7
2
(K7a
4
)
1
Arc Sequences
5 7 6 1 4 3 2
Solving Sequence
1,5
2 4 7 3 6
c
1
c
4
c
7
c
2
c
6
c
3
, c
5
Representation Ideals
I = I
u
1
I
u
1
= hu
5
+ u
4
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
u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
4
=
u
u
a
7
=
u
2
+ 1
u
4
a
3
=
u
4
u
2
+ 1
u
4
+ u
3
u
2
+ 1
a
6
=
u
3
u
3
+ u
a
6
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
+ 4u
2
+ 4u 10
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.935538 0.903908I
10.95829 3.33174I 2.08126 + 2.36228I
u = 0.935538 + 0.903908I
10.95829 + 3.33174I 2.08126 2.36228I
u = 0.645200
0.882183 11.6088
u = 0.758138 0.584034I
1.81981 + 2.21397I 3.11432 4.22289I
u = 0.758138 + 0.584034I
1.81981 2.21397I 3.11432 + 4.22289I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u
5
+ u
4
u
2
+ u + 1)
c
2
, c
3
, c
5
c
6
, c
7
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
5
y
4
+ 4y
3
3y
2
+ 3y 1)
c
2
, c
3
, c
5
c
6
, c
7
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y 1)
5