10
74
(K10a
62
)
A knot diagram
1
Linearized knot diagam
7 8 9 1 10 2 6 4 3 5
Solving Sequence
1,4 5,9
3 8 2 10 6 7
c
4
c
3
c
8
c
2
c
10
c
5
c
7
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, u
9
+ 4u
7
+ 3u
5
− 5u
3
+ u
2
+ 2a − 3u + 1, u
10
− u
9
+ 6u
8
− 6u
7
+ 13u
6
− 13u
5
+ 11u
4
− 10u
3
+ 2u
2
+ 1i
I
u
2
= hu
5
+ 2u
3
+ u
2
+ b + u + 1, −u
7
− 3u
5
− 2u
4
− 2u
3
− 4u
2
+ 2a − u − 1,
u
8
+ 3u
6
+ 2u
5
+ 2u
4
+ 4u
3
+ u
2
+ u + 2i
I
u
3
= hu
5
+ 2u
3
− u
2
+ b + 2u − 1, −u
5
+ u
4
− 2u
3
+ 2u
2
+ a − 2u + 2, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
I
u
4
= hb
2
+ bu + u
2
+ 1, −u
2
+ a − u − 2, u
3
+ u
2
+ 2u + 1i
I
u
5
= hb − u, a + 2u + 2, u
3
+ u
2
+ 2u + 1i
I
u
6
= hb + u, a − u − 1, u
2
+ 1i
* 6 irreducible components of dim
C
= 0, with total 35 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
9
+ 4u
7
+ 3u
5
− 5u
3
+ u
2
+ 2a − 3u + 1, u
10
− u
9
+ · · · + 2u
2
+ 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
−
1
2
u
9
− 2u
7
+ ··· +
3
2
u −
1
2
u
a
3
=
1
2
u
9
− u
8
+ ··· +
1
2
u +
1
2
−u
2
a
8
=
−
1
2
u
9
− 2u
7
+ ··· +
5
2
u −
1
2
u
a
2
=
1
2
u
9
− u
8
+ ··· +
1
2
u +
1
2
−u
4
− 2u
2
a
10
=
u
u
3
+ u
a
6
=
u
2
+ 1
u
4
+ 2u
2
a
7
=
−
1
2
u
9
− 3u
7
+ ··· +
7
2
u −
1
2
−
1
2
u
9
− 3u
7
+ ··· +
3
2
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
8
− 2u
7
+ 20u
6
− 10u
5
+ 32u
4
− 20u
3
+ 12u
2
− 14u − 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
10
+ 2u
9
+ 4u
8
+ 4u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 5u
2
+ 3u + 2
c
2
u
10
− 2u
9
+ u
8
− 4u
7
+ 10u
6
− 2u
5
+ 27u
4
− 66u
3
+ 32u
2
+ 4u + 8
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
10
− u
9
+ 6u
8
− 6u
7
+ 13u
6
− 13u
5
+ 11u
4
− 10u
3
+ 2u
2
+ 1
c
7
u
10
+ 4u
9
+ 10u
8
+ 14u
7
+ 15u
6
+ 10u
5
+ 7u
4
+ 5u
3
+ 11u
2
+ 11u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
10
+ 4y
9
+ 10y
8
+ 14y
7
+ 15y
6
+ 10y
5
+ 7y
4
+ 5y
3
+ 11y
2
+ 11y + 4
c
2
y
10
− 2y
9
+ ··· + 496y + 64
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
10
+ 11y
9
+ ··· + 4y + 1
c
7
y
10
+ 4y
9
+ ··· − 33y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.748770 + 0.138462I
a = 0.977962 + 0.048097I
b = 0.748770 + 0.138462I
−4.02991 − 3.81695I −11.33347 + 4.73761I
u = 0.748770 − 0.138462I
a = 0.977962 − 0.048097I
b = 0.748770 − 0.138462I
−4.02991 + 3.81695I −11.33347 − 4.73761I
u = 0.28433 + 1.41260I
a = 1.18060 − 2.05212I
b = 0.28433 + 1.41260I
8.47865 − 6.45670I 1.02275 + 3.64794I
u = 0.28433 − 1.41260I
a = 1.18060 + 2.05212I
b = 0.28433 − 1.41260I
8.47865 + 6.45670I 1.02275 − 3.64794I
u = −0.35489 + 1.40814I
a = −1.27311 − 1.80165I
b = −0.35489 + 1.40814I
5.86173 + 12.00600I −2.08626 − 7.39232I
u = −0.35489 − 1.40814I
a = −1.27311 + 1.80165I
b = −0.35489 − 1.40814I
5.86173 − 12.00600I −2.08626 + 7.39232I
u = 0.05139 + 1.48296I
a = 0.22617 − 2.44997I
b = 0.05139 + 1.48296I
11.63700 − 2.88363I 2.09026 + 2.85464I
u = 0.05139 − 1.48296I
a = 0.22617 + 2.44997I
b = 0.05139 − 1.48296I
11.63700 + 2.88363I 2.09026 − 2.85464I
u = −0.229588 + 0.355227I
a = −0.611625 + 0.659121I
b = −0.229588 + 0.355227I
−0.563291 + 1.057730I −7.69328 − 6.23330I
u = −0.229588 − 0.355227I
a = −0.611625 − 0.659121I
b = −0.229588 − 0.355227I
−0.563291 − 1.057730I −7.69328 + 6.23330I
5
II. I
u
2
= hu
5
+ 2u
3
+ u
2
+ b + u + 1, −u
7
− 3u
5
− 2u
4
− 2u
3
− 4u
2
+ 2a − u −
1, u
8
+ 3u
6
+ 2u
5
+ 2u
4
+ 4u
3
+ u
2
+ u + 2i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
1
2
u
7
+
3
2
u
5
+ ··· +
1
2
u +
1
2
−u
5
− 2u
3
− u
2
− u − 1
a
3
=
1
2
u
7
+
3
2
u
5
+ u
3
−
1
2
u +
1
2
−u
6
− 2u
4
− u
3
− u + 1
a
8
=
1
2
u
7
+
1
2
u
5
+ ··· −
1
2
u −
1
2
−u
5
− 2u
3
− u
2
− u − 1
a
2
=
1
2
u
7
+
1
2
u
5
+ ··· −
1
2
u −
1
2
−u
6
− u
5
− u
4
− 3u
3
− u − 1
a
10
=
u
u
3
+ u
a
6
=
u
2
+ 1
u
4
+ 2u
2
a
7
=
1
2
u
7
+
3
2
u
5
+ u
3
−
1
2
u +
1
2
u
7
+ 2u
5
+ u
3
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
6
− 4u
5
+ 8u
4
+ 8u − 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
4
+ u
2
+ u + 1)
2
c
2
(u
4
+ 3u
3
+ 4u
2
+ 3u + 2)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
8
+ 3u
6
+ 2u
5
+ 2u
4
+ 4u
3
+ u
2
+ u + 2
c
7
(u
4
+ 2u
3
+ 3u
2
+ u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
4
+ 2y
3
+ 3y
2
+ y + 1)
2
c
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
8
+ 6y
7
+ 13y
6
+ 10y
5
− 2y
4
− 4y
3
+ y
2
+ 3y + 4
c
7
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.856926 + 0.228629I
a = 1.089410 + 0.290658I
b = 0.309502 − 1.349500I
0.66484 + 7.64338I −5.77019 − 6.51087I
u = −0.856926 − 0.228629I
a = 1.089410 − 0.290658I
b = 0.309502 + 1.349500I
0.66484 − 7.64338I −5.77019 + 6.51087I
u = 0.511330 + 0.719091I
a = −0.656772 + 0.923628I
b = 0.036094 − 1.304740I
4.26996 − 1.39709I −0.22981 + 3.86736I
u = 0.511330 − 0.719091I
a = −0.656772 − 0.923628I
b = 0.036094 + 1.304740I
4.26996 + 1.39709I −0.22981 − 3.86736I
u = 0.036094 + 1.304740I
a = −0.021186 + 0.765848I
b = 0.511330 − 0.719091I
4.26996 + 1.39709I −0.22981 − 3.86736I
u = 0.036094 − 1.304740I
a = −0.021186 − 0.765848I
b = 0.511330 + 0.719091I
4.26996 − 1.39709I −0.22981 + 3.86736I
u = 0.309502 + 1.349500I
a = −0.161456 + 0.703984I
b = −0.856926 − 0.228629I
0.66484 − 7.64338I −5.77019 + 6.51087I
u = 0.309502 − 1.349500I
a = −0.161456 − 0.703984I
b = −0.856926 + 0.228629I
0.66484 + 7.64338I −5.77019 − 6.51087I
9
III. I
u
3
= hu
5
+ 2u
3
− u
2
+ b + 2u − 1, −u
5
+ u
4
− 2u
3
+ 2u
2
+ a − 2u +
2, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
u
5
− u
4
+ 2u
3
− 2u
2
+ 2u − 2
−u
5
− 2u
3
+ u
2
− 2u + 1
a
3
=
−u
5
− 2u
3
− u + 1
−u
5
− u
3
+ u
2
− u + 2
a
8
=
−u
4
− u
2
− 1
−u
5
− 2u
3
+ u
2
− 2u + 1
a
2
=
u
u
3
+ u
a
10
=
u
u
3
+ u
a
6
=
u
2
+ 1
u
4
+ 2u
2
a
7
=
−1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
+ 4u − 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
10
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
2
(u
3
− u
2
+ 1)
2
c
3
, c
8
, c
9
(u
3
+ u
2
+ 2u + 1)
2
c
7
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
10
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
2
(y
3
− y
2
+ 2y −1)
2
c
3
, c
8
, c
9
(y
3
+ 3y
2
+ 2y −1)
2
c
7
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498832 + 1.001300I
a = 0.398606 + 0.800120I
b = −0.215080 − 1.307140I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = −0.498832 − 1.001300I
a = 0.398606 − 0.800120I
b = −0.215080 + 1.307140I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = 0.284920 + 1.115140I
a = −0.215080 + 0.841795I
b = −0.569840
−1.11345 −9.01951 + 0.I
u = 0.284920 − 1.115140I
a = −0.215080 − 0.841795I
b = −0.569840
−1.11345 −9.01951 + 0.I
u = 0.713912 + 0.305839I
a = −1.183530 + 0.507021I
b = −0.215080 − 1.307140I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = 0.713912 − 0.305839I
a = −1.183530 − 0.507021I
b = −0.215080 + 1.307140I
3.02413 + 2.82812I −2.49024 − 2.97945I
13
IV. I
u
4
= hb
2
+ bu + u
2
+ 1, −u
2
+ a − u − 2, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
u
2
+ u + 2
b
a
3
=
−u
2
b − bu − 2b + 1
bu + u
2
+ 1
a
8
=
u
2
+ b + u + 2
b
a
2
=
u
2
− b + 2
u
2
b + 2bu + u
2
+ b + 1
a
10
=
u
−u
2
− u − 1
a
6
=
u
2
+ 1
u
2
+ u + 1
a
7
=
u
2
b + bu + 2b
bu + 2b − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
− 4u − 10
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
8
, c
9
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
2
(u
3
− u
2
+ 1)
2
c
4
, c
5
, c
10
(u
3
+ u
2
+ 2u + 1)
2
c
7
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
, c
9
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
2
(y
3
− y
2
+ 2y −1)
2
c
4
, c
5
, c
10
(y
3
+ 3y
2
+ 2y −1)
2
c
7
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0.122561 + 0.744862I
b = −0.498832 − 1.001300I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = −0.215080 + 1.307140I
a = 0.122561 + 0.744862I
b = 0.713912 − 0.305839I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = −0.215080 − 1.307140I
a = 0.122561 − 0.744862I
b = −0.498832 + 1.001300I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = −0.215080 − 1.307140I
a = 0.122561 − 0.744862I
b = 0.713912 + 0.305839I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = −0.569840
a = 1.75488
b = 0.284920 + 1.115140I
−1.11345 −9.01950
u = −0.569840
a = 1.75488
b = 0.284920 − 1.115140I
−1.11345 −9.01950
17
V. I
u
5
= hb − u, a + 2u + 2, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
−2u − 2
u
a
3
=
2u
2
+ 2u + 1
−u
2
a
8
=
−u − 2
u
a
2
=
u
−u
2
− u − 1
a
10
=
u
−u
2
− u − 1
a
6
=
u
2
+ 1
u
2
+ u + 1
a
7
=
−1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
− 4u − 10
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
10
u
3
+ u
2
+ 2u + 1
c
2
u
3
− u
2
+ 1
c
7
u
3
+ 3u
2
+ 2u − 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
10
y
3
+ 3y
2
+ 2y −1
c
2
y
3
− y
2
+ 2y −1
c
7
y
3
− 5y
2
+ 10y −1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −1.56984 − 2.61428I
b = −0.215080 + 1.307140I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = −0.215080 − 1.307140I
a = −1.56984 + 2.61428I
b = −0.215080 − 1.307140I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = −0.569840
a = −0.860319
b = −0.569840
−1.11345 −9.01950
21
VI. I
u
6
= hb + u, a − u − 1, u
2
+ 1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
5
=
1
−1
a
9
=
u + 1
−u
a
3
=
u
1
a
8
=
1
−u
a
2
=
u
1
a
10
=
u
0
a
6
=
0
−1
a
7
=
1
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
10
u
2
+ 1
c
2
u
2
c
7
(u + 1)
2
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
10
(y + 1)
2
c
2
y
2
c
7
(y −1)
2
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 1.00000 + 1.00000I
b = − 1.000000I
1.64493 −4.00000
u = − 1.000000I
a = 1.00000 − 1.00000I
b = 1.000000I
1.64493 −4.00000
25
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
+ 1)(u
3
+ u
2
+ 2u + 1)(u
4
+ u
2
+ u + 1)
2
· (u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
2
· (u
10
+ 2u
9
+ 4u
8
+ 4u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 5u
2
+ 3u + 2)
c
2
u
2
(u
3
− u
2
+ 1)
5
(u
4
+ 3u
3
+ 4u
2
+ 3u + 2)
2
· (u
10
− 2u
9
+ u
8
− 4u
7
+ 10u
6
− 2u
5
+ 27u
4
− 66u
3
+ 32u
2
+ 4u + 8)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(u
2
+ 1)(u
3
+ u
2
+ 2u + 1)
3
(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
8
+ 3u
6
+ 2u
5
+ 2u
4
+ 4u
3
+ u
2
+ u + 2)
· (u
10
− u
9
+ 6u
8
− 6u
7
+ 13u
6
− 13u
5
+ 11u
4
− 10u
3
+ 2u
2
+ 1)
c
7
(u + 1)
2
(u
3
+ 3u
2
+ 2u − 1)(u
4
+ 2u
3
+ 3u
2
+ u + 1)
2
· (u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
2
· (u
10
+ 4u
9
+ 10u
8
+ 14u
7
+ 15u
6
+ 10u
5
+ 7u
4
+ 5u
3
+ 11u
2
+ 11u + 4)
26
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y + 1)
2
(y
3
+ 3y
2
+ 2y −1)(y
4
+ 2y
3
+ 3y
2
+ y + 1)
2
· (y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
2
· (y
10
+ 4y
9
+ 10y
8
+ 14y
7
+ 15y
6
+ 10y
5
+ 7y
4
+ 5y
3
+ 11y
2
+ 11y + 4)
c
2
y
2
(y
3
− y
2
+ 2y −1)
5
(y
4
− y
3
+ 2y
2
+ 7y + 4)
2
· (y
10
− 2y
9
+ ··· + 496y + 64)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 1)
2
(y
3
+ 3y
2
+ 2y −1)
3
(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
8
+ 6y
7
+ 13y
6
+ 10y
5
− 2y
4
− 4y
3
+ y
2
+ 3y + 4)
· (y
10
+ 11y
9
+ ··· + 4y + 1)
c
7
(y −1)
2
(y
3
− 5y
2
+ 10y −1)(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)
2
· ((y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
2
)(y
10
+ 4y
9
+ ··· − 33y + 16)
27
