10
153
(K10n
10
)
A knot diagram
1
Linearized knot diagam
9 6 10 7 3 8 5 1 3 8
Solving Sequence
5,7 8,10
1 4 3 6 2 9
c
7
c
10
c
4
c
3
c
6
c
2
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
2
+ b + u + 1, −u
4
+ 6u
3
− 11u
2
+ 2a + u + 11, u
5
− 5u
4
+ 7u
3
+ 2u
2
− 8u − 1i
I
u
2
= hu
2
+ b − u + 1, u
2
+ a − u + 1, u
3
− u
2
+ 1i
I
u
3
= hb − 1, a
2
− a − 1, 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
= h−u
2
+b+u+ 1 , −u
4
+6u
3
−11u
2
+2a+u+ 11 , u
5
−5u
4
+7u
3
+2u
2
−8u−1i
(i) Arc colorings
a
5
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
10
=
1
2
u
4
− 3u
3
+ ··· −
1
2
u −
11
2
u
2
− u − 1
a
1
=
u
4
− 5u
3
+ 7u
2
+ 2u − 6
1
2
u
4
− 2u
3
+
7
2
u
2
+
7
2
u −
1
2
a
4
=
u
u
a
3
=
u
3
− 3u
2
+ u + 3
5
2
u
4
− 7u
3
+
1
2
u
2
+
19
2
u +
3
2
a
6
=
−u
2
+ 1
−u
4
a
2
=
3u
4
− 9u
3
+ 15u + 5
27
2
u
4
− 47u
3
+ ··· +
139
2
u +
17
2
a
9
=
−
1
2
u
4
+ u
3
+ ··· −
11
2
u −
9
2
−3u
4
+ 11u
3
− 7u
2
− 20u − 3
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
4
+ 10u
3
− 15u
2
+ 2u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
, c
10
u
5
+ 6u
4
+ 11u
3
+ u
2
− 12u + 1
c
2
, c
5
u
5
− u
4
− 4u
3
+ 23u
2
+ 4u − 4
c
3
, c
9
u
5
− u
4
− 7u
3
+ 52u
2
− 12u − 8
c
4
, c
7
u
5
− 5u
4
+ 7u
3
+ 2u
2
− 8u − 1
c
6
u
5
+ 11u
4
+ 53u
3
+ 126u
2
+ 68u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
y
5
− 14y
4
+ 85y
3
− 277y
2
+ 142y −1
c
2
, c
5
y
5
− 9y
4
+ 70y
3
− 569y
2
+ 200y −16
c
3
, c
9
y
5
− 15y
4
+ 129y
3
− 2552y
2
+ 976y −64
c
4
, c
7
y
5
− 11y
4
+ 53y
3
− 126y
2
+ 68y −1
c
6
y
5
− 15y
4
+ 173y
3
− 8690y
2
+ 4372y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.844155
a = 0.899891
b = 0.556753
−1.21003 −9.40830
u = −0.122993
a = −5.34961
b = −0.861880
1.12640 9.50800
u = 1.88542 + 0.91135I
a = 0.333114 + 0.921118I
b = −0.16115 + 2.52520I
−14.3433 − 7.3743I 1.72840 + 2.44716I
u = 1.88542 − 0.91135I
a = 0.333114 − 0.921118I
b = −0.16115 − 2.52520I
−14.3433 + 7.3743I 1.72840 − 2.44716I
u = 2.19630
a = −0.216510
b = 1.62743
5.74119 1.44340
5
II. I
u
2
= hu
2
+ b − u + 1, u
2
+ a − u + 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
5
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
10
=
−u
2
+ u − 1
−u
2
+ u − 1
a
1
=
−u
2
+ u − 2
−2u
2
+ u − 1
a
4
=
u
u
a
3
=
u
u
a
6
=
−u
2
+ 1
−u
2
+ u + 1
a
2
=
−1
−u
2
a
9
=
−u
2
+ u − 1
−u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
+ 8u − 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
(u − 1)
3
c
2
, c
6
u
3
− u
2
+ 2u − 1
c
3
, c
9
u
3
c
4
u
3
+ u
2
− 1
c
5
u
3
+ u
2
+ 2u + 1
c
7
u
3
− u
2
+ 1
c
8
(u + 1)
3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
(y −1)
3
c
2
, c
5
, c
6
y
3
+ 3y
2
+ 2y −1
c
3
, c
9
y
3
c
4
, c
7
y
3
− y
2
+ 2y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.337641 − 0.562280I
b = −0.337641 − 0.562280I
4.66906 − 2.82812I 2.80443 + 4.65175I
u = 0.877439 − 0.744862I
a = −0.337641 + 0.562280I
b = −0.337641 + 0.562280I
4.66906 + 2.82812I 2.80443 − 4.65175I
u = −0.754878
a = −2.32472
b = −2.32472
0.531480 −10.6090
9
III. I
u
3
= hb − 1, a
2
− a − 1, u + 1i
(i) Arc colorings
a
5
=
0
−1
a
7
=
1
0
a
8
=
1
1
a
10
=
a
1
a
1
=
1
−a + 2
a
4
=
−1
−1
a
3
=
0
a − 2
a
6
=
0
−1
a
2
=
0
a − 2
a
9
=
a
2a − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 9
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
10
u
2
+ u − 1
c
2
, c
5
u
2
c
4
, c
6
(u − 1)
2
c
7
(u + 1)
2
c
8
, c
9
u
2
− u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
8
c
9
, c
10
y
2
− 3y + 1
c
2
, c
5
y
2
c
4
, c
6
, c
7
(y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.618034
b = 1.00000
7.23771 9.00000
u = −1.00000
a = 1.61803
b = 1.00000
−0.657974 9.00000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
(u − 1)
3
(u
2
+ u − 1)(u
5
+ 6u
4
+ 11u
3
+ u
2
− 12u + 1)
c
2
u
2
(u
3
− u
2
+ 2u − 1)(u
5
− u
4
− 4u
3
+ 23u
2
+ 4u − 4)
c
3
u
3
(u
2
+ u − 1)(u
5
− u
4
− 7u
3
+ 52u
2
− 12u − 8)
c
4
(u − 1)
2
(u
3
+ u
2
− 1)(u
5
− 5u
4
+ 7u
3
+ 2u
2
− 8u − 1)
c
5
u
2
(u
3
+ u
2
+ 2u + 1)(u
5
− u
4
− 4u
3
+ 23u
2
+ 4u − 4)
c
6
(u − 1)
2
(u
3
− u
2
+ 2u − 1)(u
5
+ 11u
4
+ 53u
3
+ 126u
2
+ 68u + 1)
c
7
(u + 1)
2
(u
3
− u
2
+ 1)(u
5
− 5u
4
+ 7u
3
+ 2u
2
− 8u − 1)
c
8
(u + 1)
3
(u
2
− u − 1)(u
5
+ 6u
4
+ 11u
3
+ u
2
− 12u + 1)
c
9
u
3
(u
2
− u − 1)(u
5
− u
4
− 7u
3
+ 52u
2
− 12u − 8)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
(y −1)
3
(y
2
− 3y + 1)(y
5
− 14y
4
+ 85y
3
− 277y
2
+ 142y −1)
c
2
, c
5
y
2
(y
3
+ 3y
2
+ 2y −1)(y
5
− 9y
4
+ 70y
3
− 569y
2
+ 200y −16)
c
3
, c
9
y
3
(y
2
− 3y + 1)(y
5
− 15y
4
+ 129y
3
− 2552y
2
+ 976y −64)
c
4
, c
7
(y −1)
2
(y
3
− y
2
+ 2y −1)(y
5
− 11y
4
+ 53y
3
− 126y
2
+ 68y −1)
c
6
((y −1)
2
)(y
3
+ 3y
2
+ 2y −1)(y
5
− 15y
4
+ ··· + 4372y −1)
15
