9
42
(K9n
4
)
1
Arc Sequences
3 5 6 8 2 9 5 6 7
Solving Sequence
6,8
9
2,7
5 3 1 4
c
8
c
6
c
5
c
2
c
1
c
4
c
3
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, a + u + 1, b u 1i
I
u
2
= hu
5
2u
4
+ 2u
3
+ u 1, u
2
+ b, u
4
u
3
+ u
2
+ a + u + 1i
There are 2 irreducible components with 7 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 + 1, a + u + 1, b u 1i
(i) Arc colorings
a
6
=
0
u
a
8
=
u 1
u + 1
a
9
=
u 1
1
a
2
=
1
0
a
7
=
u 1
u + 1
a
5
=
u
u
a
3
=
u
u + 1
a
1
=
0
u
a
4
=
u
u
a
4
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 5
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 0.500000 0.866025I
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0.500000 + 0.866025I
1.64493 2.02988I 3.00000 + 3.46410I
3
II. I
u
2
= hu
5
2u
4
+ 2u
3
+ u 1, u
2
+ b, u
4
u
3
+ u
2
+ a + u + 1i
(i) Arc colorings
a
6
=
0
u
a
8
=
u
4
+ u
3
u
2
u 1
u
2
a
9
=
u
4
+ u
3
u
2
u 1
u
4
u
3
+ u
2
+ 1
a
2
=
1
0
a
7
=
u
4
2u
3
+ u
2
+ u + 2
u
3
u
2
1
a
5
=
u
u
a
3
=
u
2
+ 1
u
2
a
1
=
u
4
+ u
2
+ 1
u
4
a
4
=
u
2
+ 1
u
4
a
4
=
u
2
+ 1
u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
u
3
2u
2
+ 5u + 2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.436447 0.655029I
a = 0.423679 + 0.262806I
b = 0.238576 + 0.571771I
0.057511 + 1.373618I 0.45374 4.59823I
u = 0.436447 + 0.655029I
a = 0.423679 0.262806I
b = 0.238576 0.571771I
0.057511 1.373618I 0.45374 + 4.59823I
u = 0.668466
a = 2.01628
b = 0.446847
2.55277 4.34961
u = 1.10221 1.09532I
a = 1.084463 + 0.905094I
b = 0.01515 2.41455I
17.6979 4.0569I 4.27894 + 1.95729I
u = 1.10221 + 1.09532I
a = 1.084463 0.905094I
b = 0.01515 + 2.41455I
17.6979 + 4.0569I 4.27894 1.95729I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)(u
5
+ 6u
3
+ u + 1)
c
2
(u
2
+ u + 1)(u
5
+ 2u
4
+ 2u
3
+ u + 1)
c
3
(u
2
u + 1)(u
5
+ 2u
4
+ 14u
3
16u
2
+ 9u 9)
c
4
, c
7
u
2
(u
5
+ u
4
+ 8u
3
+ u
2
4u + 4)
c
5
(u
2
u + 1)(u
5
+ 2u
4
+ 2u
3
+ u + 1)
c
6
(u + 1)
2
(u
5
+ 3u
4
u
3
6u
2
1)
c
8
, c
9
(u 1)
2
(u
5
+ 3u
4
u
3
6u
2
1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
2
+ y + 1)(y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y 1)
c
2
, c
5
(y
2
+ y + 1)(y
5
+ 6y
3
+ y 1)
c
3
(y
2
+ y + 1)(y
5
+ 24y
4
+ 278y
3
+ 32y
2
207y 81)
c
4
, c
7
y
2
(y
5
+ 15y
4
+ 54y
3
73y
2
+ 8y 16)
c
6
, c
8
, c
9
(y 1)
2
(y
5
11y
4
+ 37y
3
30y
2
12y 1)
7