9
10
(K9a
39
)
1
Arc Sequences
7 6 9 8 1 2 3 4 5
Solving Sequence
3,6
2 7 8 1 5 4 9
c
2
c
6
c
7
c
1
c
5
c
4
c
9
c
3
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
6
+ 3u
4
+ u
3
+ 2u
2
+ 2u 1i
I
u
2
= hu
10
u
9
+ 4u
8
4u
7
+ 6u
6
6u
5
+ 3u
4
3u
3
+ 1i
There are 2 irreducible components with 16 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
6
+ 3u
4
+ u
3
+ 2u
2
+ 2u 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
8
=
u
3
2u
u
3
+ u
a
1
=
u
2
+ 1
u
4
2u
2
a
5
=
u
5
+ 2u
3
+ u
u
4
+ 2u
2
a
4
=
u
5
+ u
4
+ 2u
3
+ 2u
2
+ u
u
2
a
9
=
u
5
u
4
+ 2u
3
u
2
+ 1
u
a
9
=
u
5
u
4
+ 2u
3
u
2
+ 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
5
4u
4
8u
3
12u
2
4u 14
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.841864
6.52764 14.6818
u = 0.126468 1.352399I
8.36373 3.39374I 1.63982 + 3.51762I
u = 0.126468 + 1.352399I
8.36373 + 3.39374I 1.63982 3.51762I
u = 0.341865
0.576591 17.1629
u = 0.376468 1.319676I
1.76812 + 8.77346I 6.43784 5.90094I
u = 0.376468 + 1.319676I
1.76812 8.77346I 6.43784 + 5.90094I
3
II. I
u
2
= hu
10
u
9
+ 4u
8
4u
7
+ 6u
6
6u
5
+ 3u
4
3u
3
+ 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
8
=
u
3
2u
u
3
+ u
a
1
=
u
2
+ 1
u
4
2u
2
a
5
=
u
5
+ 2u
3
+ u
u
7
3u
5
2u
3
+ u
a
4
=
2u
9
7u
7
8u
5
+ u
2
+ 4u + 3
u
9
+ 3u
7
+ 3u
5
u
2
u 2
a
9
=
u
8
3u
6
3u
4
+ 1
u
9
+ 4u
7
u
6
+ 6u
5
3u
4
+ 3u
3
3u
2
1
a
9
=
u
8
3u
6
3u
4
+ 1
u
9
+ 4u
7
u
6
+ 6u
5
3u
4
+ 3u
3
3u
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
9
+ 12u
7
+ 12u
5
4u
3
8u 10
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.429649 0.392970I
2.96077 1.53058I 6.51511 + 4.43065I
u = 0.429649 + 0.392970I
2.96077 + 1.53058I 6.51511 4.43065I
u = 0.383851 1.270633I
2.58269 4.40083I 10.74431 + 3.49859I
u = 0.383851 + 1.270633I
2.58269 + 4.40083I 10.74431 3.49859I
u = 0.090539 1.215345I
2.96077 + 1.53058I 6.51511 4.43065I
u = 0.090539 + 1.215345I
2.96077 1.53058I 6.51511 + 4.43065I
u = 0.383413 1.200423I
0.888787 7.48114
u = 0.383413 + 1.200423I
0.888787 7.48114
u = 0.839548 0.070481I
2.58269 + 4.40083I 10.74431 3.49859I
u = 0.839548 + 0.070481I
2.58269 4.40083I 10.74431 + 3.49859I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
3
c
4
, c
6
, c
8
(u
6
+ 3u
4
u
3
+ 2u
2
2u 1)
(u
10
+ u
9
+ 4u
8
+ 4u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 3u
3
+ 1)
c
5
, c
7
, c
9
(u
5
u
4
2u
3
+ u
2
+ u + 1)
2
(u
6
+ 3u
5
+ 2u
4
+ u
3
+ 5u
2
+ 3u 2)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
3
c
4
, c
6
, c
8
(y
6
+ 6y
5
+ 13y
4
+ 9y
3
6y
2
8y + 1)
(y
10
+ 7y
9
+ 20y
8
+ 26y
7
+ 6y
6
22y
5
19y
4
+ 3y
3
+ 6y
2
+ 1)
c
5
, c
7
, c
9
(y
5
5y
4
+ 8y
3
3y
2
y 1)
2
(y
6
5y
5
+ 8y
4
3y
3
+ 11y
2
29y + 4)
7