7
2
(K7a
4
)
A knot diagram
1
Linearized knot diagam
5 7 6 1 4 3 2
Solving Sequence
1,5
2 4 6 3 7
c
1
c
4
c
5
c
3
c
7
c
2
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
5
− u
4
+ u
2
+ u − 1i
* 1 irreducible components of dim
C
= 0, with total 5 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= hu
5
− u
4
+ u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
4
=
u
u
a
6
=
−u
3
−u
3
+ u
a
3
=
u
4
− u
2
+ 1
u
4
− u
3
− u
2
+ 1
a
7
=
−u
2
+ 1
−u
4
a
7
=
−u
2
+ 1
−u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
+ 4u
2
− 4u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
5
+ u
4
− u
2
+ u + 1
c
2
, c
3
, c
5
c
6
, c
7
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
2
, c
3
, c
5
c
6
, c
7
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.758138 + 0.584034I
1.81981 + 2.21397I −3.11432 − 4.22289I
u = −0.758138 − 0.584034I
1.81981 − 2.21397I −3.11432 + 4.22289I
u = 0.935538 + 0.903908I
10.95830 − 3.33174I −2.08126 + 2.36228I
u = 0.935538 − 0.903908I
10.95830 + 3.33174I −2.08126 − 2.36228I
u = 0.645200
−0.882183 −11.6090
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
u
5
+ u
4
− u
2
+ u + 1
c
2
, c
3
, c
5
c
6
, c
7
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
2
, c
3
, c
5
c
6
, c
7
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
7
