7
5
(K7a
3
)
1
Arc Sequences
5 1 7 6 2 3 4
Solving Sequence
2,5
6 1 3 4 7
c
5
c
1
c
2
c
4
c
7
c
3
, c
6
Representation Ideals
I = I
u
1
I
u
1
= hu
8
+ u
7
u
6
2u
5
+ u
4
+ 2u
3
2u 1i
There are 1 irreducible components with 8 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
8
+ u
7
u
6
2u
5
+ u
4
+ 2u
3
2u 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
4
=
u
3
u
3
+ u
a
7
=
u
7
2u
5
+ 2u
3
2u
u
7
u
6
+ 2u
5
+ u
4
2u
3
2u
2
+ 2u + 1
a
7
=
u
7
2u
5
+ 2u
3
2u
u
7
u
6
+ 2u
5
+ u
4
2u
3
2u
2
+ 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
7
8u
5
4u
4
+ 8u
3
+ 4u
2
4u 14
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.031807 0.655470I
2.37968 6.44354I 9.42845 + 5.29417I
u = 1.031807 + 0.655470I
2.37968 + 6.44354I 9.42845 5.29417I
u = 0.603304
0.845036 11.8945
u = 0.570868 0.730671I
1.04066 + 1.13123I 7.41522 0.51079I
u = 0.570868 + 0.730671I
1.04066 1.13123I 7.41522 + 0.51079I
u = 0.855237 0.665892I
2.15941 + 2.57849I 4.27708 3.56796I
u = 0.855237 + 0.665892I
2.15941 2.57849I 4.27708 + 3.56796I
u = 1.09818
6.50273 13.8640
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
8
+ u
7
u
6
2u
5
+ u
4
+ 2u
3
2u 1)
c
2
, c
4
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
c
3
, c
6
, c
7
(u
8
+ u
7
3u
6
2u
5
+ 3u
4
+ 2u 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
8
3y
7
+ 7y
6
10y
5
+ 11y
4
10y
3
+ 6y
2
4y + 1)
c
2
, c
4
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
17y
4
34y
3
22y
2
4y + 1)
c
3
, c
6
, c
7
(y
8
7y
7
+ 19y
6
22y
5
+ 3y
4
+ 14y
3
6y
2
4y + 1)
5