10
154
(K10n
7
)
A knot diagram
1
Linearized knot diagam
9 4 1 7 4 9 5 2 6 3
Solving Sequence
3,10
1 4
2,6
5 9 7 8
c
10
c
3
c
2
c
5
c
9
c
6
c
7
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
5
+ 2u
4
+ u
3
− 2u
2
+ b − u, u
3
+ 2u
2
+ a + 2u, u
6
+ 3u
5
+ 3u
4
− 2u
3
− 4u
2
− u + 1i
I
u
2
= hb, u
2
+ a + 2u + 1, u
3
+ u
2
− 1i
I
u
3
= h−a
2
+ b − 3a − 1, a
3
+ 3a
2
+ 2a + 1, u − 1i
I
u
4
= hu
4
+ 2u
3
+ u
2
+ 2b − u − 1, −u
5
− 3u
4
− 7u
3
− 4u
2
+ 4a − 2u + 5, u
6
+ 2u
5
+ 4u
4
+ u
3
+ 2u
2
− 3u + 1i
* 4 irreducible components of dim
C
= 0, with total 18 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
5
+2u
4
+u
3
−2u
2
+b−u, u
3
+2u
2
+a+2u, u
6
+3u
5
+3u
4
−2u
3
−4u
2
−u+1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
u
2
a
4
=
−u
−u
3
+ u
a
2
=
u
3
u
5
− u
3
+ u
a
6
=
−u
3
− 2u
2
− 2u
−u
5
− 2u
4
− u
3
+ 2u
2
+ u
a
5
=
−2u
3
− 2u
2
− 2u
−2u
5
− 2u
4
+ 2u
2
+ u
a
9
=
−u
3
− 2u
2
+ 2
u
3
− u
a
7
=
−2u
2
− 2u − 1
−2u
4
− 2u
3
+ u
2
+ 2u
a
8
=
2u
4
+ 2u
3
− 2u − 1
−4u
5
− 8u
4
+ 8u
2
+ 4u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u
5
− 24u
4
− 32u
3
− 8u
2
− 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
c
9
u
6
− u
5
+ 5u
4
+ 2u
3
+ 4u
2
+ u − 1
c
2
, c
5
u
6
+ 3u
5
+ 13u
4
+ 20u
3
+ 18u
2
+ 9u + 1
c
3
, c
4
, c
7
c
10
u
6
− 3u
5
+ 3u
4
+ 2u
3
− 4u
2
+ u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
c
9
y
6
+ 9y
5
+ 37y
4
+ 36y
3
+ 2y
2
− 9y + 1
c
2
, c
5
y
6
+ 17y
5
+ 85y
4
+ 16y
3
− 10y
2
− 45y + 1
c
3
, c
4
, c
7
c
10
y
6
− 3y
5
+ 13y
4
− 20y
3
+ 18y
2
− 9y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.822978 + 0.498752I
a = 0.732119 − 0.244993I
b = 0.203480 − 0.959760I
1.78286 + 4.10821I −5.51333 − 7.68125I
u = −0.822978 − 0.498752I
a = 0.732119 + 0.244993I
b = 0.203480 + 0.959760I
1.78286 − 4.10821I −5.51333 + 7.68125I
u = 0.931750
a = −4.40872
b = −0.350492
−3.00199 −62.5370
u = 0.385643
a = −1.12608
b = 0.572966
−0.943503 −9.62410
u = −1.33572 + 1.10504I
a = −0.964719 − 0.871256I
b = −0.81472 + 2.12358I
14.9943 + 9.2499I −8.40611 − 3.97593I
u = −1.33572 − 1.10504I
a = −0.964719 + 0.871256I
b = −0.81472 − 2.12358I
14.9943 − 9.2499I −8.40611 + 3.97593I
5
II. I
u
2
= hb, u
2
+ a + 2u + 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
u
2
a
4
=
−u
u
2
+ u − 1
a
2
=
−u
2
+ 1
u
2
a
6
=
−u
2
− 2u − 1
0
a
5
=
−u
2
− 3u − 1
u
2
+ u − 1
a
9
=
1
0
a
7
=
−u
2
− 2u − 1
0
a
8
=
u
−u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
− 8
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ u
2
+ 2u + 1
c
2
, c
8
u
3
− u
2
+ 2u − 1
c
3
u
3
− u
2
+ 1
c
4
(u − 1)
3
c
5
, c
7
(u + 1)
3
c
6
, c
9
u
3
c
10
u
3
+ u
2
− 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
8
y
3
+ 3y
2
+ 2y −1
c
3
, c
10
y
3
− y
2
+ 2y −1
c
4
, c
5
, c
7
(y −1)
3
c
6
, c
9
y
3
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.539798 − 0.182582I
b = 0
1.37919 + 2.82812I −7.78492 − 1.30714I
u = −0.877439 − 0.744862I
a = 0.539798 + 0.182582I
b = 0
1.37919 − 2.82812I −7.78492 + 1.30714I
u = 0.754878
a = −3.07960
b = 0
−2.75839 −7.43020
9
III. I
u
3
= h−a
2
+ b − 3a − 1, a
3
+ 3a
2
+ 2a + 1, u − 1i
(i) Arc colorings
a
3
=
0
1
a
10
=
1
0
a
1
=
1
1
a
4
=
−1
0
a
2
=
1
1
a
6
=
a
a
2
+ 3a + 1
a
5
=
a
2
+ 4a + 1
a
2
+ 3a + 1
a
9
=
a + 2
a + 2
a
7
=
−2a
2
− 4a − 1
−a
2
− 2a
a
8
=
a + 2
a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −a
2
− 3a − 9
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
3
c
2
, c
10
(u − 1)
3
c
3
(u + 1)
3
c
4
u
3
+ u
2
− 1
c
5
, c
9
u
3
+ u
2
+ 2u + 1
c
6
u
3
− u
2
+ 2u − 1
c
7
u
3
− u
2
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
3
c
2
, c
3
, c
10
(y −1)
3
c
4
, c
7
y
3
− y
2
+ 2y −1
c
5
, c
6
, c
9
y
3
+ 3y
2
+ 2y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.337641 + 0.562280I
b = −0.215080 + 1.307140I
1.37919 + 2.82812I −7.78492 − 1.30714I
u = 1.00000
a = −0.337641 − 0.562280I
b = −0.215080 − 1.307140I
1.37919 − 2.82812I −7.78492 + 1.30714I
u = 1.00000
a = −2.32472
b = −0.569840
−2.75839 −7.43020
13
IV. I
u
4
= hu
4
+ 2u
3
+ u
2
+ 2b − u − 1, −u
5
− 3u
4
− 7u
3
− 4u
2
+ 4a − 2u +
5, u
6
+ 2u
5
+ 4u
4
+ u
3
+ 2u
2
− 3u + 1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
1
=
1
u
2
a
4
=
−u
−u
3
+ u
a
2
=
u
3
u
5
− u
3
+ u
a
6
=
1
4
u
5
+
3
4
u
4
+ ··· +
1
2
u −
5
4
−
1
2
u
4
− u
3
−
1
2
u
2
+
1
2
u +
1
2
a
5
=
1
4
u
5
+
1
4
u
4
+ ··· + 2u −
7
4
u
5
+
1
2
u
4
+ ··· −
5
2
u +
3
2
a
9
=
−
3
4
u
5
−
5
4
u
4
+ ··· −
1
2
u +
7
4
3
4
u
5
+
1
4
u
4
+ ··· +
3
2
u −
3
4
a
7
=
1
2
u
5
+
3
2
u
4
+ ··· + u −
5
2
−
1
4
u
5
−
1
4
u
4
+ ··· + u +
3
4
a
8
=
−
3
4
u
5
−
1
4
u
4
+ ··· +
3
2
u −
9
4
7
4
u
5
−
3
4
u
4
+ ··· +
5
2
u +
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1
2
u
5
+ u
4
+
3
2
u
3
−
1
2
u
2
−
1
2
u − 9
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
c
9
u
6
− u
5
+ 8u
4
− u
3
+ 8u
2
+ 20u + 8
c
2
, c
5
u
6
− 4u
5
+ 16u
4
− 29u
3
+ 18u
2
+ 5u + 1
c
3
, c
4
, c
7
c
10
u
6
− 2u
5
+ 4u
4
− u
3
+ 2u
2
+ 3u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
c
9
y
6
+ 15y
5
+ 78y
4
+ 183y
3
+ 232y
2
− 272y + 64
c
2
, c
5
y
6
+ 16y
5
+ 60y
4
− 223y
3
+ 646y
2
+ 11y + 1
c
3
, c
4
, c
7
c
10
y
6
+ 4y
5
+ 16y
4
+ 29y
3
+ 18y
2
− 5y + 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.277479 + 1.215720I
a = −0.222521 − 0.974928I
b = −0.90097 + 1.51597I
4.69981 −6.19806 + 0.I
u = −0.277479 − 1.215720I
a = −0.222521 + 0.974928I
b = −0.90097 − 1.51597I
4.69981 −6.19806 + 0.I
u = 0.400969 + 0.193096I
a = −0.900969 + 0.433884I
b = 0.623490 − 0.085936I
−0.939962 −9.24698 + 0.I
u = 0.400969 − 0.193096I
a = −0.900969 − 0.433884I
b = 0.623490 + 0.085936I
−0.939962 −9.24698 + 0.I
u = −1.12349 + 1.40881I
a = 0.623490 + 0.781831I
b = −0.22252 − 2.53859I
15.9794 −7.55496 + 0.I
u = −1.12349 − 1.40881I
a = 0.623490 − 0.781831I
b = −0.22252 + 2.53859I
15.9794 −7.55496 + 0.I
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
9
u
3
(u
3
+ u
2
+ 2u + 1)(u
6
− u
5
+ 5u
4
+ 2u
3
+ 4u
2
+ u − 1)
· (u
6
− u
5
+ 8u
4
− u
3
+ 8u
2
+ 20u + 8)
c
2
((u − 1)
3
)(u
3
− u
2
+ 2u − 1)(u
6
− 4u
5
+ ··· + 5u + 1)
· (u
6
+ 3u
5
+ 13u
4
+ 20u
3
+ 18u
2
+ 9u + 1)
c
3
, c
7
(u + 1)
3
(u
3
− u
2
+ 1)(u
6
− 3u
5
+ 3u
4
+ 2u
3
− 4u
2
+ u + 1)
· (u
6
− 2u
5
+ 4u
4
− u
3
+ 2u
2
+ 3u + 1)
c
4
, c
10
(u − 1)
3
(u
3
+ u
2
− 1)(u
6
− 3u
5
+ 3u
4
+ 2u
3
− 4u
2
+ u + 1)
· (u
6
− 2u
5
+ 4u
4
− u
3
+ 2u
2
+ 3u + 1)
c
5
((u + 1)
3
)(u
3
+ u
2
+ 2u + 1)(u
6
− 4u
5
+ ··· + 5u + 1)
· (u
6
+ 3u
5
+ 13u
4
+ 20u
3
+ 18u
2
+ 9u + 1)
c
6
, c
8
u
3
(u
3
− u
2
+ 2u − 1)(u
6
− u
5
+ 5u
4
+ 2u
3
+ 4u
2
+ u − 1)
· (u
6
− u
5
+ 8u
4
− u
3
+ 8u
2
+ 20u + 8)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
c
9
y
3
(y
3
+ 3y
2
+ 2y −1)(y
6
+ 9y
5
+ 37y
4
+ 36y
3
+ 2y
2
− 9y + 1)
· (y
6
+ 15y
5
+ 78y
4
+ 183y
3
+ 232y
2
− 272y + 64)
c
2
, c
5
(y −1)
3
(y
3
+ 3y
2
+ 2y −1)
· (y
6
+ 16y
5
+ 60y
4
− 223y
3
+ 646y
2
+ 11y + 1)
· (y
6
+ 17y
5
+ 85y
4
+ 16y
3
− 10y
2
− 45y + 1)
c
3
, c
4
, c
7
c
10
((y −1)
3
)(y
3
− y
2
+ 2y −1)(y
6
− 3y
5
+ ··· − 9y + 1)
· (y
6
+ 4y
5
+ 16y
4
+ 29y
3
+ 18y
2
− 5y + 1)
19
