10
10
(K10a
64
)
1
Arc Sequences
6 10 9 8 7 2 1 4 3 5
Solving Sequence
2,6
7 1 8 5 4 9 3 10
c
6
c
1
c
7
c
5
c
4
c
8
c
3
c
10
c
2
, c
9
Representation Ideals
I = I
u
1
I
u
1
= hu
22
+ u
21
+ ··· + u + 1i
There are 1 irreducible components with 22 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
22
+ u
21
5u
20
6u
19
+ 12u
18
+ 17u
17
15u
16
28u
15
+ 8u
14
+
28u
13
+ 4u
12
16u
11
8u
10
+ 5u
9
+ 5u
8
2u
7
u
6
+ 3u
5
+ u
4
2u
3
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
7
=
u
u
a
1
=
1
u
2
a
8
=
u
3
u
5
u
3
+ u
a
5
=
u
3
u
3
+ u
a
4
=
u
11
+ 2u
9
2u
7
+ u
3
u
13
3u
11
+ 5u
9
4u
7
+ 2u
5
u
3
+ u
a
9
=
u
19
+ 4u
17
8u
15
+ 8u
13
3u
11
2u
9
+ 2u
7
u
3
u
21
5u
19
+ 13u
17
20u
15
+ 20u
13
13u
11
+ 7u
9
4u
7
+ 3u
5
u
3
+ u
a
3
=
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
10
=
u
8
u
6
+ u
4
+ 1
u
8
+ 2u
6
2u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
20
20u
18
4u
17
+ 52u
16
+ 16u
15
76u
14
32u
13
+ 68u
12
+
32u
11
32u
10
16u
9
+ 12u
8
+ 4u
7
8u
6
8u
5
+ 12u
4
+ 8u
3
4u
2
4u + 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.158641 0.550804I
14.2580 9.1806I 3.12638 + 5.65206I
u = 1.158641 + 0.550804I
14.2580 + 9.1806I 3.12638 5.65206I
u = 1.113887 0.304376I
6.14010 0.06031I 5.06623 0.22454I
u = 1.113887 + 0.304376I
6.14010 + 0.06031I 5.06623 + 0.22454I
u = 1.069939 0.505718I
0.85422 4.78547I 3.13676 + 6.89182I
u = 1.069939 + 0.505718I
0.85422 + 4.78547I 3.13676 6.89182I
u = 0.771950 0.627160I
9.01082 2.42790I 0.88720 + 3.18483I
u = 0.771950 + 0.627160I
9.01082 + 2.42790I 0.88720 3.18483I
u = 0.386678 0.542882I
1.115747 + 0.498475I 8.47948 1.93150I
u = 0.386678 + 0.542882I
1.115747 0.498475I 8.47948 + 1.93150I
u = 0.254770 0.794582I
11.59403 + 4.17420I 0.08704 2.12766I
u = 0.254770 + 0.794582I
11.59403 4.17420I 0.08704 + 2.12766I
u = 0.271243 0.702058I
2.15136 2.90283I 0.96971 + 3.73642I
u = 0.271243 + 0.702058I
2.15136 + 2.90283I 0.96971 3.73642I
u = 0.660123 0.489854I
0.62516 + 1.83614I 2.06876 5.29489I
u = 0.660123 + 0.489854I
0.62516 1.83614I 2.06876 + 5.29489I
u = 1.013803 0.421442I
1.60921 + 1.77285I 0.315441 + 0.247623I
u = 1.013803 + 0.421442I
1.60921 1.77285I 0.315441 0.247623I
u = 1.124835 0.532465I
4.60646 + 7.61506I 2.18846 7.28240I
u = 1.124835 + 0.532465I
4.60646 7.61506I 2.18846 + 7.28240I
u = 1.185860 0.285971I
16.0637 0.8225I 5.38923 0.37902I
u = 1.185860 + 0.285971I
16.0637 + 0.8225I 5.38923 + 0.37902I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
22
+ u
21
+ ··· + u + 1)
c
2
, c
3
, c
4
c
8
, c
9
(u
22
+ u
21
+ ··· + u + 1)
c
5
(u
22
+ 11u
21
+ ··· + u + 1)
c
7
(u
22
+ 3u
21
+ ··· + 9u + 8)
c
10
(u
22
+ u
21
+ ··· + 4u + 4)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
22
11y
21
+ ··· y + 1)
c
2
, c
3
, c
4
c
8
, c
9
(y
22
+ 29y
21
+ ··· y + 1)
c
5
(y
22
+ y
21
+ ··· + 11y + 1)
c
7
(y
22
+ 9y
21
+ ··· + 239y + 64)
c
10
(y
22
+ 5y
21
+ ··· 40y + 16)
5