10
142
(K10n
30
)
A knot diagram
1
Linearized knot diagam
8 6 9 8 2 3 1 4 5 2
Solving Sequence
2,5
6
3,8
1 4 7 10 9
c
5
c
2
c
1
c
4
c
7
c
10
c
9
c
3
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
5
+ u
4
+ 4u
3
− 5u
2
+ 2b − u, a − 1, u
6
− u
5
− 5u
4
+ 4u
3
+ 5u
2
+ u − 1i
I
u
2
= h−u
3
+ b + u + 2, −u
3
− 2u
2
+ 3a + 2u + 6, u
4
− u
3
− 2u
2
+ 3i
I
u
3
= hb, a + 1, u + 1i
I
u
4
= hb
2
+ 2, a + 1, 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
= h−u
5
+ u
4
+ 4u
3
− 5u
2
+ 2b − u, a − 1, u
6
− u
5
− 5u
4
+ 4u
3
+ 5u
2
+ u − 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
1
1
2
u
5
−
1
2
u
4
− 2u
3
+
5
2
u
2
+
1
2
u
a
1
=
u
1
2
u
4
+
1
2
u
3
− 2u
2
+
1
2
u +
1
2
a
4
=
−
1
2
u
5
+
1
2
u
4
+ ··· −
1
2
u + 1
−
1
2
u
5
+
5
2
u
3
+ ··· − 2u +
1
2
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
u
1
2
u
4
−
1
2
u
3
− 2u
2
+
1
2
u +
1
2
a
9
=
1
2
u
4
−
1
2
u
3
− 2u
2
+
3
2
u +
1
2
1
2
u
4
−
1
2
u
3
− 2u
2
+
1
2
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
4
+ 3u
3
+ 4u
2
− 11u − 13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
u
6
+ u
5
− 5u
4
− 4u
3
+ 5u
2
− u − 1
c
3
, c
4
, c
8
u
6
+ 3u
5
+ 7u
4
+ 10u
3
+ 10u
2
+ 8u + 2
c
9
u
6
− 3u
5
− 11u
4
+ 32u
3
− 2u
2
+ 16u + 10
c
10
u
6
+ 11u
5
+ 43u
4
+ 66u
3
+ 27u
2
+ 11u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
y
6
− 11y
5
+ 43y
4
− 66y
3
+ 27y
2
− 11y + 1
c
3
, c
4
, c
8
y
6
+ 5y
5
+ 9y
4
− 4y
3
− 32y
2
− 24y + 4
c
9
y
6
− 31y
5
+ 309y
4
− 864y
3
− 1240y
2
− 296y + 100
c
10
y
6
− 35y
5
+ 451y
4
− 2274y
3
− 637y
2
− 67y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.526900 + 0.379519I
a = 1.00000
b = 0.036498 − 1.278320I
3.26038 + 1.42716I −6.28345 − 4.88332I
u = −0.526900 − 0.379519I
a = 1.00000
b = 0.036498 + 1.278320I
3.26038 − 1.42716I −6.28345 + 4.88332I
u = 0.338910
a = 1.00000
b = 0.374390
−0.610583 −16.1650
u = 1.85126 + 0.30576I
a = 1.00000
b = 0.63990 + 1.46861I
−13.0621 − 6.7708I −12.38492 + 2.96218I
u = 1.85126 − 0.30576I
a = 1.00000
b = 0.63990 − 1.46861I
−13.0621 + 6.7708I −12.38492 − 2.96218I
u = −1.98762
a = 1.00000
b = 1.27282
−17.6195 −14.4980
5
II. I
u
2
= h−u
3
+ b + u + 2, −u
3
− 2u
2
+ 3a + 2u + 6, u
4
− u
3
− 2u
2
+ 3i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
1
3
u
3
+
2
3
u
2
−
2
3
u − 2
u
3
− u − 2
a
1
=
−
4
3
u
3
+
1
3
u
2
+
5
3
u + 1
−u
3
− u
2
+ 3u + 2
a
4
=
2
3
u
3
+
1
3
u
2
−
1
3
u − 1
u
3
− u − 1
a
7
=
−u
2
+ 1
−u
3
+ 3
a
10
=
−
4
3
u
3
+
1
3
u
2
+
5
3
u + 1
u
3
− u − 1
a
9
=
−
1
3
u
3
+
1
3
u
2
+
2
3
u
u
3
− u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
+ 4u − 6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
u
4
+ u
3
− 2u
2
+ 3
c
3
, c
4
, c
8
(u
2
− u + 1)
2
c
9
(u
2
+ u + 1)
2
c
10
u
4
+ 5u
3
+ 10u
2
+ 12u + 9
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
y
4
− 5y
3
+ 10y
2
− 12y + 9
c
3
, c
4
, c
8
c
9
(y
2
+ y + 1)
2
c
10
y
4
− 5y
3
− 2y
2
+ 36y + 81
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.953264 + 0.702911I
a = −0.905826 − 0.839043I
b = −0.500000 + 0.866025I
−3.28987 + 2.02988I −12.00000 − 3.46410I
u = −0.953264 − 0.702911I
a = −0.905826 + 0.839043I
b = −0.500000 − 0.866025I
−3.28987 − 2.02988I −12.00000 + 3.46410I
u = 1.45326 + 0.16311I
a = −0.594174 + 0.550367I
b = −0.500000 + 0.866025I
−3.28987 + 2.02988I −12.00000 − 3.46410I
u = 1.45326 − 0.16311I
a = −0.594174 − 0.550367I
b = −0.500000 − 0.866025I
−3.28987 − 2.02988I −12.00000 + 3.46410I
9
III. I
u
3
= hb, a + 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
3
=
1
0
a
8
=
−1
0
a
1
=
−1
−1
a
4
=
1
0
a
7
=
0
1
a
10
=
−1
0
a
9
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u + 1
c
2
, c
7
, c
10
u − 1
c
3
, c
4
, c
8
c
9
u
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
10
y −1
c
3
, c
4
, c
8
c
9
y
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
13
IV. I
u
4
= hb
2
+ 2, a + 1, u − 1i
(i) Arc colorings
a
2
=
0
1
a
5
=
1
0
a
6
=
1
1
a
3
=
−1
0
a
8
=
−1
b
a
1
=
1
−b + 1
a
4
=
b + 1
2
a
7
=
0
1
a
10
=
1
−b
a
9
=
−b + 1
−b
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
(u − 1)
2
c
2
, c
7
(u + 1)
2
c
3
, c
4
, c
8
c
9
u
2
+ 2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
10
(y −1)
2
c
3
, c
4
, c
8
c
9
(y + 2)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 1.414210I
1.64493 −12.0000
u = 1.00000
a = −1.00000
b = − 1.414210I
1.64493 −12.0000
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
((u − 1)
2
)(u + 1)(u
4
+ u
3
− 2u
2
+ 3)(u
6
+ u
5
+ ··· − u − 1)
c
2
, c
7
(u − 1)(u + 1)
2
(u
4
+ u
3
− 2u
2
+ 3)(u
6
+ u
5
+ ··· − u − 1)
c
3
, c
4
, c
8
u(u
2
+ 2)(u
2
− u + 1)
2
(u
6
+ 3u
5
+ 7u
4
+ 10u
3
+ 10u
2
+ 8u + 2)
c
9
u(u
2
+ 2)(u
2
+ u + 1)
2
(u
6
− 3u
5
− 11u
4
+ 32u
3
− 2u
2
+ 16u + 10)
c
10
(u − 1)
3
(u
4
+ 5u
3
+ 10u
2
+ 12u + 9)
· (u
6
+ 11u
5
+ 43u
4
+ 66u
3
+ 27u
2
+ 11u + 1)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
(y −1)
3
(y
4
− 5y
3
+ 10y
2
− 12y + 9)
· (y
6
− 11y
5
+ 43y
4
− 66y
3
+ 27y
2
− 11y + 1)
c
3
, c
4
, c
8
y(y + 2)
2
(y
2
+ y + 1)
2
(y
6
+ 5y
5
+ 9y
4
− 4y
3
− 32y
2
− 24y + 4)
c
9
y(y + 2)
2
(y
2
+ y + 1)
2
· (y
6
− 31y
5
+ 309y
4
− 864y
3
− 1240y
2
− 296y + 100)
c
10
(y −1)
3
(y
4
− 5y
3
− 2y
2
+ 36y + 81)
· (y
6
− 35y
5
+ 451y
4
− 2274y
3
− 637y
2
− 67y + 1)
19
