8
21
(K8n
2
)
1
Arc Sequences
4 7 5 2 7 1 2 6
Solving Sequence
1,4 2,7
3 6 5 8
c
1
c
2
c
6
c
5
c
8
c
3
, c
4
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu 1, a + 1, b + 1i
I
u
2
= hu
4
u
3
+ u + 1, b u, u
3
u
2
+ a + ui
I
u
3
= hu
4
u
3
+ 2u 1, u
3
+ b + 1, u
3
u
2
+ a + 2i
There are 3 irreducible components with 9 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, a + 1, b + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
7
=
1
1
a
3
=
1
1
a
6
=
2
1
a
5
=
1
0
a
8
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
3.28987 12.0000
3
II. I
u
2
= hu
4
u
3
+ u + 1, b u, u
3
u
2
+ a + ui
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
7
=
u
3
+ u
2
u
u
a
3
=
u
3
u
2
+ u + 1
a
6
=
u
3
+ u
2
u
a
5
=
u
u
3
+ u
a
8
=
u
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
3
+ 6u
2
2u 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.566121 0.458821I
a = 0.50000 + 1.32288I
b = 0.566121 0.458821I
0.66070 1.45022I 4.56010 + 4.72374I
u = 0.566121 + 0.458821I
a = 0.50000 1.32288I
b = 0.566121 + 0.458821I
0.66070 + 1.45022I 4.56010 4.72374I
u = 1.066121 0.864054I
a = 0.50000 + 1.32288I
b = 1.066121 0.864054I
4.77303 + 6.78371I 3.43990 4.72374I
u = 1.066121 + 0.864054I
a = 0.50000 1.32288I
b = 1.066121 + 0.864054I
4.77303 6.78371I 3.43990 + 4.72374I
5
III. I
u
3
= hu
4
u
3
+ 2u 1, u
3
+ b + 1, u
3
u
2
+ a + 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
7
=
u
3
+ u
2
2
u
3
1
a
3
=
u
3
2u
2
1
a
6
=
2u
3
+ u
2
3
u
3
1
a
5
=
u
u
3
+ u
a
8
=
2u
3
+ u
2
+ u 3
u
3
+ u
2
+ u 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.15372
a = 0.866760
b = 0.535687
2.30291 2.00000
u = 0.535687
a = 1.86676
b = 1.15372
2.30291 2.00000
u = 0.809017 0.981593I
a = 0.500000 0.606658I
b = 0.809017 + 0.981593I
5.59278 2.00000
u = 0.809017 + 0.981593I
a = 0.500000 + 0.606658I
b = 0.809017 0.981593I
5.59278 2.00000
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u + 1)(u
4
+ u
3
2u 1)(u
4
+ u
3
u + 1)
c
2
, c
7
u(u
2
u 1)
2
(u
4
+ 3u
3
+ 3u
2
+ 2u + 2)
c
3
(u 1)(u
4
+ u
3
+ 2u
2
+ 4u + 1)(u
4
+ u
3
+ 4u
2
+ u + 1)
c
5
(u + 1)(u
4
+ u
3
+ 2u
2
+ 4u + 1)(u
4
+ u
3
+ 4u
2
+ u + 1)
c
6
, c
8
(u 1)(u
4
+ u
3
2u 1)(u
4
+ u
3
u + 1)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
6
c
8
(y 1)(y
4
y
3
+ 2y
2
4y + 1)(y
4
y
3
+ 4y
2
y + 1)
c
2
, c
7
y(y
2
3y + 1)
2
(y
4
3y
3
+ y
2
+ 8y + 4)
c
3
, c
5
(y 1)(y
4
+ 3y
3
2y
2
12y + 1)(y
4
+ 7y
3
+ 16y
2
+ 7y + 1)
9