5
2
(K5a
1
)
1
Arc Sequences
3 5 1 2 4
Solving Sequence
2,5
3 1 4
c
2
c
1
c
4
c
3
, c
5
Representation Ideals
I = I
u
1
I
u
1
= hu
3
u
2
+ 1i
There are 1 irreducible components with 3 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
3
u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
1
=
u
2
+ 1
u
2
+ u + 1
a
4
=
u
u
a
4
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.754878
1.11345 9.01951
u = 0.877439 0.744862I
3.02413 + 2.82812I 2.49024 2.97945I
u = 0.877439 + 0.744862I
3.02413 2.82812I 2.49024 + 2.97945I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
5
(u
3
+ u
2
+ 2u + 1)
c
2
, c
4
(u
3
+ u
2
1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
5
(y
3
+ 3y
2
+ 2y 1)
c
2
, c
4
(y
3
y
2
+ 2y 1)
5