7
5
(K7a
3
)
A knot diagram
1
Linearized knot diagam
5 1 7 6 2 3 4
Solving Sequence
3,6
7 4 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
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
* 1 irreducible components of dim
C
= 0, with total 8 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
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
4
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
u
3
− 2u
−u
3
+ u
a
5
=
u
3
− 2u
−u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
+ 12u
4
+ 4u
3
− 8u
2
− 8u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
2
, c
4
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
3
, c
6
, c
7
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
2
, c
4
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
3
, c
6
, c
7
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.180120 + 0.268597I
−1.04066 + 1.13123I −7.41522 − 0.51079I
u = −1.180120 − 0.268597I
−1.04066 − 1.13123I −7.41522 + 0.51079I
u = −0.108090 + 0.747508I
2.15941 + 2.57849I −4.27708 − 3.56796I
u = −0.108090 − 0.747508I
2.15941 − 2.57849I −4.27708 + 3.56796I
u = 1.37100
−6.50273 −13.8640
u = 1.334530 + 0.318930I
−2.37968 − 6.44354I −9.42845 + 5.29417I
u = 1.334530 − 0.318930I
−2.37968 + 6.44354I −9.42845 − 5.29417I
u = −0.463640
−0.845036 −11.8940
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
2
, c
4
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
3
, c
6
, c
7
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
2
, c
4
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
3
, c
6
, c
7
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
7
