7
4
(K7a
6
)
1
Arc Sequences
5 6 7 2 1 4 3
Solving Sequence
1,6
5 2 3 4 7
c
5
c
1
c
2
c
4
c
7
c
3
, c
6
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
3
+ 2u + 1i
I
u
2
= hu
4
u
3
+ 2u
2
2u + 1i
There are 2 irreducible components with 7 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
+ 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
5
=
u
u
a
2
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
a
4
=
1
u 1
a
7
=
u
u
2
a
7
=
u
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 4u 10
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.453398
0.787199 12.6359
u = 0.22670 1.46771I
9.44074 + 5.13794I 0.68207 3.20902I
u = 0.22670 + 1.46771I
9.44074 5.13794I 0.68207 + 3.20902I
3
II. I
u
2
= hu
4
u
3
+ 2u
2
2u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
5
=
u
u
a
2
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
3
+ 2u 1
a
4
=
u
3
+ 2u
u
3
+ u
a
7
=
2u
3
+ u
2
3u + 3
u
3
+ u
2
u + 2
a
7
=
2u
3
+ u
2
3u + 3
u
3
+ u
2
u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 4u 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.121744 1.306622I
3.28987 2.02988I 4.00000 + 3.46410I
u = 0.121744 + 1.306622I
3.28987 + 2.02988I 4.00000 3.46410I
u = 0.621744 0.440597I
3.28987 + 2.02988I 4.00000 3.46410I
u = 0.621744 + 0.440597I
3.28987 2.02988I 4.00000 + 3.46410I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
c
5
, c
6
, c
7
(u
3
+ 2u 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)
c
2
(u
2
u + 1)
2
(u
3
+ 3u
2
+ 5u + 2)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
5
, c
6
, c
7
(y
3
+ 4y
2
+ 4y 1)(y
4
+ 3y
3
+ 2y
2
+ 1)
c
2
(y
2
+ y + 1)
2
(y
3
+ y
2
+ 13y 4)
7