9
24
(K9a
7
)
1
Arc Sequences
7 4 1 8 9 2 6 5 3
Solving Sequence
1,7 2,4
3 6 8 9 5
c
1
c
3
c
6
c
7
c
9
c
5
c
2
, c
4
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu 1, b, a 1i
I
u
2
= hu
6
2u
4
u
3
+ u
2
+ u + 1, u
3
+ b u 1, u
4
+ u
3
+ u
2
+ a ui
I
u
3
= hu
17
+ 2u
16
+ ··· u 1,
u
16
u
15
+ 3u
14
+ 5u
13
4u
12
10u
11
u
10
+ 10u
9
+ 6u
8
3u
7
7u
6
u
5
+ 2u
4
2u
2
+ b + u + 1,
3u
16
4u
15
+ ··· + a + 3i
There are 3 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 1, b, a 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
1
0
a
2
=
1
0
a
4
=
0
1
a
3
=
1
1
a
6
=
1
0
a
8
=
1
0
a
9
=
0
1
a
5
=
1
1
a
5
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
0 0
3
II. I
u
2
= hu
6
2u
4
u
3
+ u
2
+ u + 1, u
3
+ b u 1, u
4
+ u
3
+ u
2
+ a ui
(i) Arc colorings
a
1
=
1
0
a
7
=
u
4
u
3
u
2
+ u
u
3
+ u + 1
a
2
=
u
u
3
+ u
a
4
=
0
u
a
3
=
u
u
a
6
=
u
3
+ 2u
u
3
+ u
a
8
=
1
0
a
9
=
u
2
+ 1
u
2
a
5
=
u
u
a
5
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 4u + 2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.033348 0.428825I
a = 1.38805 + 1.54652I
b = 0.500000 + 0.866025I
2.02988I 3.46410I
u = 1.033348 + 0.428825I
a = 1.38805 1.54652I
b = 0.500000 0.866025I
2.02988I 3.46410I
u = 0.218964 0.666188I
a = 0.032546 1.388748I
b = 0.500000 0.866025I
2.02988I 3.46410I
u = 0.218964 + 0.666188I
a = 0.032546 + 1.388748I
b = 0.500000 + 0.866025I
2.02988I 3.46410I
u = 1.252312 0.237364I
a = 0.079407 0.337191I
b = 0.500000 + 0.866025I
2.02988I 3.46410I
u = 1.252312 + 0.237364I
a = 0.079407 + 0.337191I
b = 0.500000 0.866025I
2.02988I 3.46410I
5
III.
I
u
3
= hu
17
+2u
16
+· · ·u1, u
16
u
15
+· · ·+b+1, 3u
16
4u
15
+· · ·+a+3i
(i) Arc colorings
a
1
=
1
0
a
7
=
3u
16
+ 4u
15
+ ··· u 3
u
16
+ u
15
+ ··· u 1
a
2
=
u
u
3
+ u
a
4
=
0
u
a
3
=
u
u
a
6
=
2u
16
+ 3u
15
+ ··· u 2
u
16
+ u
15
+ ··· u 1
a
8
=
u
16
+ 2u
15
+ ··· + u
2
1
u
16
+ 2u
15
+ ··· u 1
a
9
=
u
2
+ 1
u
2
a
5
=
u
16
+ u
15
+ ··· 2u
4
4u
3
u
14
4u
12
2u
11
+ 7u
10
+ 6u
9
4u
8
8u
7
2u
6
+ 4u
5
+ 4u
4
u
2
a
5
=
u
16
+ u
15
+ ··· 2u
4
4u
3
u
14
4u
12
2u
11
+ 7u
10
+ 6u
9
4u
8
8u
7
2u
6
+ 4u
5
+ 4u
4
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
15
4u
14
+ 2u
13
+ 16u
12
+ 8u
11
24u
10
30u
9
+ 12u
8
+
32u
7
+ 12u
6
16u
5
12u
4
4u
3
+ 4u
2
6u
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.172116 0.583556I
a = 1.23887 + 1.13514I
b = 0.718492 + 1.129371I
2.40324 10.83373I 0.89378 + 7.41261I
u = 1.172116 + 0.583556I
a = 1.23887 1.13514I
b = 0.718492 1.129371I
2.40324 + 10.83373I 0.89378 7.41261I
u = 1.130682 0.513073I
a = 1.32904 1.25195I
b = 0.656745 1.004701I
2.57978 6.57063I 3.26005 + 6.43452I
u = 1.130682 + 0.513073I
a = 1.32904 + 1.25195I
b = 0.656745 + 1.004701I
2.57978 + 6.57063I 3.26005 6.43452I
u = 0.796399 0.723427I
a = 0.70217 1.37913I
b = 0.110097 1.246508I
8.03468 2.71165I 5.84242 + 3.13710I
u = 0.796399 + 0.723427I
a = 0.70217 + 1.37913I
b = 0.110097 + 1.246508I
8.03468 + 2.71165I 5.84242 3.13710I
u = 0.621791 0.419413I
a = 0.48878 + 1.92765I
b = 0.003992 + 0.842342I
1.30982 1.46955I 3.63583 + 4.66528I
u = 0.621791 + 0.419413I
a = 0.48878 1.92765I
b = 0.003992 0.842342I
1.30982 + 1.46955I 3.63583 4.66528I
u = 0.288739 0.863831I
a = 0.138052 + 1.226062I
b = 0.578864 + 1.116300I
5.04981 + 5.51158I 4.25126 3.84490I
u = 0.288739 + 0.863831I
a = 0.138052 1.226062I
b = 0.578864 1.116300I
5.04981 5.51158I 4.25126 + 3.84490I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.374678 0.520641I
a = 0.500608 + 0.782532I
b = 0.834865 + 0.265014I
2.61956 0.43387I 2.56834 0.87540I
u = 0.374678 + 0.520641I
a = 0.500608 0.782532I
b = 0.834865 0.265014I
2.61956 + 0.43387I 2.56834 + 0.87540I
u = 0.867068
a = 0.286555
b = 0.463897
1.25812 8.68792
u = 1.072950 0.498433I
a = 0.164644 0.415457I
b = 0.976738 + 0.562668I
0.61043 + 4.64771I 0.43915 4.11695I
u = 1.072950 + 0.498433I
a = 0.164644 + 0.415457I
b = 0.976738 0.562668I
0.61043 4.64771I 0.43915 + 4.11695I
u = 1.128565 0.359117I
a = 0.141801 + 0.359295I
b = 0.742615 0.650908I
3.65923 + 1.22724I 6.14847 0.85505I
u = 1.128565 + 0.359117I
a = 0.141801 0.359295I
b = 0.742615 + 0.650908I
3.65923 1.22724I 6.14847 + 0.85505I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u)(u
2
+ u + 1)
3
(u
17
2u
16
+ ··· 2u + 2)
c
2
(u + 1)(u
6
+ 4u
5
+ ··· u + 1)(u
17
+ 8u
16
+ ··· + 3u + 1)
c
3
, c
9
(u 1)(u
6
2u
4
+ ··· + u + 1)(u
17
+ 2u
16
+ ··· u 1)
c
4
, c
8
(u 1)(u
6
2u
4
+ ··· u + 1)(u
17
+ 2u
16
+ ··· + 3u + 1)
c
5
(u + 1)(u
6
2u
4
+ ··· u + 1)(u
17
+ 2u
16
+ ··· + 3u + 1)
c
7
(u)(u
2
+ u + 1)
3
(u
17
+ 6u
16
+ ··· + 8u 4)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y)(y
2
+ y + 1)
3
(y
17
+ 6y
16
+ ··· + 8y 4)
c
2
(y 1)(y
6
4y
5
+ 10y
4
11y
3
+ 19y
2
3y + 1)
(y
17
+ 4y
16
+ ··· 13y 1)
c
3
, c
9
(y 1)(y
6
4y
5
+ ··· + y + 1)(y
17
8y
16
+ ··· + 3y 1)
c
4
, c
5
, c
8
(y 1)(y
6
4y
5
+ ··· + y + 1)(y
17
16y
16
+ ··· + 19y 1)
c
7
(y)(y
2
+ y + 1)
3
(y
17
+ 6y
16
+ ··· + 376y 16)
10