7
3
(K7a
5
)
A knot diagram
1
Linearized knot diagam
5 6 1 7 2 3 4
Solving Sequence
1,3
4 7 5 6 2
c
3
c
7
c
4
c
6
c
2
c
1
, c
5
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 1 irreducible components of dim
C
= 0, with total 6 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
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
(i) Arc colorings
a
1
=
0
u
a
3
=
1
0
a
4
=
1
u
2
a
7
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
u
3
+ 2u
u
3
+ u
a
2
=
−u
5
− 2u
3
− u
−u
5
+ u
4
− 2u
3
+ u
2
− u − 1
a
2
=
−u
5
− 2u
3
− u
−u
5
+ u
4
− 2u
3
+ u
2
− u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
+ 4u
3
− 8u
2
+ 4u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
3
, c
4
, c
7
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
3
, c
4
, c
7
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.873214
−7.66009 −12.2690
u = −0.138835 + 1.234450I
2.96024 + 1.97241I −4.57572 − 3.68478I
u = −0.138835 − 1.234450I
2.96024 − 1.97241I −4.57572 + 3.68478I
u = 0.408802 + 1.276380I
−3.69558 − 4.59213I −8.58114 + 3.20482I
u = 0.408802 − 1.276380I
−3.69558 + 4.59213I −8.58114 − 3.20482I
u = −0.413150
−0.738851 −13.4170
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
3
, c
4
, c
7
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
3
, c
4
, c
7
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
7
