10
42
(K10a
31
)
1
Arc Sequences
5 10 8 6 2 1 9 3 7 4
Solving Sequence
2,6
5 1 7 4 10 3 9 8
c
5
c
1
c
6
c
4
c
10
c
2
c
9
c
8
c
3
, c
7
Representation Ideals
I = I
u
1
I
u
1
= hu
40
+ u
39
+ ··· + 2u + 1i
There are 1 irreducible components with 40 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= hu
40
+ u
39
+ · · · + 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
7
=
u
5
2u
3
+ u
u
5
u
3
+ u
a
4
=
u
u
3
+ u
a
10
=
u
6
u
4
+ 1
u
8
+ 2u
6
2u
4
a
3
=
u
14
3u
12
+ 4u
10
u
8
2u
6
+ 2u
4
+ 1
u
16
+ 4u
14
8u
12
+ 8u
10
4u
8
a
9
=
u
18
+ 5u
16
12u
14
+ 17u
12
15u
10
+ 9u
8
4u
6
+ 2u
4
u
2
+ 1
u
18
+ 4u
16
9u
14
+ 12u
12
11u
10
+ 6u
8
2u
6
u
2
a
8
=
u
31
8u
29
+ ··· + 12u
7
4u
5
u
31
7u
29
+ ··· + 4u
7
+ 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) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.120574 0.575970I
1.06923 13.23985I 0.19703 + 9.63322I
u = 1.120574 + 0.575970I
1.06923 + 13.23985I 0.19703 9.63322I
u = 1.116799 0.540554I
4.40573 7.54884I 5.84455 + 7.16323I
u = 1.116799 + 0.540554I
4.40573 + 7.54884I 5.84455 7.16323I
u = 1.093864 0.474186I
2.46460 1.67611I 4.01967 + 0.72581I
u = 1.093864 + 0.474186I
2.46460 + 1.67611I 4.01967 0.72581I
u = 1.085732 0.202553I
0.384043 + 0.065531I 1.65195 + 0.65182I
u = 1.085732 + 0.202553I
0.384043 0.065531I 1.65195 0.65182I
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 = 0.964797 0.581212I
3.52924 + 1.02826I 3.02738 0.15735I
u = 0.964797 + 0.581212I
3.52924 1.02826I 3.02738 + 0.15735I
u = 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 = 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 = 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 = 0.294391 0.695895I
2.04511 + 2.81020I 2.71121 3.60415I
u = 0.294391 + 0.695895I
2.04511 2.81020I 2.71121 + 3.60415I
u = 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
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 = 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 = 0.571687 0.673264I
5.18330 + 0.15085I 6.02823 0.49618I
u = 0.571687 + 0.673264I
5.18330 0.15085I 6.02823 + 0.49618I
u = 0.991959 0.580881I
3.94345 + 4.71182I 3.76114 5.41408I
u = 0.991959 + 0.580881I
3.94345 4.71182I 3.76114 + 5.41408I
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.075659 0.536322I
0.51656 + 5.28641I 1.70674 5.92677I
u = 1.075659 + 0.536322I
0.51656 5.28641I 1.70674 + 5.92677I
u = 1.109468 0.575876I
2.04477 + 7.46361I 1.61835 4.86663I
u = 1.109468 + 0.575876I
2.04477 7.46361I 1.61835 + 4.86663I
u = 1.110952 0.292389I
6.07985 + 0.03674I 9.04849 + 0.16943I
u = 1.110952 + 0.292389I
6.07985 0.03674I 9.04849 0.16943I
u = 1.112775 0.379878I
3.07700 + 5.78108I 4.88901 6.61715I
u = 1.112775 + 0.379878I
3.07700 5.78108I 4.88901 + 6.61715I
u = 1.120657 0.212549I
1.32786 5.57768I 3.43862 + 4.39035I
u = 1.120657 + 0.212549I
1.32786 + 5.57768I 3.43862 4.39035I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
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)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
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)
5