6
2
(K6a
2
)
1
Arc Sequences
5 6 1 2 4 3
Solving Sequence
1,5
2 4 6 3
c
1
c
4
c
5
c
3
c
2
, c
6
Representation Ideals
I = I
u
1
I
u
1
= hu
5
u
4
+ 2u
3
u
2
+ u 1i
There are 1 irreducible components with 5 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
5
u
4
+ 2u
3
u
2
+ u 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
4
=
u
u
3
+ u
a
6
=
u
3
u
4
u
3
+ u
2
+ 1
a
3
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
4u
2
+ 4u 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.339110 0.822375I
0.32910 + 1.53058I 2.51511 4.43065I
u = 0.339110 + 0.822375I
0.32910 1.53058I 2.51511 + 4.43065I
u = 0.455697 1.200152I
5.87256 4.40083I 6.74431 + 3.49859I
u = 0.455697 + 1.200152I
5.87256 + 4.40083I 6.74431 3.49859I
u = 0.766826
2.40108 3.48114
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
c
2
, c
3
, c
6
(u
5
+ u
4
2u
3
u
2
+ u 1)
c
5
(u
5
+ 3u
4
+ 4u
3
+ u
2
u 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
c
2
, c
3
, c
6
(y
5
5y
4
+ 8y
3
3y
2
y 1)
c
5
(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)
5