10
1
(K10a
75
)
A knot diagram
1
Linearized knot diagam
6 10 9 8 7 1 5 4 3 2
Solving Sequence
4,9
3 10 2 1 8 5 7 6
c
3
c
9
c
2
c
10
c
8
c
4
c
7
c
6
c
1
, c
5
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
− u
7
+ 7u
6
− 6u
5
+ 15u
4
− 10u
3
+ 10u
2
− 4u + 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
+ 7u
6
− 6u
5
+ 15u
4
− 10u
3
+ 10u
2
− 4u + 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
10
=
−u
u
3
+ u
a
2
=
u
2
+ 1
−u
4
− 2u
2
a
1
=
−u
3
− 2u
u
5
+ 3u
3
+ u
a
8
=
u
u
a
5
=
u
2
+ 1
u
2
a
7
=
u
3
+ 2u
u
3
+ u
a
6
=
u
4
+ 3u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
7
− 4u
6
+ 28u
5
− 24u
4
+ 60u
3
− 40u
2
+ 40u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
8
+ u
7
+ u
6
+ 3u
4
+ 2u
3
+ 2u
2
+ 1
c
2
, c
3
, c
4
c
5
, c
7
, c
8
c
9
, c
10
u
8
+ u
7
+ 7u
6
+ 6u
5
+ 15u
4
+ 10u
3
+ 10u
2
+ 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
8
+ y
7
+ 7y
6
+ 6y
5
+ 15y
4
+ 10y
3
+ 10y
2
+ 4y + 1
c
2
, c
3
, c
4
c
5
, c
7
, c
8
c
9
, c
10
y
8
+ 13y
7
+ 67y
6
+ 174y
5
+ 239y
4
+ 166y
3
+ 50y
2
+ 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.147789 + 0.913548I
3.50819 − 2.28803I 1.30973 + 4.26686I
u = 0.147789 − 0.913548I
3.50819 + 2.28803I 1.30973 − 4.26686I
u = 0.06403 + 1.48479I
11.71740 − 3.09309I 1.88403 + 2.68898I
u = 0.06403 − 1.48479I
11.71740 + 3.09309I 1.88403 − 2.68898I
u = 0.272222 + 0.278653I
−0.267684 − 0.921357I −5.17544 + 7.34493I
u = 0.272222 − 0.278653I
−0.267684 + 0.921357I −5.17544 − 7.34493I
u = 0.01595 + 1.86641I
−14.9579 − 3.5262I 1.98168 + 2.14300I
u = 0.01595 − 1.86641I
−14.9579 + 3.5262I 1.98168 − 2.14300I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u
8
+ u
7
+ u
6
+ 3u
4
+ 2u
3
+ 2u
2
+ 1
c
2
, c
3
, c
4
c
5
, c
7
, c
8
c
9
, c
10
u
8
+ u
7
+ 7u
6
+ 6u
5
+ 15u
4
+ 10u
3
+ 10u
2
+ 4u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
8
+ y
7
+ 7y
6
+ 6y
5
+ 15y
4
+ 10y
3
+ 10y
2
+ 4y + 1
c
2
, c
3
, c
4
c
5
, c
7
, c
8
c
9
, c
10
y
8
+ 13y
7
+ 67y
6
+ 174y
5
+ 239y
4
+ 166y
3
+ 50y
2
+ 4y + 1
7
