7
7
(K7a
1
)
1
Arc Sequences
6 7 1 3 2 5 4
Solving Sequence
1,4
3 5 7 2 6
c
3
c
4
c
7
c
2
c
6
c
1
, c
5
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
4
+ u
2
u + 1i
I
u
2
= hu
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1i
There are 2 irreducible components with 10 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
2
u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
3
=
u
u
a
5
=
u
3
u
3
+ u
a
7
=
1
u
2
a
2
=
u
3
u
2
a
6
=
u
3
+ u
2
u + 1
u
a
6
=
u
3
+ u
2
u + 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
4u
2
+ 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.547424 1.120873I
2.62503 + 7.64338I 1.77019 6.51087I
u = 0.547424 + 1.120873I
2.62503 7.64338I 1.77019 + 6.51087I
u = 0.547424 0.585652I
0.98010 1.39709I 3.77019 + 3.86736I
u = 0.547424 + 0.585652I
0.98010 + 1.39709I 3.77019 3.86736I
3
II. I
u
2
= hu
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
3
=
u
u
a
5
=
u
3
u
3
+ u
a
7
=
1
u
2
a
2
=
u
3
u
5
+ u
3
+ u
a
6
=
u
4
+ u
2
+ u + 1
u
5
+ 2u
3
+ u
2
+ u + 1
a
6
=
u
4
+ u
2
+ u + 1
u
5
+ 2u
3
+ u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
4u 2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.713912 0.305839I
0.26574 2.82812I 1.50976 + 2.97945I
u = 0.713912 + 0.305839I
0.26574 + 2.82812I 1.50976 2.97945I
u = 0.284920 1.115141I
4.40332 5.01951
u = 0.284920 + 1.115141I
4.40332 5.01951
u = 0.498832 1.001303I
0.26574 2.82812I 1.50976 + 2.97945I
u = 0.498832 + 1.001303I
0.26574 + 2.82812I 1.50976 2.97945I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
5
c
7
(u
4
+ u
2
u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
c
2
(u
3
u
2
+ 1)
2
(u
4
+ 3u
3
+ 4u
2
+ 3u + 2)
c
4
, c
6
(u
4
+ 2u
3
+ 3u
2
+ u + 1)(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
5
c
7
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
c
2
(y
3
y
2
+ 2y 1)
2
(y
4
y
3
+ 2y
2
+ 7y + 4)
c
4
, c
6
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
y
5
+ 4y
4
2y
3
+ 8y
2
+ 1)
7