10
37
(K10a
49
)
1
Arc Sequences
5 1 9 6 2 4 10 3 8 7
Solving Sequence
1,5
2 3 6 4 7 10 8 9
c
1
c
2
c
5
c
4
c
6
c
10
c
7
c
9
c
3
, c
8
Representation Ideals
I = I
u
1
I
u
1
= hu
26
+ u
25
+ ··· u + 1i
There are 1 irreducible components with 26 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
26
+ u
25
+ · · · u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
4
=
u
3
u
5
u
3
+ u
a
7
=
u
5
u
u
7
+ u
5
2u
3
+ u
a
10
=
u
12
u
10
+ 3u
8
2u
6
+ 2u
4
u
2
+ 1
u
14
2u
12
+ 5u
10
6u
8
+ 6u
6
4u
4
+ u
2
a
8
=
u
19
+ 2u
17
6u
15
+ 8u
13
11u
11
+ 10u
9
8u
7
+ 4u
5
3u
3
u
21
+ 3u
19
+ ··· u
3
+ u
a
9
=
u
25
+ 4u
23
+ ··· 6u
3
+ u
u
25
3u
23
+ ··· 7u
5
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
25
+ 16u
23
+ 4u
22
52u
21
12u
20
+ 116u
19
+ 36u
18
204u
17
64u
16
+ 292u
15
+ 96u
14
328u
13
104u
12
+ 296u
11
+ 88u
10
200u
9
40u
8
+
88u
7
+ 8u
6
8u
5
+ 20u
4
8u
3
12u
2
+ 12u + 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.012158 0.254718I
7.03067 0.01867I 5.98123 + 1.03882I
u = 1.012158 + 0.254718I
7.03067 + 0.01867I 5.98123 1.03882I
u = 0.987090 0.785195I
10.9658I 7.61359I
u = 0.987090 + 0.785195I
10.9658I 7.61359I
u = 0.929921 0.812975I
6.82471 6.16497I 5.29314 + 6.39075I
u = 0.929921 + 0.812975I
6.82471 + 6.16497I 5.29314 6.39075I
u = 0.863693 0.835096I
7.03067 + 0.01867I 5.98123 1.03882I
u = 0.863693 + 0.835096I
7.03067 0.01867I 5.98123 + 1.03882I
u = 0.783473 0.854699I
0.63001 + 4.85595I 1.10716 2.80733I
u = 0.783473 + 0.854699I
0.63001 4.85595I 1.10716 + 2.80733I
u = 0.779118 0.130510I
1.314854 0.335766I 6.85384 + 0.55767I
u = 0.779118 + 0.130510I
1.314854 + 0.335766I 6.85384 0.55767I
u = 0.034282 0.657607I
3.75047 2.94952I 0.57746 + 2.74210I
u = 0.034282 + 0.657607I
3.75047 + 2.94952I 0.57746 2.74210I
u = 0.352654 0.410519I
1.314854 0.335766I 6.85384 + 0.55767I
u = 0.352654 + 0.410519I
1.314854 + 0.335766I 6.85384 0.55767I
u = 0.773091 0.826946I
1.11937I 2.31583I
u = 0.773091 + 0.826946I
1.11937I 2.31583I
u = 0.813977 0.362129I
3.36877I 8.60580I
u = 0.813977 + 0.362129I
3.36877I 8.60580I
u = 0.887854 0.783648I
3.75047 + 2.94952I 0.57746 2.74210I
u = 0.887854 + 0.783648I
3.75047 2.94952I 0.57746 + 2.74210I
u = 0.979820 0.768887I
0.63001 + 4.85595I 1.10716 2.80733I
u = 0.979820 + 0.768887I
0.63001 4.85595I 1.10716 + 2.80733I
u = 1.013775 0.289330I
6.82471 + 6.16497I 5.29314 6.39075I
u = 1.013775 + 0.289330I
6.82471 6.16497I 5.29314 + 6.39075I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
5
c
8
(u
26
+ u
25
+ ··· u + 1)
c
2
, c
4
, c
6
c
7
, c
9
, c
10
(u
26
+ 7u
25
+ ··· + 3u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
5
c
8
(y
26
7y
25
+ ··· 3y + 1)
c
2
, c
4
, c
6
c
7
, c
9
, c
10
(y
26
+ 25y
25
+ ··· + 13y + 1)
5