7
7
(K7a
1
)
A knot diagram
1
Linearized knot diagam
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
Ideals for irreducible components
2
of X
par
I
u
1
= hu
4
+ u
2
− u + 1i
I
u
2
= hu
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 10 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
4
+ u
2
− u + 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
3
=
1
u
2
a
5
=
u
2
+ 1
−u
2
+ u − 1
a
7
=
−u
u
a
2
=
u
u
2
− u + 1
a
6
=
−1
u
3
+ 1
a
6
=
−1
u
3
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
− 4u
2
+ 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
u
4
+ u
2
− u + 1
c
2
u
4
− 3u
3
+ 4u
2
− 3u + 2
c
4
, c
6
u
4
+ 2u
3
+ 3u
2
+ u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
2
y
4
− y
3
+ 2y
2
+ 7y + 4
c
4
, c
6
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
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
u = −0.547424 + 1.120870I
−2.62503 − 7.64338I −1.77019 + 6.51087I
u = −0.547424 − 1.120870I
−2.62503 + 7.64338I −1.77019 − 6.51087I
5
II. I
u
2
= hu
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
3
=
1
u
2
a
5
=
u
2
+ 1
u
4
a
7
=
−u
u
a
2
=
u
4
+ u
2
+ 1
−u
4
a
6
=
u
5
+ 2u
3
+ u + 1
u
5
+ u
3
+ u
2
+ u
a
6
=
u
5
+ 2u
3
+ u + 1
u
5
+ u
3
+ u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
− 4u − 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
6
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
2
(y
3
− y
2
+ 2y −1)
2
c
4
, c
6
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.498832 + 1.001300I
−0.26574 + 2.82812I 1.50976 − 2.97945I
u = 0.498832 − 1.001300I
−0.26574 − 2.82812I 1.50976 + 2.97945I
u = −0.284920 + 1.115140I
−4.40332 −5.01951 + 0.I
u = −0.284920 − 1.115140I
−4.40332 −5.01951 + 0.I
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
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
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)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
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)
11
