9
35
(K9a
40
)
1
Arc Sequences
6 8 7 9 2 1 3 5 4
Solving Sequence
2,8 3,5
6 9 1 4 7
c
2
c
5
c
8
c
1
c
4
c
7
c
3
, c
6
, c
9
Representation Ideals
I =
5
\
i=1
I
u
i
I
u
1
= hu
2
+ 1, a + 1, b + ui
I
u
2
= hu
3
+ u
2
+ 3u + 1, b u, a 1i
I
u
3
= hu
4
+ 2u
3
+ 3u
2
+ 3u + 2, u
3
+ 2a u + 1, u
3
+ u
2
+ b + 2u + 1i
I
u
4
= hu
4
u
3
+ 3u
2
2u + 1, a 1, u
3
u
2
+ b + 3u 1i
I
u
5
= hu
4
u
3
+ 3u
2
2u + 1, b u, u
3
+ a + 2u + 1i
There are 5 irreducible components with 17 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
2
+ 1, a + 1, b + ui
(i) Arc colorings
a
2
=
1
0
a
8
=
1
u
a
3
=
u + 1
1
a
5
=
0
u
a
6
=
u
u
a
9
=
1
u + 1
a
1
=
0
1
a
4
=
u
1
a
7
=
u
0
a
7
=
u
0
(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 = 1.00000I
a = 1.00000
b = 1.00000I
4.93480 0
u = 1.00000I
a = 1.00000
b = 1.00000I
4.93480 0
3
II. I
u
2
= hu
3
+ u
2
+ 3u + 1, b u, a 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
1
u
a
3
=
u + 1
u
2
a
5
=
0
u
a
6
=
u
u
a
9
=
1
u
2
+ u
a
1
=
u
2
+ 1
u
2
a
4
=
u
2u 1
a
7
=
u
2
+ u + 1
u
2
2u 1
a
7
=
u
2
+ u + 1
u
2
2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
2
6u 18
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.361103
a = 1.00000
b = 0.361103
0.595615 16.6158
u = 0.31945 1.63317I
a = 1.00000
b = 0.31945 1.63317I
17.5696 7.9406I 0.69212 + 3.53846I
u = 0.31945 + 1.63317I
a = 1.00000
b = 0.31945 + 1.63317I
17.5696 + 7.9406I 0.69212 3.53846I
5
III. I
u
3
= hu
4
+ 2u
3
+ 3u
2
+ 3u + 2, u
3
+ 2a u + 1, u
3
+ u
2
+ b + 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
1
2
u
3
+
1
2
u
1
2
u
3
u
2
2u 1
a
3
=
3
2
u
3
+ 2u
2
+
5
2
u +
7
2
u
3
2u
2
2u 3
a
5
=
0
u
a
6
=
u
u
a
9
=
1
2
u
3
+
1
2
u
1
2
1
a
1
=
u
2
+ 1
u
2
a
4
=
3
2
u
3
+ 2u
2
+
5
2
u +
5
2
u
3
u
2
u 1
a
7
=
u
3
2u
u
3
+ u
a
7
=
u
3
2u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u 6
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.956685 0.641200I
a = 0.826150 1.069072I
b = 0.10488 + 1.55249I
10.08061 3.16396I 2.17326 + 2.56480I
u = 0.956685 + 0.641200I
a = 0.826150 + 1.069072I
b = 0.10488 1.55249I
10.08061 + 3.16396I 2.17326 2.56480I
u = 0.043315 1.227186I
a = 0.423850 + 0.307015I
b = 0.395123 + 0.506844I
3.07886 1.41510I 5.82674 + 4.90874I
u = 0.043315 + 1.227186I
a = 0.423850 0.307015I
b = 0.395123 0.506844I
3.07886 + 1.41510I 5.82674 4.90874I
7
IV. I
u
4
= hu
4
u
3
+ 3u
2
2u + 1, a 1, u
3
u
2
+ b + 3u 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
1
u
3
+ u
2
3u + 1
a
3
=
u
3
+ u
2
3u + 2
u
2
+ u 2
a
5
=
0
u
a
6
=
u
u
a
9
=
1
u
3
+ 2u
2
3u + 1
a
1
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ 1
a
7
=
u
3
2u
u
3
+ u
a
7
=
u
3
2u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
4u
2
+ 12u 10
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.10488 1.55249I
a = 1.00000
b = 0.956685 + 0.641200I
10.08061 + 3.16396I 2.17326 2.56480I
u = 0.10488 + 1.55249I
a = 1.00000
b = 0.956685 0.641200I
10.08061 3.16396I 2.17326 + 2.56480I
u = 0.395123 0.506844I
a = 1.00000
b = 0.043315 + 1.227186I
3.07886 + 1.41510I 5.82674 4.90874I
u = 0.395123 + 0.506844I
a = 1.00000
b = 0.043315 1.227186I
3.07886 1.41510I 5.82674 + 4.90874I
9
V. I
u
5
= hu
4
u
3
+ 3u
2
2u + 1, b u, u
3
+ a + 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
u
3
2u 1
u
a
3
=
u
3
+ u
2
3u + 2
u
2
a
5
=
0
u
a
6
=
u
u
a
9
=
u
3
2u 1
1
a
1
=
u
2
+ 1
u
2
a
4
=
2u
3
2u
2
+ 5u 3
u
3
+ u
2
2u + 1
a
7
=
u
3
2u
u
3
+ u
a
7
=
u
3
2u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
4u
2
+ 12u 10
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.10488 1.55249I
a = 0.452576 0.585652I
b = 0.10488 1.55249I
10.08061 + 3.16396I 2.17326 2.56480I
u = 0.10488 + 1.55249I
a = 0.452576 + 0.585652I
b = 0.10488 + 1.55249I
10.08061 3.16396I 2.17326 + 2.56480I
u = 0.395123 0.506844I
a = 1.54742 + 1.12087I
b = 0.395123 0.506844I
3.07886 + 1.41510I 5.82674 4.90874I
u = 0.395123 + 0.506844I
a = 1.54742 1.12087I
b = 0.395123 + 0.506844I
3.07886 1.41510I 5.82674 + 4.90874I
11
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
(u
2
+ 1)(u
3
+ u
2
+ 3u + 1)(u
4
u
3
+ 3u
2
2u + 1)
2
(u
4
+ 2u
3
+ 3u
2
+ 3u + 2)
12
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
(y + 1)
2
(y
3
+ 5y
2
+ 7y 1)(y
4
+ 2y
3
+ y
2
+ 3y + 4)
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
13