9
18
(K9a
24
)
1
Arc Sequences
6 1 8 7 9 2 4 5 3
Solving Sequence
1,6
2 3 7 9 5 4 8
c
1
c
2
c
6
c
9
c
5
c
4
c
8
c
3
, c
7
Representation Ideals
I = I
u
1
I
u
1
= hu
20
+ u
19
+ ··· + 3u
2
1i
There are 1 irreducible components with 20 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
20
+ u
19
3u
18
4u
17
+ 7u
16
+ 10u
15
10u
14
18u
13
+ 10u
12
+
23u
11
7u
10
24u
9
+ u
8
+ 18u
7
+ 2u
6
10u
5
3u
4
+ 5u
3
+ 3u
2
1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
7
=
u
u
3
+ u
a
9
=
u
4
u
2
+ 1
u
4
a
5
=
u
9
2u
7
+ 3u
5
2u
3
+ u
u
9
+ u
7
u
5
+ u
a
4
=
u
13
+ 2u
11
3u
9
+ 2u
7
+ u
u
15
+ 3u
13
6u
11
+ 7u
9
6u
7
+ 4u
5
2u
3
+ u
a
8
=
u
14
+ 3u
12
6u
10
+ 7u
8
6u
6
+ 4u
4
2u
2
+ 1
u
14
2u
12
+ 3u
10
2u
8
u
2
a
8
=
u
14
+ 3u
12
6u
10
+ 7u
8
6u
6
+ 4u
4
2u
2
+ 1
u
14
2u
12
+ 3u
10
2u
8
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
18
+ 12u
16
+ 4u
15
28u
14
8u
13
+ 40u
12
+ 20u
11
44u
10
24u
9
+ 36u
8
+ 28u
7
20u
6
24u
5
+ 8u
4
+ 12u
3
12u 10
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.06645
5.55788 16.4403
u = 1.025289 0.696947I
3.28242 10.05773I 6.70834 + 7.26612I
u = 1.025289 + 0.696947I
3.28242 + 10.05773I 6.70834 7.26612I
u = 0.970835 0.584854I
1.26889 2.13456I 8.50898 + 2.16962I
u = 0.970835 + 0.584854I
1.26889 + 2.13456I 8.50898 2.16962I
u = 0.823242 0.609121I
1.66654 2.35832I 6.35225 + 4.49783I
u = 0.823242 + 0.609121I
1.66654 + 2.35832I 6.35225 4.49783I
u = 0.643405 0.793221I
4.43062 + 4.43308I 4.68370 2.52728I
u = 0.643405 + 0.793221I
4.43062 4.43308I 4.68370 + 2.52728I
u = 0.316707 0.594243I
2.76418 2.16136I 4.73748 + 3.31855I
u = 0.316707 + 0.594243I
2.76418 + 2.16136I 4.73748 3.31855I
u = 0.497807
0.680181 14.7621
u = 0.614354 0.699743I
0.324511 0.815726I 9.67172 + 1.07888I
u = 0.614354 + 0.699743I
0.324511 + 0.815726I 9.67172 1.07888I
u = 0.865641 0.753542I
7.97473 + 2.84648I 2.39002 2.97861I
u = 0.865641 + 0.753542I
7.97473 2.84648I 2.39002 + 2.97861I
u = 1.011234 0.657862I
1.48284 + 6.07240I 11.45285 5.87540I
u = 1.011234 + 0.657862I
1.48284 6.07240I 11.45285 + 5.87540I
u = 1.072572 0.081778I
1.65658 + 3.96853I 11.89349 3.79787I
u = 1.072572 + 0.081778I
1.65658 3.96853I 11.89349 + 3.79787I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
20
+ u
19
+ ··· + 3u
2
1)
c
2
, c
9
(u
20
+ 7u
19
+ ··· + 6u + 1)
c
3
, c
4
, c
7
(u
20
+ u
19
+ ··· + 2u 1)
c
5
, c
8
(u
20
+ u
19
+ ··· 4u 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
20
7y
19
+ ··· 6y + 1)
c
2
, c
9
(y
20
+ 13y
19
+ ··· 6y + 1)
c
3
, c
4
, c
7
(y
20
+ 17y
19
+ ··· 6y + 1)
c
5
, c
8
(y
20
11y
19
+ ··· 6y + 1)
5