8
20
(K8n
1
)
1
Arc Sequences
4 7 5 2 8 5 2 6
Solving Sequence
2,7 5,8
4 1 3 6
c
7
c
4
c
1
c
3
c
6
c
2
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu 1, b, a 1i
I
u
2
= hu
5
+ 2u
4
2u
3
3u
2
+ 3u + 1, u
3
+ a 2u + 2, u
4
3u
2
+ b + u + 1i
There are 2 irreducible components with 6 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
2
=
1
0
a
7
=
1
0
a
5
=
0
1
a
8
=
1
0
a
4
=
1
1
a
1
=
0
1
a
3
=
1
0
a
6
=
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
5
+ 2u
4
2u
3
3u
2
+ 3u + 1, u
3
+ a 2u + 2, u
4
3u
2
+ b + u + 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
u
3
+ 2u 2
u
4
+ 3u
2
u 1
a
5
=
0
u
a
8
=
u
4
u
3
+ 3u
2
+ u 3
u
4
+ 3u
2
u 1
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
u
3
+ u
a
6
=
u
3
+ 2u 2
u
4
2u
2
+ 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
+ 6u
3
8u
2
6u + 8
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.81245 0.17314I
a = 0.165924 + 1.354819I
b = 0.37638 2.02979I
11.90992 4.12490I 1.10604 + 2.15443I
u = 1.81245 + 0.17314I
a = 0.165924 1.354819I
b = 0.37638 + 2.02979I
11.90992 + 4.12490I 1.10604 2.15443I
u = 0.274898
a = 2.52902
b = 0.504107
1.20365 8.94304
u = 0.949895 0.441667I
a = 0.401414 + 0.226060I
b = 0.375669 0.888717I
1.85138 + 1.10891I 2.36548 2.04112I
u = 0.949895 + 0.441667I
a = 0.401414 0.226060I
b = 0.375669 + 0.888717I
1.85138 1.10891I 2.36548 + 2.04112I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u 1)(u
5
+ 2u
4
2u
3
3u
2
+ 3u + 1)
c
2
, c
7
u(u
5
+ u
4
+ 5u
3
+ u
2
+ 2u 2)
c
3
(u + 1)(u
5
+ 8u
4
+ 22u
3
+ 25u
2
+ 15u + 1)
c
5
, c
8
(u 1)(u
5
+ 2u
4
+ 2u
3
u
2
u 1)
c
6
(u + 1)(u
5
+ 6u
3
+ u
2
u + 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)(y
5
8y
4
+ 22y
3
25y
2
+ 15y 1)
c
2
, c
7
y(y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y 4)
c
3
(y 1)(y
5
20y
4
+ 114y
3
+ 19y
2
+ 175y 1)
c
5
, c
8
(y 1)(y
5
+ 6y
3
y
2
y 1)
c
6
(y 1)(y
5
+ 12y
4
+ 34y
3
13y
2
y 1)
7