6
1
(K6a
3
)
1
Arc Sequences
4 6 2 1 3 5
Solving Sequence
3,6
2 5 1 4
c
2
c
5
c
6
c
4
c
1
, c
3
Representation Ideals
I = I
u
1
I
u
1
= hu
4
u
3
+ u
2
+ 1i
There are 1 irreducible components with 4 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
4
u
3
+ u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
5
=
u
u
a
1
=
u
3
u
3
+ u
a
4
=
u
2
+ 1
u
3
u
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 4u 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.351808 0.720342I
0.21101 + 1.41510I 1.82674 4.90874I
u = 0.351808 + 0.720342I
0.21101 1.41510I 1.82674 + 4.90874I
u = 0.851808 0.911292I
6.79074 3.16396I 1.82674 + 2.56480I
u = 0.851808 + 0.911292I
6.79074 + 3.16396I 1.82674 2.56480I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
c
6
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
c
2
, c
5
(u
4
+ u
3
+ u
2
+ 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
6
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
c
2
, c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
5