8
1
(K8a
11
)
1
Arc Sequences
5 4 7 2 1 8 3 6
Solving Sequence
3,7
4 8 2 5 1 6
c
3
c
7
c
2
c
4
c
1
c
6
c
5
, c
8
Representation Ideals
I = I
u
1
I
u
1
= hu
6
u
5
+ u
4
+ 2u
2
u + 1i
There are 1 irreducible components with 6 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
6
u
5
+ u
4
+ 2u
2
u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
8
=
u
u
a
2
=
u
2
+ 1
u
4
a
5
=
u
4
+ u
2
+ 1
u
5
+ u
4
+ u
2
u + 1
a
1
=
u
5
+ u
u
5
+ u
3
+ u
a
6
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
5
4u
2
8u 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.716019 0.809696I
4.64282 + 2.65597I 1.58115 3.39809I
u = 0.716019 + 0.809696I
4.64282 2.65597I 1.58115 + 3.39809I
u = 0.283231 0.633899I
0.258090 1.108706I 3.53615 + 6.18117I
u = 0.283231 + 0.633899I
0.258090 + 1.108706I 3.53615 6.18117I
u = 0.932789 0.951611I
15.3545 3.4272I 1.95500 + 2.25224I
u = 0.932789 + 0.951611I
15.3545 + 3.4272I 1.95500 2.25224I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
4
c
5
, c
6
, c
8
(u
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 3u + 1)
c
3
, c
7
(u
6
+ u
5
+ u
4
+ 2u
2
+ u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
4
c
5
, c
6
, c
8
(y
6
+ 9y
5
+ 29y
4
+ 40y
3
+ 22y
2
+ 3y + 1)
c
3
, c
7
(y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1)
5