10
28
(K10a
44
)
1
Arc Sequences
5 9 8 6 2 1 10 3 4 7
Solving Sequence
2,5
6 1 7 4 10 8 3 9
c
5
c
1
c
6
c
4
c
10
c
7
c
3
c
9
c
2
, 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
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
1
u
2
a
7
=
u
3
u
5
u
3
+ u
a
4
=
u
3
u
3
+ u
a
10
=
u
6
u
4
+ 1
u
8
+ 2u
6
2u
4
a
8
=
u
9
+ 2u
7
u
5
2u
3
+ u
u
11
3u
9
+ 4u
7
u
5
u
3
+ u
a
3
=
u
23
+ 6u
21
16u
19
+ 20u
17
4u
15
22u
13
+ 26u
11
6u
9
9u
7
+ 6u
5
u
25
7u
23
+ ··· 2u
3
+ u
a
9
=
u
14
3u
12
+ 4u
10
u
8
+ 1
u
14
+ 4u
12
7u
10
+ 4u
8
+ 2u
6
4u
4
+ u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
25
32u
23
+ 4u
22
+ 116u
21
28u
20
228u
19
+ 88u
18
+
220u
17
144u
16
+ 16u
15
+ 100u
14
284u
13
+ 52u
12
+ 268u
11
148u
10
20u
9
+ 84u
8
116u
7
+ 20u
6
+ 60u
5
36u
4
+ 4u
3
+ 8u
2
4u 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.260646 0.436852I
14.6036 + 0.7042I 4.80376 + 0.14810I
u = 1.260646 + 0.436852I
14.6036 0.7042I 4.80376 0.14810I
u = 1.232475 0.474736I
7.90858 6.86486I 0.85861 + 6.16378I
u = 1.232475 + 0.474736I
7.90858 + 6.86486I 0.85861 6.16378I
u = 1.098979 0.206450I
7.37246 + 0.32949I 5.60033 + 0.20899I
u = 1.098979 + 0.206450I
7.37246 0.32949I 5.60033 0.20899I
u = 0.963114 0.429790I
0.36195 3.85582I 3.97718 + 7.89236I
u = 0.963114 + 0.429790I
0.36195 + 3.85582I 3.97718 7.89236I
u = 0.445071 0.389205I
1.050413 + 0.215716I 9.69812 1.13318I
u = 0.445071 + 0.389205I
1.050413 0.215716I 9.69812 + 1.13318I
u = 0.027215 0.843903I
4.30846 + 2.13264I 2.18965 3.16032I
u = 0.027215 + 0.843903I
4.30846 2.13264I 2.18965 + 3.16032I
u = 0.051158 0.880772I
10.59634 5.33673I 1.16942 + 2.96646I
u = 0.051158 + 0.880772I
10.59634 + 5.33673I 1.16942 2.96646I
u = 0.311125 0.584230I
3.40769 2.56217I 2.05300 + 2.97329I
u = 0.311125 + 0.584230I
3.40769 + 2.56217I 2.05300 2.97329I
u = 0.720594 0.453573I
2.18139 + 1.93104I 3.25405 4.18474I
u = 0.720594 + 0.453573I
2.18139 1.93104I 3.25405 + 4.18474I
u = 0.932207 0.261463I
1.57798 + 1.00473I 1.82896 0.57498I
u = 0.932207 + 0.261463I
1.57798 1.00473I 1.82896 + 0.57498I
u = 1.030408 0.480033I
5.39158 + 6.75127I 1.33497 7.43906I
u = 1.030408 + 0.480033I
5.39158 6.75127I 1.33497 + 7.43906I
u = 1.237146 0.448499I
8.09804 + 2.43962I 1.44223 0.17519I
u = 1.237146 + 0.448499I
8.09804 2.43962I 1.44223 + 0.17519I
u = 1.244861 0.491994I
14.1992 + 10.2647I 4.13372 5.98641I
u = 1.244861 + 0.491994I
14.1992 10.2647I 4.13372 + 5.98641I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
26
+ u
25
+ ··· + u + 1)
c
2
, c
3
, c
8
(u
26
+ u
25
+ ··· + u + 1)
c
4
(u
26
+ 15u
25
+ ··· + 3u + 1)
c
6
, c
7
, c
10
(u
26
+ 3u
25
+ ··· + 11u + 3)
c
9
(u
26
+ u
25
+ ··· + 13u + 17)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
26
15y
25
+ ··· 3y + 1)
c
2
, c
3
, c
8
(y
26
+ 25y
25
+ ··· 3y + 1)
c
4
(y
26
7y
25
+ ··· + 13y + 1)
c
6
, c
7
, c
10
(y
26
+ 29y
25
+ ··· + 65y + 9)
c
9
(y
26
+ 13y
25
+ ··· + 3129y + 289)
5