8
4
(K8a
17
)
1
Arc Sequences
6 5 8 7 1 2 4 3
Solving Sequence
3,8
4 1 7 5 6 2
c
3
c
8
c
7
c
4
c
5
c
2
c
1
, c
6
Representation Ideals
I = I
u
1
I
u
1
= hu
9
u
8
+ 6u
7
5u
6
+ 11u
5
7u
4
+ 6u
3
2u
2
+ u 1i
There are 1 irreducible components with 9 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
9
u
8
+ 6u
7
5u
6
+ 11u
5
7u
4
+ 6u
3
2u
2
+ u 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
1
=
u
u
a
7
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
u
6
+ 3u
4
+ 2u
2
+ 1
u
6
2u
4
+ u
2
a
2
=
u
6
+ 3u
4
+ 2u
2
+ 1
u
8
+ 4u
6
+ 4u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
7
+ 4u
6
20u
5
+ 16u
4
28u
3
+ 16u
2
8u + 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.290170 0.487341I
0.035384 1.109693I 0.55374 + 6.23947I
u = 0.290170 + 0.487341I
0.035384 + 1.109693I 0.55374 6.23947I
u = 0.05587 1.55975I
7.06362 2.21388I 4.24115 + 3.04598I
u = 0.05587 + 1.55975I
7.06362 + 2.21388I 4.24115 3.04598I
u = 0.12170 1.63384I
13.4612 + 5.5005I 7.48937 2.97298I
u = 0.12170 + 1.63384I
13.4612 5.5005I 7.48937 + 2.97298I
u = 0.429032 0.787939I
5.16280 + 3.41073I 5.88238 4.39642I
u = 0.429032 + 0.787939I
5.16280 3.41073I 5.88238 + 4.39642I
u = 0.590618
2.83680 1.66670
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
(u
9
+ u
8
4u
7
3u
6
+ 5u
5
+ u
4
2u
3
+ 2u
2
+ u + 1)
c
2
(u
9
+ 3u
8
+ 2u
7
5u
6
u
5
+ 13u
4
+ 10u
3
2u
2
+ u + 3)
c
3
, c
4
, c
7
c
8
(u
9
+ u
8
+ 6u
7
+ 5u
6
+ 11u
5
+ 7u
4
+ 6u
3
+ 2u
2
+ u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
(y
9
9y
8
+ 32y
7
55y
6
+ 45y
5
19y
4
+ 16y
3
10y
2
3y 1)
c
2
(y
9
5y
8
+ 32y
7
87y
6
+ 185y
5
223y
4
+ 180y
3
62y
2
+ 13y 9)
c
3
, c
4
, c
7
c
8
(y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
6y
2
3y 1)
5