9
27
(K9a
12
)
1
Arc Sequences
5 4 7 8 9 2 3 1 6
Solving Sequence
3,8
7 4 5 2 1 6 9
c
7
c
3
c
4
c
2
c
1
c
6
c
9
c
5
, c
8
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
8
=
0
u
a
7
=
u
u
a
4
=
u
2
+ 1
u
2
a
5
=
u
2
+ 1
u
4
a
2
=
u
4
+ u
2
+ 1
u
4
a
1
=
u
10
3u
8
4u
6
u
4
+ u
2
+ 1
u
12
+ 2u
10
+ 2u
8
+ u
4
a
6
=
u
7
2u
5
2u
3
u
7
u
5
+ u
a
9
=
u
21
6u
19
17u
17
26u
15
20u
13
+ 13u
9
+ 10u
7
+ u
5
2u
3
u
u
23
+ 5u
21
+ 12u
19
+ 15u
17
+ 10u
15
+ 2u
13
u
9
u
7
u
5
+ u
a
9
=
u
21
6u
19
17u
17
26u
15
20u
13
+ 13u
9
+ 10u
7
+ u
5
2u
3
u
u
23
+ 5u
21
+ 12u
19
+ 15u
17
+ 10u
15
+ 2u
13
u
9
u
7
u
5
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
23
4u
22
24u
21
20u
20
68u
19
52u
18
108u
17
80u
16
96u
15
84u
14
32u
13
52u
12
+ 24u
11
8u
10
+ 32u
9
+ 28u
8
+ 16u
7
+ 20u
6
+ 4u
4
+ 4u
3
4u
2
4u 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.796432 0.144602I
0.79700 + 6.17959I 1.78521 5.04555I
u = 0.796432 + 0.144602I
0.79700 6.17959I 1.78521 + 5.04555I
u = 0.580381 0.259924I
1.74298 0.40841I 5.87200 + 0.75563I
u = 0.580381 + 0.259924I
1.74298 + 0.40841I 5.87200 0.75563I
u = 0.512242 1.189925I
3.87224 10.99998I 1.31825 + 8.05284I
u = 0.512242 + 1.189925I
3.87224 + 10.99998I 1.31825 8.05284I
u = 0.472424 1.121720I
0.74814 3.77265I 1.89193 + 3.49106I
u = 0.472424 + 1.121720I
0.74814 + 3.77265I 1.89193 3.49106I
u = 0.414627 0.808476I
0.05596 1.77225I 0.01088 + 4.04184I
u = 0.414627 + 0.808476I
0.05596 + 1.77225I 0.01088 4.04184I
u = 0.376287 1.204934I
4.82981 + 2.24524I 3.02697 1.89383I
u = 0.376287 + 1.204934I
4.82981 2.24524I 3.02697 + 1.89383I
u = 0.096397 0.986281I
1.74384 2.05721I 4.27298 + 4.01793I
u = 0.096397 + 0.986281I
1.74384 + 2.05721I 4.27298 4.01793I
u = 0.413902 1.197930I
6.35994 + 2.92383I 5.29020 3.29300I
u = 0.413902 + 1.197930I
6.35994 2.92383I 5.29020 + 3.29300I
u = 0.486243 1.189527I
5.84506 + 5.78082I 4.37527 3.72629I
u = 0.486243 + 1.189527I
5.84506 5.78082I 4.37527 + 3.72629I
u = 0.539628 0.849352I
2.43992 + 5.71321I 4.10823 7.50361I
u = 0.539628 + 0.849352I
2.43992 5.71321I 4.10823 + 7.50361I
u = 0.542169 0.664263I
2.96425 1.34320I 6.02964 + 0.62000I
u = 0.542169 + 0.664263I
2.96425 + 1.34320I 6.02964 0.62000I
u = 0.766849 0.083191I
2.63437 1.18290I 1.39246 + 0.39910I
u = 0.766849 + 0.083191I
2.63437 + 1.18290I 1.39246 0.39910I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
24
+ 3u
23
+ ··· + 4u + 1)
c
2
(u
24
+ 13u
23
+ ··· 2u
2
+ 1)
c
3
, c
7
(u
24
+ u
23
+ ··· + 2u + 1)
c
4
, c
6
(u
24
+ u
23
+ ··· + 10u + 1)
c
5
, c
9
(u
24
+ u
23
+ ··· + 2u
3
+ 1)
c
8
(u
24
+ 11u
23
+ ··· 2u
2
+ 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
24
+ y
23
+ ··· + 20y + 1)
c
2
(y
24
3y
23
+ ··· 4y + 1)
c
3
, c
7
(y
24
+ 13y
23
+ ··· 2y
2
+ 1)
c
4
, c
6
(y
24
19y
23
+ ··· 48y + 1)
c
5
, c
9
(y
24
11y
23
+ ··· 2y
2
+ 1)
c
8
(y
24
+ 5y
23
+ ··· 4y + 1)
5