10
124
(K10n
21
)
A knot diagram
1
Linearized knot diagam
5 6 7 10 2 3 10 7 4 8
Solving Sequence
3,7 4,10
8 6 2 5 1 9
c
3
c
7
c
6
c
2
c
5
c
1
c
9
c
4
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, a + u + 1, u
2
+ u − 1i
I
u
2
= hb − u, a − u + 1, u
2
− 3u + 1i
* 2 irreducible components of dim
C
= 0, with total 4 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
= hb + u, a + u + 1, u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u + 1
a
10
=
−u − 1
−u
a
8
=
−u − 1
0
a
6
=
u
u
a
2
=
u
u − 1
a
5
=
1
−u + 1
a
1
=
0
−u
a
9
=
−u − 1
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −15
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
u
2
+ u − 1
c
4
, c
9
u
2
c
5
, c
6
u
2
− u − 1
c
7
(u − 1)
2
c
8
, c
10
(u + 1)
2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
y
2
− 3y + 1
c
4
, c
9
y
2
c
7
, c
8
, c
10
(y −1)
2
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.61803
b = −0.618034
−2.63189 −15.0000
u = −1.61803
a = 0.618034
b = 1.61803
−10.5276 −15.0000
5
II. I
u
2
= hb − u, a − u + 1, u
2
− 3u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
3u − 1
a
10
=
u − 1
u
a
8
=
−3u + 1
−4u + 2
a
6
=
u
u
a
2
=
−3u + 2
−3u + 1
a
5
=
−6u + 3
−7u + 3
a
1
=
12u − 4
15u − 6
a
9
=
−3u + 1
−25u + 10
(ii) Obstruction class = −1
(iii) Cusp Shapes = −15
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
u
2
+ 3u + 1
c
4
, c
9
u
2
− 8u − 4
c
7
, c
10
u
2
− 4u − 1
c
8
u
2
+ 18u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
y
2
− 7y + 1
c
4
, c
9
y
2
− 72y + 16
c
7
, c
10
y
2
− 18y + 1
c
8
y
2
− 322y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.381966
a = −0.618034
b = 0.381966
−0.657974 −15.0000
u = 2.61803
a = 1.61803
b = 2.61803
7.23771 −15.0000
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
(u
2
+ u − 1)(u
2
+ 3u + 1)
c
4
, c
9
u
2
(u
2
− 8u − 4)
c
5
, c
6
(u
2
− u − 1)(u
2
+ 3u + 1)
c
7
(u − 1)
2
(u
2
− 4u − 1)
c
8
(u + 1)
2
(u
2
+ 18u + 1)
c
10
(u + 1)
2
(u
2
− 4u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
(y
2
− 7y + 1)(y
2
− 3y + 1)
c
4
, c
9
y
2
(y
2
− 72y + 16)
c
7
, c
10
(y −1)
2
(y
2
− 18y + 1)
c
8
(y −1)
2
(y
2
− 322y + 1)
11
