10
35
(K10a
23
)
1
Arc Sequences
6 10 2 8 7 1 5 4 3 9
Solving Sequence
3,10
2 4 9 1 8 5 7 6
c
2
c
3
c
9
c
10
c
8
c
4
c
7
c
5
c
1
, c
6
Representation Ideals
I = I
u
1
I
u
1
= hu
24
u
23
+ ··· + 2u + 1i
There are 1 irreducible components with 24 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
24
u
23
+ · · · + 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
2
=
1
u
2
a
4
=
u
2
+ 1
u
4
a
9
=
u
u
a
1
=
u
3
u
3
+ u
a
8
=
u
7
2u
5
2u
3
2u
u
9
u
7
u
5
+ u
a
5
=
u
12
+ 3u
10
+ 5u
8
+ 6u
6
+ 4u
4
+ 3u
2
+ 1
u
14
+ 2u
12
+ 3u
10
+ 2u
8
u
2
a
7
=
u
17
4u
15
9u
13
14u
11
15u
9
14u
7
10u
5
6u
3
3u
u
19
3u
17
6u
15
7u
13
5u
11
3u
9
+ u
3
+ u
a
6
=
u
22
+ 5u
20
+ ··· + 6u
2
+ 1
u
23
u
22
+ ··· 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
22
+ 4u
21
16u
20
+ 16u
19
44u
18
+ 40u
17
76u
16
+ 68u
15
100u
14
+ 84u
13
104u
12
+ 92u
11
84u
10
+ 80u
9
68u
8
+ 68u
7
32u
6
+ 48u
5
16u
4
+ 20u
3
+ 16u + 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.796791 0.499576I
6.63583 + 1.57218I 0.12166 2.29522I
u = 0.796791 + 0.499576I
6.63583 1.57218I 0.12166 + 2.29522I
u = 0.646340 1.071004I
8.32116 + 3.84160I 2.22402 2.38554I
u = 0.646340 + 1.071004I
8.32116 3.84160I 2.22402 + 2.38554I
u = 0.594609 0.955867I
0.63403 + 3.08008I 2.04297 2.82964I
u = 0.594609 + 0.955867I
0.63403 3.08008I 2.04297 + 2.82964I
u = 0.528833 0.750384I
0.10636 + 1.48443I 1.33713 3.68159I
u = 0.528833 + 0.750384I
0.10636 1.48443I 1.33713 + 3.68159I
u = 0.363279 0.314457I
0.204139 + 1.110190I 3.08627 5.87957I
u = 0.363279 + 0.314457I
0.204139 1.110190I 3.08627 + 5.87957I
u = 0.072875 0.970800I
3.49325 + 2.24409I 5.16388 4.25877I
u = 0.072875 + 0.970800I
3.49325 2.24409I 5.16388 + 4.25877I
u = 0.010886 1.154687I
12.40929 + 3.30322I 5.60088 2.43434I
u = 0.010886 + 1.154687I
12.40929 3.30322I 5.60088 + 2.43434I
u = 0.661510 1.069177I
8.06054 10.39450I 1.68269 + 7.07233I
u = 0.661510 + 1.069177I
8.06054 + 10.39450I 1.68269 7.07233I
u = 0.664372 0.974834I
0.78944 7.34378I 2.03585 + 8.70536I
u = 0.664372 + 0.974834I
0.78944 + 7.34378I 2.03585 8.70536I
u = 0.679491 0.850026I
3.76737 2.61939I 8.11481 + 3.60921I
u = 0.679491 + 0.850026I
3.76737 + 2.61939I 8.11481 3.60921I
u = 0.698704 0.680933I
1.66329 + 2.08350I 4.24893 3.59251I
u = 0.698704 + 0.680933I
1.66329 2.08350I 4.24893 + 3.59251I
u = 0.809538 0.528973I
6.45491 + 4.87894I 0.44407 2.58342I
u = 0.809538 + 0.528973I
6.45491 4.87894I 0.44407 + 2.58342I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
24
+ u
23
+ ··· + 2u + 1)
c
2
, c
9
(u
24
+ u
23
+ ··· 2u + 1)
c
3
, c
10
(u
24
+ 9u
23
+ ··· + 4u + 1)
c
4
, c
5
, c
7
c
8
(u
24
+ 5u
23
+ ··· + 4u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
24
+ 5y
23
+ ··· + 4y + 1)
c
2
, c
9
(y
24
+ 9y
23
+ ··· + 4y + 1)
c
3
, c
10
(y
24
+ 13y
23
+ ··· + 44y + 1)
c
4
, c
5
, c
7
c
8
(y
24
+ 29y
23
+ ··· + 20y + 1)
5