8
11
(K8a
9
)
1
Arc Sequences
6 7 8 1 2 5 4 3
Solving Sequence
1,6
2 5 7 3 4 8
c
1
c
5
c
6
c
2
c
4
c
8
c
3
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
3
+ u + 1i
I
u
2
= hu
10
u
9
+ 3u
8
3u
7
+ 5u
6
5u
5
+ 4u
4
4u
3
+ 3u
2
2u + 1i
There are 2 irreducible components with 13 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 + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
5
=
u
1
a
7
=
u + 1
u
2
+ u
a
3
=
u
u
a
4
=
u 1
1
a
8
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.682328
1.64493 6.00000
u = 0.341164 1.161541I
1.64493 6.00000
u = 0.341164 + 1.161541I
1.64493 6.00000
3
II. I
u
2
= hu
10
u
9
+ 3u
8
3u
7
+ 5u
6
5u
5
+ 4u
4
4u
3
+ 3u
2
2u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
7
=
u
3
u
5
+ u
3
+ u
a
3
=
u
6
u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
4
=
u
3
u
3
+ u
a
8
=
u
5
2u
3
+ u
2
u + 1
u
9
2u
7
2u
5
+ u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
9
+ 4u
8
8u
7
+ 8u
6
8u
5
+ 12u
4
+ 4u
2
4u 2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.584958 0.771492I
4.93719 + 2.31006I 0.86369 3.52133I
u = 0.584958 + 0.771492I
4.93719 2.31006I 0.86369 + 3.52133I
u = 0.449566 1.164785I
4.87665 + 4.14585I 8.98134 3.97600I
u = 0.449566 + 1.164785I
4.87665 4.14585I 8.98134 + 3.97600I
u = 0.248527 0.782547I
0.448055 1.231689I 4.90177 + 5.44908I
u = 0.248527 + 0.782547I
0.448055 + 1.231689I 4.90177 5.44908I
u = 0.524355 1.163405I
0.38115 8.28632I 4.17560 + 6.14881I
u = 0.524355 + 1.163405I
0.38115 + 8.28632I 4.17560 6.14881I
u = 0.761643 0.208049I
2.41360 + 3.47839I 0.80497 2.79515I
u = 0.761643 + 0.208049I
2.41360 3.47839I 0.80497 + 2.79515I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
3
+ u 1)(u
10
+ u
9
+ ··· + 2u + 1)
c
2
, c
4
(u + 1)
3
(u
10
2u
9
+ ··· u + 2)
c
3
, c
7
, c
8
(u
3
+ u 1)(u
10
+ u
9
+ 5u
8
+ 5u
7
+ 9u
6
+ 9u
5
+ 6u
4
+ 6u
3
+ u
2
+ 1)
c
6
(u
3
+ 2u
2
+ u 1)
(u
10
+ 5u
9
+ 13u
8
+ 19u
7
+ 17u
6
+ 7u
5
2u
3
+ u
2
+ 2u + 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
3
+ 2y
2
+ y 1)
(y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
2y
3
+ y
2
+ 2y + 1)
c
2
, c
4
(y 1)
3
(y
10
6y
9
+ ··· + 19y + 4)
c
3
, c
7
, c
8
(y
3
+ 2y
2
+ y 1)
(y
10
+ 9y
9
+ 33y
8
+ 59y
7
+ 41y
6
21y
5
44y
4
6y
3
+ 13y
2
+ 2y + 1)
c
6
(y
3
2y
2
+ 5y 1)
(y
10
+ y
9
+ 13y
8
+ 11y
7
+ 45y
6
+ 35y
5
+ 12y
4
+ 2y
3
+ 9y
2
2y + 1)
7