = 0.374958 + 0.750172I
4.20581 + 2.43691I 4.87403 − 0.79132I
u = 0.374958 − 0.750172I
4.20581 − 2.43691I 4.87403 + 0.79132I
u = 1.112780 + 0.379878I
−3.07700 − 5.78108I −4.88901 + 6.61715I
u = 1.112780 − 0.379878I
−3.07700 + 5.78108I −4.88901 − 6.61715I
u = −1.093860 + 0.474186I
−2.46460 + 1.67611I −4.01967 − 0.72581I
u = −1.093860 − 0.474186I
−2.46460 − 1.67611I −4.01967 + 0.72581I
u = 1.075660 + 0.536322I
−0.51656 − 5.28641I 1.70674 + 5.92677I
u = 1.075660 − 0.536322I
−0.51656 + 5.28641I 1.70674 − 5.92677I
u = −0.626259 + 0.461310I
−0.75320 + 1.72242I −1.30257 − 5.15094I
u = −0.626259 − 0.461310I
−0.75320 − 1.72242I −1.30257 + 5.15094I
u = −1.116800 + 0.540554I
−4.40573 + 7.54884I −5.84455 − 7.16323I
u = −1.116800 − 0.540554I
−4.40573 − 7.54884I −5.84455 + 7.16323I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.294391 + 0.695895I
−2.04511 − 2.81020I −2.71121 + 3.60415I
u = −0.294391 − 0.695895I
−2.04511 + 2.81020I −2.71121 − 3.60415I
u = 1.109470 + 0.575876I
2.04477 − 7.46361I 1.61835 + 4.86663I
u = 1.109470 − 0.575876I
2.04477 + 7.46361I 1.61835 − 4.86663I
u = −1.120570 + 0.575970I
1.06923 + 13.23980I 0. − 9.63322I
u = −1.120570 − 0.575970I
1.06923 − 13.23980I 0. + 9.63322I
u = 0.404022 + 0.614715I
1.43625 + 0.71721I 6.03452 − 1.24829I
u = 0.404022 − 0.614715I
1.43625 − 0.71721I 6.03452 + 1.24829I
u = −0.079510 + 0.604610I
0.18870 + 2.31784I 0.10490 − 3.06865I
u = −0.079510 − 0.604610I
0.18870 − 2.31784I 0.10490 + 3.06865I
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
40
+ u
39
+ ··· + 2u + 1
c
2
u
40
+ 5u
39
+ ··· + 12u + 1
c
3
, c
8
u
40
− u
39
+ ··· − 2u
3
+ 1
c
4
u
40
+ 19u
39
+ ··· + 2u
2
+ 1
c
6
u
40
+ 3u
39
+ ··· + 61u + 16
c
7
, c
9
u
40
+ 13u
39
+ ··· − 2u
2
+ 1
c
10
u
40
− u
39
+ ··· + 70u + 25
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
40
− 19y
39
+ ··· + 2y
2
+ 1
c
2
y
40
+ y
39
+ ··· + 12y + 1
c
3
, c
8
y
40
+ 13y
39
+ ··· − 2y
2
+ 1
c
4
y
40
+ 5y
39
+ ··· + 4y + 1
c
6
y
40
+ 9y
39
+ ··· + 4695y + 256
c
7
, c
9
y
40
+ 29y
39
+ ··· − 4y + 1
c
10
y
40
− 11y
39
+ ··· − 11300y + 625
8
