10
140
(K10n
29
)
A knot diagram
1
Linearized knot diagam
8 6 7 9 8 3 1 5 6 2
Solving Sequence
2,6 3,8
1 5 7 10 9 4
c
2
c
1
c
5
c
7
c
10
c
9
c
4
c
3
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
6
− 7u
5
− 14u
4
+ 39u
3
+ 32u
2
+ 29b − 47u + 4,
− 25u
6
+ 44u
5
+ 146u
4
− 183u
3
− 255u
2
+ 174a + 167u − 108,
u
7
− 2u
6
− 5u
5
+ 9u
4
+ 9u
3
− 14u
2
+ 3u + 3i
I
u
2
= hb − 1, a
2
+ 2, u − 1i
I
u
3
= hb + 1, a, u + 1i
* 3 irreducible components of dim
C
= 0, with total 10 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
= h2u
6
− 7u
5
+ · · · + 29b + 4, −25u
6
+ 44u
5
+ · · · + 174a − 108, u
7
−
2u
6
− 5u
5
+ 9u
4
+ 9u
3
− 14u
2
+ 3u + 3i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
−u
2
a
8
=
0.143678u
6
− 0.252874u
5
+ ··· − 0.959770u + 0.620690
−0.0689655u
6
+ 0.241379u
5
+ ··· + 1.62069u − 0.137931
a
1
=
0.0804598u
6
− 0.281609u
5
+ ··· − 1.55747u + 1.32759
0.103448u
6
+ 0.137931u
5
+ ··· + 0.0689655u − 0.793103
a
5
=
0.0632184u
6
+ 0.0287356u
5
+ ··· − 0.402299u − 0.706897
0.120690u
6
− 0.172414u
5
+ ··· + 0.913793u + 0.241379
a
7
=
u
−u
3
+ u
a
10
=
0.183908u
6
− 0.143678u
5
+ ··· − 1.48851u + 0.534483
0.103448u
6
+ 0.137931u
5
+ ··· + 0.0689655u − 0.793103
a
9
=
0.183908u
6
− 0.143678u
5
+ ··· − 1.48851u + 0.534483
−0.310345u
6
+ 0.0862069u
5
+ ··· + 1.29310u − 0.120690
a
4
=
u
2
− 1
−u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
43
29
u
6
+
78
29
u
5
+
214
29
u
4
−
331
29
u
3
−
369
29
u
2
+
445
29
u −
144
29
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
7
+ 2u
6
+ 3u
5
+ u
4
+ 5u
3
− 2u
2
− u + 3
c
2
, c
3
, c
6
u
7
− 2u
6
− 5u
5
+ 9u
4
+ 9u
3
− 14u
2
+ 3u + 3
c
4
, c
5
, c
8
u
7
− u
6
+ 7u
5
− 3u
4
+ 12u
3
+ 2u
2
+ 4u + 2
c
9
u
7
+ 10u
6
+ 70u
5
+ 250u
4
+ 410u
3
+ 180u
2
+ 56u + 16
c
10
u
7
− 2u
6
+ 15u
5
− 35u
4
+ 11u
3
+ 20u
2
+ 13u + 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
7
+ 2y
6
+ 15y
5
+ 35y
4
+ 11y
3
− 20y
2
+ 13y −9
c
2
, c
3
, c
6
y
7
− 14y
6
+ 79y
5
− 221y
4
+ 315y
3
− 196y
2
+ 93y −9
c
4
, c
5
, c
8
y
7
+ 13y
6
+ 67y
5
+ 171y
4
+ 216y
3
+ 104y
2
+ 8y −4
c
9
y
7
+ 40y
6
+ 720y
5
− 8588y
4
+ 85620y
3
+ 5520y
2
− 2624y −256
c
10
y
7
+ 26y
6
+ 107y
5
− 789y
4
+ 1947y
3
+ 516y
2
− 191y −81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.673944 + 0.445187I
a = 0.544144 + 0.706219I
b = 0.593853 − 0.464339I
1.22231 + 1.45738I 0.50826 − 4.10370I
u = 0.673944 − 0.445187I
a = 0.544144 − 0.706219I
b = 0.593853 + 0.464339I
1.22231 − 1.45738I 0.50826 + 4.10370I
u = −0.350429
a = 1.08068
b = −0.777623
−1.01758 −11.3200
u = −1.61248 + 0.50127I
a = −0.519526 + 0.799826I
b = 0.227371 − 1.297870I
8.76077 + 1.03782I 1.54723 − 0.70964I
u = −1.61248 − 0.50127I
a = −0.519526 − 0.799826I
b = 0.227371 + 1.297870I
8.76077 − 1.03782I 1.54723 + 0.70964I
u = 2.11375 + 0.36632I
a = −0.064957 − 0.921422I
b = −1.43241 + 1.36324I
−17.6990 + 5.2126I 0.60442 − 1.93466I
u = 2.11375 − 0.36632I
a = −0.064957 + 0.921422I
b = −1.43241 − 1.36324I
−17.6990 − 5.2126I 0.60442 + 1.93466I
5
II. I
u
2
= hb − 1, a
2
+ 2, u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
−1
a
8
=
a
1
a
1
=
−a + 1
−1
a
5
=
−2
a + 1
a
7
=
1
0
a
10
=
−a
−1
a
9
=
−a
a − 1
a
4
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
(u − 1)
2
c
4
, c
5
, c
8
c
9
u
2
+ 2
c
6
, c
7
(u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
, c
7
, c
10
(y −1)
2
c
4
, c
5
, c
8
c
9
(y + 2)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.414210I
b = 1.00000
4.93480 0
u = 1.00000
a = − 1.414210I
b = 1.00000
4.93480 0
9
III. I
u
3
= hb + 1, a, u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
−1
a
3
=
1
−1
a
8
=
0
−1
a
1
=
1
−1
a
5
=
0
−1
a
7
=
−1
0
a
10
=
0
−1
a
9
=
0
−1
a
4
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
u + 1
c
4
, c
5
, c
8
c
9
u
c
6
, c
7
, c
10
u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
, c
7
, c
10
y −1
c
4
, c
5
, 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 = 0
b = −1.00000
0 0
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
(u + 1)(u
7
+ 2u
6
+ 3u
5
+ u
4
+ 5u
3
− 2u
2
− u + 3)
c
2
, c
3
(u − 1)
2
(u + 1)(u
7
− 2u
6
− 5u
5
+ 9u
4
+ 9u
3
− 14u
2
+ 3u + 3)
c
4
, c
5
, c
8
u(u
2
+ 2)(u
7
− u
6
+ 7u
5
− 3u
4
+ 12u
3
+ 2u
2
+ 4u + 2)
c
6
(u − 1)(u + 1)
2
(u
7
− 2u
6
− 5u
5
+ 9u
4
+ 9u
3
− 14u
2
+ 3u + 3)
c
7
(u − 1)(u + 1)
2
(u
7
+ 2u
6
+ 3u
5
+ u
4
+ 5u
3
− 2u
2
− u + 3)
c
9
u(u
2
+ 2)(u
7
+ 10u
6
+ 70u
5
+ 250u
4
+ 410u
3
+ 180u
2
+ 56u + 16)
c
10
(u − 1)
3
(u
7
− 2u
6
+ 15u
5
− 35u
4
+ 11u
3
+ 20u
2
+ 13u + 9)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y −1)
3
(y
7
+ 2y
6
+ 15y
5
+ 35y
4
+ 11y
3
− 20y
2
+ 13y −9)
c
2
, c
3
, c
6
(y −1)
3
(y
7
− 14y
6
+ 79y
5
− 221y
4
+ 315y
3
− 196y
2
+ 93y −9)
c
4
, c
5
, c
8
y(y + 2)
2
(y
7
+ 13y
6
+ 67y
5
+ 171y
4
+ 216y
3
+ 104y
2
+ 8y −4)
c
9
y(y + 2)
2
· (y
7
+ 40y
6
+ 720y
5
− 8588y
4
+ 85620y
3
+ 5520y
2
− 2624y −256)
c
10
((y −1)
3
)(y
7
+ 26y
6
+ ··· − 191y −81)
15
