10
152
(K10n
36
)
A knot diagram
1
Linearized knot diagam
4 5 10 2 8 10 5 6 3 7
Solving Sequence
1,4
2
5,7
8 10 3 6 9
c
1
c
4
c
7
c
10
c
3
c
6
c
8
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
4
+ 2u
3
+ b − 2u, u
2
+ a + 2u + 1, u
5
+ 3u
4
+ 2u
3
− 3u
2
− 3u + 1i
I
u
2
= hb, a + u + 2, u
2
+ u − 1i
I
u
3
= hb − a − 1, a
2
+ a − 1, u − 1i
I
u
4
= hu
3
+ 2u
2
+ 2b − 1, −u
3
− 2u
2
+ 2a − 2u + 1, u
4
+ u
3
+ 2u
2
− u + 1i
* 4 irreducible components of dim
C
= 0, with total 13 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
+ 2u
3
+ b − 2u, u
2
+ a + 2u + 1, u
5
+ 3u
4
+ 2u
3
− 3u
2
− 3u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
7
=
−u
2
− 2u − 1
−u
4
− 2u
3
+ 2u
a
8
=
−2u − 1
−2u
3
− u
2
+ 2u
a
10
=
−u
3
− 2u
2
+ 2
u
3
− u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
−2u
2
− 2u
−2u
4
− 2u
3
+ 2u
2
+ u
a
9
=
2u
3
+ 2u
2
− 2u − 1
−4u
4
− 8u
3
+ 4u
2
+ 8u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u
4
− 24u
3
− 24u
2
− 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
u
5
− 3u
4
+ 2u
3
+ 3u
2
− 3u − 1
c
3
, c
6
, c
9
c
10
u
5
+ u
4
+ 2u
3
− 5u
2
− u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
y
5
− 5y
4
+ 16y
3
− 27y
2
+ 15y − 1
c
3
, c
6
, c
9
c
10
y
5
+ 3y
4
+ 12y
3
− 31y
2
+ 11y − 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.912859
a = −3.65903
b = −0.390081
−2.96486 −53.8110
u = −1.39373
a = −0.155021
b = −1.14610
−11.4408 −21.8300
u = −1.39814 + 0.93867I
a = 0.722590 + 0.747455I
b = 1.01518 − 1.84157I
5.12323 + 8.53607I −12.97824 − 4.17771I
u = −1.39814 − 0.93867I
a = 0.722590 − 0.747455I
b = 1.01518 + 1.84157I
5.12323 − 8.53607I −12.97824 + 4.17771I
u = 0.277157
a = −1.63113
b = 0.505833
−0.775637 −12.4020
5
II. I
u
2
= hb, a + u + 2, u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
−u + 1
a
5
=
−u
−u + 1
a
7
=
−u − 2
0
a
8
=
−2
u − 1
a
10
=
1
0
a
3
=
u
u
a
6
=
−u − 2
0
a
9
=
u
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −11
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
9
u
2
+ u − 1
c
3
, c
4
u
2
− u − 1
c
5
(u − 1)
2
c
6
, c
10
u
2
c
7
, c
8
(u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
9
y
2
− 3y + 1
c
5
, c
7
, c
8
(y − 1)
2
c
6
, c
10
y
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −2.61803
b = 0
−2.63189 −11.0000
u = −1.61803
a = −0.381966
b = 0
−10.5276 −11.0000
9
III. I
u
3
= hb − a − 1, a
2
+ a − 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
7
=
a
a + 1
a
8
=
−1
a + 1
a
10
=
0
−a − 2
a
3
=
0
1
a
6
=
a
−a − 2
a
9
=
0
−a − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −11
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
9
u
2
c
4
(u + 1)
2
c
5
, c
6
u
2
+ u − 1
c
7
, c
8
, c
10
u
2
− u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
2
c
3
, c
9
y
2
c
5
, c
6
, c
7
c
8
, c
10
y
2
− 3y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.618034
b = 1.61803
−10.5276 −11.0000
u = 1.00000
a = −1.61803
b = −0.618034
−2.63189 −11.0000
13
IV. I
u
4
= hu
3
+ 2u
2
+ 2b − 1, −u
3
− 2u
2
+ 2a − 2u + 1, u
4
+ u
3
+ 2u
2
− u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
7
=
1
2
u
3
+ u
2
+ u −
1
2
−
1
2
u
3
− u
2
+
1
2
a
8
=
u
3
+ u
2
+ u
−
3
2
u
3
+ u
2
− u +
1
2
a
10
=
−u
3
− u
2
− u + 1
3
2
u
3
+ u −
1
2
a
3
=
−u
2
+ 1
u
3
+ 4u
2
− u + 1
a
6
=
−
1
2
u
3
+ u
2
− u +
1
2
−
3
2
u
3
− 4u
2
+ 3u −
3
2
a
9
=
2u
3
+ 3u
2
+ 1
−
13
2
u
3
−
3
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −11
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
u
4
− u
3
+ 2u
2
+ u + 1
c
3
, c
6
, c
9
c
10
u
4
+ u
3
+ 5u
2
+ 8u + 4
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
y
4
+ 3y
3
+ 8y
2
+ 3y + 1
c
3
, c
6
, c
9
c
10
y
4
+ 9y
3
+ 17y
2
− 24y + 16
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.309017 + 0.535233I
a = −0.500000 + 0.866025I
b = 0.809017 − 0.330792I
−0.657974 −11.0000
u = 0.309017 − 0.535233I
a = −0.500000 − 0.866025I
b = 0.809017 + 0.330792I
−0.657974 −11.0000
u = −0.80902 + 1.40126I
a = −0.500000 − 0.866025I
b = −0.30902 + 2.26728I
7.23771 −11.0000
u = −0.80902 − 1.40126I
a = −0.500000 + 0.866025I
b = −0.30902 − 2.26728I
7.23771 −11.0000
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
((u − 1)
2
)(u
2
+ u − 1)(u
4
− u
3
+ ··· + u + 1)(u
5
− 3u
4
+ ··· − 3u − 1)
c
3
, c
10
u
2
(u
2
− u − 1)(u
4
+ u
3
+ ··· + 8u + 4)(u
5
+ u
4
+ ··· − u + 1)
c
4
, c
7
, c
8
((u + 1)
2
)(u
2
− u − 1)(u
4
− u
3
+ ··· + u + 1)(u
5
− 3u
4
+ ··· − 3u − 1)
c
6
, c
9
u
2
(u
2
+ u − 1)(u
4
+ u
3
+ ··· + 8u + 4)(u
5
+ u
4
+ ··· − u + 1)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
(y − 1)
2
(y
2
− 3y + 1)(y
4
+ 3y
3
+ 8y
2
+ 3y + 1)
· (y
5
− 5y
4
+ 16y
3
− 27y
2
+ 15y − 1)
c
3
, c
6
, c
9
c
10
y
2
(y
2
− 3y + 1)(y
4
+ 9y
3
+ 17y
2
− 24y + 16)
· (y
5
+ 3y
4
+ 12y
3
− 31y
2
+ 11y − 1)
19
