9
8
(K9a
8
)
1
Arc Sequences
9 8 6 1 7 4 3 2 5
Solving Sequence
4,6
7 3 8 2 9 1 5
c
6
c
3
c
7
c
2
c
8
c
1
c
5
c
4
, c
9
Representation Ideals
I = I
u
1
I
u
1
= hu
15
u
14
4u
13
+ 5u
12
+ 6u
11
10u
10
+ 7u
8
8u
7
+ 4u
6
+ 6u
5
8u
4
+ 2u
3
+ 2u
2
2u + 1i
There are 1 irreducible components with 15 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
15
u
14
4u
13
+ 5u
12
+ 6u
11
10u
10
+ 7u
8
8u
7
+ 4u
6
+ 6u
5
8u
4
+ 2u
3
+ 2u
2
2u + 1i
(i) Arc colorings
a
4
=
1
0
a
6
=
0
u
a
7
=
u
u
a
3
=
1
u
2
a
8
=
u
3
u
5
u
3
+ u
a
2
=
u
6
u
4
+ 1
u
8
+ 2u
6
2u
4
a
9
=
u
9
+ 2u
7
u
5
2u
3
+ u
u
11
3u
9
+ 4u
7
u
5
u
3
+ u
a
1
=
u
12
3u
10
+ 3u
8
+ 2u
6
4u
4
+ u
2
+ 1
u
14
+ 4u
12
7u
10
+ 4u
8
+ 2u
6
4u
4
+ u
2
a
5
=
u
3
u
3
+ u
a
5
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
13
16u
11
+ 4u
10
+ 28u
9
12u
8
12u
7
+ 16u
6
16u
5
+ 24u
3
8u
2
+ 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.268723 0.457284I
11.97598 1.54935I 7.09602 + 0.66420I
u = 1.268723 + 0.457284I
11.97598 + 1.54935I 7.09602 0.66420I
u = 1.093892 0.311098I
3.39978 1.10849I 7.51398 + 0.68443I
u = 1.093892 + 0.311098I
3.39978 + 1.10849I 7.51398 0.68443I
u = 0.863978
1.25565 8.48380
u = 0.023100 0.900040I
8.02484 3.25615I 3.67133 + 2.40088I
u = 0.023100 + 0.900040I
8.02484 + 3.25615I 3.67133 2.40088I
u = 0.193328 0.557909I
0.02424 1.73642I 0.42769 + 4.08118I
u = 0.193328 + 0.557909I
0.02424 + 1.73642I 0.42769 4.08118I
u = 0.747479 0.391613I
1.24227 + 1.75942I 2.85085 5.01461I
u = 0.747479 + 0.391613I
1.24227 1.75942I 2.85085 + 5.01461I
u = 1.070293 0.443484I
2.41352 + 5.68434I 4.20490 7.47679I
u = 1.070293 + 0.443484I
2.41352 5.68434I 4.20490 + 7.47679I
u = 1.260405 0.482704I
11.7871 + 8.1923I 6.69502 5.35870I
u = 1.260405 + 0.482704I
11.7871 8.1923I 6.69502 + 5.35870I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
7
c
8
(u
15
+ 3u
14
+ ··· + 8u
2
1)
c
3
, c
6
(u
15
+ u
14
+ ··· 2u 1)
c
4
, c
9
(u
15
+ u
14
+ ··· + 2u + 1)
c
5
(u
15
+ 9u
14
+ ··· 4u
2
+ 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
7
c
8
(y
15
+ 19y
14
+ ··· + 16y 1)
c
3
, c
6
(y
15
9y
14
+ ··· + 4y
2
1)
c
4
, c
9
(y
15
+ 3y
14
+ ··· + 8y
2
1)
c
5
(y
15
5y
14
+ ··· + 8y 1)
5