9
46
(K9n
5
)
1
Arc Sequences
5 7 9 6 1 3 6 4 3
Solving Sequence
3,6
7 8
1,2
5 4 9
c
6
c
7
c
2
c
5
c
4
c
9
c
1
, c
3
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
2
u + 2, b + 1, 2a u 1i
I
u
2
= hu
2
+ 1, a + 1, b ui
I
u
3
= hu
4
+ 4u
2
+ 2u + 1, a + 1, u
3
+ u
2
+ 2b 3u 1i
There are 3 irreducible components with 8 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
u + 2, b + 1, 2a u 1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
2
u +
1
2
1
a
7
=
1
2
u +
1
2
1
a
8
=
u + 1
u + 2
a
1
=
1
0
a
2
=
1
2
u +
3
2
1
a
5
=
1
2
u
1
2
1
a
4
=
u
2
a
9
=
1
u 2
a
9
=
1
u 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.50000 1.32288I
a = 0.750000 0.661438I
b = 1.00000
4.93480 2.00000
u = 0.50000 + 1.32288I
a = 0.750000 + 0.661438I
b = 1.00000
4.93480 2.00000
3
II. I
u
2
= hu
2
+ 1, a + 1, b ui
(i) Arc colorings
a
3
=
0
u
a
6
=
1
u
a
7
=
1
u 1
a
8
=
0
1
a
1
=
1
0
a
2
=
u
1
a
5
=
u 1
u
a
4
=
u
0
a
9
=
1
1
a
9
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.00000I
a = 1.00000
b = 1.00000I
1.64493 0
u = 1.00000I
a = 1.00000
b = 1.00000I
1.64493 0
5
III. I
u
3
= hu
4
+ 4u
2
+ 2u + 1, a + 1, u
3
+ u
2
+ 2b 3u 1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
1
2
u
3
1
2
u
2
+
3
2
u +
1
2
a
7
=
1
1
2
u
3
+
1
2
u
2
+
3
2
u +
1
2
a
8
=
u
2
1
2u
2
+ 2u + 1
a
1
=
1
0
a
2
=
u
1
2
u
3
1
2
u
2
+
1
2
u
1
2
a
5
=
1
2
u
3
1
2
u
2
+
3
2
u
1
2
1
2
u
3
1
2
u
2
+
3
2
u +
1
2
a
4
=
u
u
3
+ u
a
9
=
1
u
2
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 14u + 8
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.264316 0.422125I
a = 1.00000
b = 0.219104 0.751390I
0.426736 + 1.175633I 4.79089 5.96277I
u = 0.264316 + 0.422125I
a = 1.00000
b = 0.219104 + 0.751390I
0.426736 1.175633I 4.79089 + 5.96277I
u = 0.26432 1.99036I
a = 1.00000
b = 1.28090 + 1.27441I
16.8761 4.7517I 0.79089 + 2.00586I
u = 0.26432 + 1.99036I
a = 1.00000
b = 1.28090 1.27441I
16.8761 + 4.7517I 0.79089 2.00586I
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u 1)
2
(u
2
+ 1)(u
4
+ 3u
3
+ 5u
2
+ 3u + 2)
c
2
, c
3
, c
6
c
8
, c
9
(u
2
+ 1)(u
2
+ u + 2)(u
4
+ 4u
2
2u + 1)
c
4
(u + 1)
4
(u
4
u
3
+ 11u
2
11u + 4)
c
7
(u + 1)
2
(u
2
+ 3u + 4)(u
4
+ 8u
3
+ 18u
2
+ 4u + 1)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)
2
(y + 1)
2
(y
4
+ y
3
+ 11y
2
+ 11y + 4)
c
2
, c
3
, c
6
c
8
, c
9
(y + 1)
2
(y
2
+ 3y + 4)(y
4
+ 8y
3
+ 18y
2
+ 4y + 1)
c
4
(y 1)
4
(y
4
+ 21y
3
+ 107y
2
33y + 16)
c
7
(y 1)
2
(y
2
y + 16)(y
4
28y
3
+ 262y
2
+ 20y + 1)
9