7
4
(K7a
6
)
A knot diagram
1
Linearized knot diagam
5 6 7 2 1 4 3
Solving Sequence
4,6
7 3 1 2 5
c
6
c
3
c
7
c
2
c
5
c
1
, c
4
Ideals for irreducible components
2
of X
par
I
u
1
= hu
3
+ 2u + 1i
I
u
2
= hu
4
− u
3
+ 2u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 7 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
3
+ 2u + 1i
(i) Arc colorings
a
4
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
u
−u − 1
a
1
=
u
2
+ 1
−u
a
2
=
−1
−u − 1
a
5
=
−u
−u
2
a
5
=
−u
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
+ 4u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
u
3
+ 2u + 1
c
2
u
3
+ 3u
2
+ 5u + 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
y
3
+ 4y
2
+ 4y −1
c
2
y
3
+ y
2
+ 13y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
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
u = −0.453398
−0.787199 −12.6360
5
II. I
u
2
= hu
4
− u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
4
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
3
+ 2u − 1
a
2
=
u
3
+ 2u
u
3
+ u
a
5
=
−2u
3
+ u
2
− 3u + 3
−u
3
+ u
2
− u + 2
a
5
=
−2u
3
+ u
2
− 3u + 3
−u
3
+ u
2
− u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
+ 4u − 6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
u
4
− u
3
+ 2u
2
− 2u + 1
c
2
(u
2
− u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
y
4
+ 3y
3
+ 2y
2
+ 1
c
2
(y
2
+ y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
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
u = −0.121744 + 1.306620I
3.28987 + 2.02988I −4.00000 − 3.46410I
u = −0.121744 − 1.306620I
3.28987 − 2.02988I −4.00000 + 3.46410I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
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)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
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)
11
