10
157
(K10n
42
)
A knot diagram
1
Linearized knot diagam
4 5 9 7 3 1 5 6 4 6
Solving Sequence
3,5 6,7
8 9 2 4 1 10
c
5
c
7
c
8
c
2
c
4
c
1
c
10
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, −u
2
+ a − 2u, u
3
+ 2u
2
+ u − 1i
I
u
2
= hu
2
a + u
2
+ b, −u
2
a + a
2
− 3u
2
+ 2a + 4u − 3, u
3
− u
2
+ 1i
I
u
3
= h3u
5
+ 5u
4
− 2u
3
− 15u
2
+ 4b − 22u − 12, −5u
5
− 5u
4
+ 4u
3
+ 21u
2
+ 8a + 28u + 12,
u
6
+ 3u
5
+ 2u
4
− 5u
3
− 14u
2
− 16u − 8i
I
u
4
= hb + u, u
2
+ a, u
3
− u + 1i
I
u
5
= hb + u, 2u
2
a + a
2
− 3au + 2u
2
− 4u + 4, u
3
− u
2
+ 1i
I
u
6
= hau + b − u + 1, −u
2
a + a
2
+ au + u
2
− a − u + 1, u
3
− u
2
+ 1i
I
u
7
= hb − u + 1, a + 2u − 2, u
2
− u − 1i
I
u
8
= hb − u − 1, a, u
2
+ u + 1i
* 8 irreducible components of dim
C
= 0, with total 34 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, −u
2
+ a − 2u, u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
3
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
7
=
u
2
+ 2u
−u
a
8
=
u
2
+ u
−u
a
9
=
1
u − 1
a
2
=
−u
u
a
4
=
u
u
2
a
1
=
u
2
+ u − 1
−u + 1
a
10
=
u
2
− 1
−2u
2
− 2u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
+ 13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
3
− 3u
2
+ 4u − 1
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
u
3
+ 2u
2
+ u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
3
− y
2
+ 10y −1
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
y
3
− 2y
2
+ 5y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.23279 + 0.79255I
a = −1.57395 − 0.36899I
b = 1.23279 − 0.79255I
1.24160 − 12.66530I 9.43351 + 7.81637I
u = −1.23279 − 0.79255I
a = −1.57395 + 0.36899I
b = 1.23279 + 0.79255I
1.24160 + 12.66530I 9.43351 − 7.81637I
u = 0.465571
a = 1.14790
b = −0.465571
0.806671 12.1330
5
II. I
u
2
= hu
2
a + u
2
+ b, −u
2
a + a
2
− 3u
2
+ 2a + 4u − 3, u
3
− u
2
+ 1i
(i) Arc colorings
a
3
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
7
=
a
−u
2
a − u
2
a
8
=
−u
2
a − u
2
+ a
−u
2
a − u
2
a
9
=
−au + u
2
− u − 1
au − u
2
+ 2u + 1
a
2
=
−u
u
a
4
=
au − 2u
2
+ a + 3u
−u
2
a − au + u
2
− 3u
a
1
=
u
2
a − au − u
2
+ u
au
a
10
=
u
2
a − au − u
2
au + u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u + 14
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
6
− 3u
4
− 2u
3
+ 6u
2
− 2u − 1
c
2
, c
5
, c
6
c
10
(u
3
− u
2
+ 1)
2
c
3
, c
4
, c
7
c
9
u
6
+ 3u
5
+ 2u
4
− 5u
3
− 14u
2
− 16u − 8
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
6
− 6y
5
+ 21y
4
− 42y
3
+ 34y
2
− 16y + 1
c
2
, c
5
, c
6
c
10
(y
3
− y
2
+ 2y −1)
2
c
3
, c
4
, c
7
c
9
y
6
− 5y
5
+ 6y
4
− y
3
+ 4y
2
− 32y + 64
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.165364 + 0.499124I
b = 0.472913 − 1.198340I
−1.11345 + 5.65624I 6.98049 − 5.95889I
u = 0.877439 + 0.744862I
a = −1.61956 + 0.80802I
b = 1.189450 + 0.636059I
−1.11345 + 5.65624I 6.98049 − 5.95889I
u = 0.877439 − 0.744862I
a = −0.165364 − 0.499124I
b = 0.472913 + 1.198340I
−1.11345 − 5.65624I 6.98049 + 5.95889I
u = 0.877439 − 0.744862I
a = −1.61956 − 0.80802I
b = 1.189450 − 0.636059I
−1.11345 − 5.65624I 6.98049 + 5.95889I
u = −0.754878
a = 2.15552
b = −1.79815
7.16171 20.0390
u = −0.754878
a = −3.58568
b = 1.47343
7.16171 20.0390
9
III. I
u
3
= h3u
5
+ 5u
4
− 2u
3
− 15u
2
+ 4b − 22u − 12, −5u
5
− 5u
4
+ 4u
3
+
21u
2
+ 8a + 28u + 12, u
6
+ 3u
5
+ 2u
4
− 5u
3
− 14u
2
− 16u − 8i
(i) Arc colorings
a
3
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
7
=
5
8
u
5
+
5
8
u
4
+ ··· −
7
2
u −
3
2
−
3
4
u
5
−
5
4
u
4
+ ··· +
11
2
u + 3
a
8
=
−
1
8
u
5
−
5
8
u
4
+ ··· + 2u +
3
2
−
3
4
u
5
−
5
4
u
4
+ ··· +
11
2
u + 3
a
9
=
−
3
8
u
5
−
3
8
u
4
+ ··· +
3
2
u +
1
2
5
4
u
5
+
7
4
u
4
+ ··· −
17
2
u − 5
a
2
=
−u
u
a
4
=
−
1
4
u
5
−
1
4
u
4
+ ··· + u +
1
2
1
2
u
5
+ u
4
+ ··· −
7
2
u − 2
a
1
=
−
1
4
u
5
−
1
2
u
4
+ ··· +
9
4
u + 2
1
4
u
5
+
1
4
u
4
−
1
4
u
2
− u − 2
a
10
=
−
1
4
u
4
−
1
4
u
3
+
5
4
u + 2
−
3
4
u
5
−
3
4
u
4
+
11
4
u
2
+ 5u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6u
5
− 10u
4
+ 4u
3
+ 30u
2
+ 44u + 38
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
6
− 3u
4
− 2u
3
+ 6u
2
− 2u − 1
c
2
, c
5
, c
6
c
10
u
6
+ 3u
5
+ 2u
4
− 5u
3
− 14u
2
− 16u − 8
c
3
, c
4
, c
7
c
9
(u
3
− u
2
+ 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
6
− 6y
5
+ 21y
4
− 42y
3
+ 34y
2
− 16y + 1
c
2
, c
5
, c
6
c
10
y
6
− 5y
5
+ 6y
4
− y
3
+ 4y
2
− 32y + 64
c
3
, c
4
, c
7
c
9
(y
3
− y
2
+ 2y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.472913 + 1.198340I
a = 0.374563 + 0.283509I
b = −0.877439 − 0.744862I
−1.11345 + 5.65624I 6.98049 − 5.95889I
u = −0.472913 − 1.198340I
a = 0.374563 − 0.283509I
b = −0.877439 + 0.744862I
−1.11345 − 5.65624I 6.98049 + 5.95889I
u = −1.189450 + 0.636059I
a = 1.49641 + 0.38207I
b = −0.877439 + 0.744862I
−1.11345 − 5.65624I 6.98049 + 5.95889I
u = −1.189450 − 0.636059I
a = 1.49641 − 0.38207I
b = −0.877439 − 0.744862I
−1.11345 + 5.65624I 6.98049 − 5.95889I
u = −1.47343
a = −1.83705
b = 0.754878
7.16171 20.0390
u = 1.79815
a = −0.904909
b = 0.754878
7.16171 20.0390
13
IV. I
u
4
= hb + u, u
2
+ a, u
3
− u + 1i
(i) Arc colorings
a
3
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
7
=
−u
2
−u
a
8
=
−u
2
− u
−u
a
9
=
−1
−u + 1
a
2
=
−u
u
a
4
=
u
u
2
a
1
=
u
2
− u − 1
u − 1
a
10
=
u
2
− 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 8u + 13
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− u
2
+ 2u − 1
c
2
, c
4
, c
6
c
9
u
3
− u − 1
c
3
, c
5
, c
7
c
10
u
3
− u + 1
c
8
u
3
+ u
2
+ 2u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
3
+ 3y
2
+ 2y −1
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
y
3
− 2y
2
+ y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.662359 + 0.562280I
a = −0.122561 − 0.744862I
b = −0.662359 − 0.562280I
1.83893 + 3.77083I 7.21088 − 7.47768I
u = 0.662359 − 0.562280I
a = −0.122561 + 0.744862I
b = −0.662359 + 0.562280I
1.83893 − 3.77083I 7.21088 + 7.47768I
u = −1.32472
a = −1.75488
b = 1.32472
9.48162 16.5780
17
V. I
u
5
= hb + u, 2u
2
a + a
2
− 3au + 2u
2
− 4u + 4, u
3
− u
2
+ 1i
(i) Arc colorings
a
3
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
7
=
a
−u
a
8
=
a − u
−u
a
9
=
−u
2
a + u
2
+ a − 1
u
2
a − au − u
2
− a + 2
a
2
=
−u
u
a
4
=
−au + 1
u
2
a
1
=
u
2
− 2u + 1
−u
2
a + au + u
2
+ a − 2
a
10
=
u
2
a − au − a − u + 2
2au + a − u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
2
a − 4au − 4u
2
− 4a + 18
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
(u
3
+ u
2
+ 2u + 1)
2
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(u
3
− u
2
+ 1)
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(y
3
− y
2
+ 2y −1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.592519 − 0.137827I
b = −0.877439 − 0.744862I
3.02413 + 2.82812I 13.50976 − 2.97945I
u = 0.877439 + 0.744862I
a = 1.60964 − 0.24187I
b = −0.877439 − 0.744862I
−1.11345 6.98049 + 0.I
u = 0.877439 − 0.744862I
a = 0.592519 + 0.137827I
b = −0.877439 + 0.744862I
3.02413 − 2.82812I 13.50976 + 2.97945I
u = 0.877439 − 0.744862I
a = 1.60964 + 0.24187I
b = −0.877439 + 0.744862I
−1.11345 6.98049 + 0.I
u = −0.754878
a = −1.70216 + 2.29387I
b = 0.754878
3.02413 − 2.82812I 13.50976 + 2.97945I
u = −0.754878
a = −1.70216 − 2.29387I
b = 0.754878
3.02413 + 2.82812I 13.50976 − 2.97945I
21
VI. I
u
6
= hau + b − u + 1, −u
2
a + a
2
+ au + u
2
− a − u + 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
3
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
7
=
a
−au + u − 1
a
8
=
−au + a + u − 1
−au + u − 1
a
9
=
1
−au − u
2
+ a + u − 2
a
2
=
−u
u
a
4
=
u
−u
2
a + au − u + 1
a
1
=
au + u
2
− u
−u
2
a + au + a + 1
a
10
=
u
2
− 1
au + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4au + 10
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(u
3
− u
2
+ 1)
2
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
6
+ 3y
5
+ 2y
4
− 25y
3
+ 8y
2
+ 48y + 64
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(y
3
− y
2
+ 2y −1)
2
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.162359 + 0.986732I
b = 0.754878
3.02413 + 2.82812I 13.50976 − 2.97945I
u = 0.877439 + 0.744862I
a = 0.500000 − 0.424452I
b = −0.877439 + 0.744862I
−1.11345 6.98049 + 0.I
u = 0.877439 − 0.744862I
a = −0.162359 − 0.986732I
b = 0.754878
3.02413 − 2.82812I 13.50976 + 2.97945I
u = 0.877439 − 0.744862I
a = 0.500000 + 0.424452I
b = −0.877439 − 0.744862I
−1.11345 6.98049 + 0.I
u = −0.754878
a = 1.16236 + 0.98673I
b = −0.877439 + 0.744862I
3.02413 − 2.82812I 13.50976 + 2.97945I
u = −0.754878
a = 1.16236 − 0.98673I
b = −0.877439 − 0.744862I
3.02413 + 2.82812I 13.50976 − 2.97945I
25
VII. I
u
7
= hb − u + 1, a + 2u − 2, u
2
− u − 1i
(i) Arc colorings
a
3
=
0
u
a
5
=
1
0
a
6
=
1
−u − 1
a
7
=
−2u + 2
u − 1
a
8
=
−u + 1
u − 1
a
9
=
−u + 2
−2
a
2
=
−u
u
a
4
=
2u − 3
−u + 2
a
1
=
u − 3
2
a
10
=
2u − 3
−2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
26
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u + 1)
2
c
2
, c
4
, c
6
c
9
u
2
+ u − 1
c
3
, c
5
, c
7
c
10
u
2
− u − 1
c
8
(u − 1)
2
27
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y −1)
2
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
y
2
− 3y + 1
28
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 3.23607
b = −1.61803
6.57974 3.00000
u = 1.61803
a = −1.23607
b = 0.618034
6.57974 3.00000
29
VIII. I
u
8
= hb − u − 1, a, u
2
+ u + 1i
(i) Arc colorings
a
3
=
0
u
a
5
=
1
0
a
6
=
1
u + 1
a
7
=
0
u + 1
a
8
=
u + 1
u + 1
a
9
=
u
0
a
2
=
−u
u
a
4
=
1
u
a
1
=
−u − 1
0
a
10
=
−2u − 1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3
30
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
(u + 1)
2
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
u
2
+ u + 1
31
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y −1)
2
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
y
2
+ y + 1
32
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = 0.500000 + 0.866025I
−3.28987 3.00000
u = −0.500000 − 0.866025I
a = 0
b = 0.500000 − 0.866025I
−3.28987 3.00000
33
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)
4
(u
3
− 3u
2
+ 4u − 1)(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
· (u
6
− 3u
4
− 2u
3
+ 6u
2
− 2u − 1)
2
· (u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8)
c
2
, c
4
, c
6
c
9
(u
2
+ u − 1)(u
2
+ u + 1)(u
3
− u − 1)(u
3
− u
2
+ 1)
6
(u
3
+ 2u
2
+ u − 1)
· (u
6
+ 3u
5
+ 2u
4
− 5u
3
− 14u
2
− 16u − 8)
c
3
, c
5
, c
7
c
10
(u
2
− u − 1)(u
2
+ u + 1)(u
3
− u + 1)(u
3
− u
2
+ 1)
6
(u
3
+ 2u
2
+ u − 1)
· (u
6
+ 3u
5
+ 2u
4
− 5u
3
− 14u
2
− 16u − 8)
c
8
(u − 1)
2
(u + 1)
2
(u
3
− 3u
2
+ 4u − 1)(u
3
+ u
2
+ 2u + 1)
3
· (u
6
− 3u
4
− 2u
3
+ 6u
2
− 2u − 1)
2
· (u
6
− 5u
5
+ 14u
4
− 25u
3
+ 28u
2
− 20u + 8)
34
X. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y −1)
4
(y
3
− y
2
+ 10y −1)(y
3
+ 3y
2
+ 2y −1)
3
· (y
6
− 6y
5
+ 21y
4
− 42y
3
+ 34y
2
− 16y + 1)
2
· (y
6
+ 3y
5
+ 2y
4
− 25y
3
+ 8y
2
+ 48y + 64)
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(y
2
− 3y + 1)(y
2
+ y + 1)(y
3
− 2y
2
+ y −1)(y
3
− 2y
2
+ 5y −1)
· (y
3
− y
2
+ 2y −1)
6
(y
6
− 5y
5
+ 6y
4
− y
3
+ 4y
2
− 32y + 64)
35
