10
141
(K10n
25
)
A knot diagram
1
Linearized knot diagam
6 5 10 8 1 2 9 5 1 4
Solving Sequence
1,5
6 2
3,8
9 4 7 10
c
5
c
1
c
2
c
8
c
4
c
7
c
10
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
+ u
5
− 3u
4
− u
3
+ 3u
2
+ b + 1, u
6
+ u
5
− 4u
4
− u
3
+ 6u
2
+ 2a, u
7
+ 3u
6
− 5u
4
+ 4u
2
+ 2u + 2i
I
u
2
= hb
2
− bu + u
2
− 1, u
2
+ a − u − 2, u
3
− u
2
− 2u + 1i
I
u
3
= hb + 1, 2a − u + 2, u
2
− 2i
I
v
1
= ha, b − 1, v + 1i
* 4 irreducible components of dim
C
= 0, with total 16 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
6
+ u
5
− 3u
4
− u
3
+ 3u
2
+ b + 1, u
6
+ u
5
− 4u
4
− u
3
+ 6u
2
+
2a, u
7
+ 3u
6
− 5u
4
+ 4u
2
+ 2u + 2i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
2
=
u
−u
3
+ u
a
3
=
−u
3
+ 2u
−u
3
+ u
a
8
=
−
1
2
u
6
−
1
2
u
5
+ 2u
4
+
1
2
u
3
− 3u
2
−u
6
− u
5
+ 3u
4
+ u
3
− 3u
2
− 1
a
9
=
1
2
u
6
+
1
2
u
5
− u
4
−
1
2
u
3
+ 1
−u
6
− u
5
+ 3u
4
+ u
3
− 3u
2
− 1
a
4
=
1
2
u
6
+
1
2
u
5
− u
4
+
1
2
u
3
+ u
2
− u + 1
u
6
+ u
5
− 2u
4
+ 2u
2
+ 1
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
1
2
u
6
+
1
2
u
5
− u
4
−
1
2
u
3
+ 1
u
6
+ u
5
− 2u
4
− u
3
+ u
2
+ u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
5
− 8u
3
+ 6u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u
7
− 3u
6
+ 5u
4
− 4u
2
+ 2u − 2
c
2
u
7
+ 9u
6
+ 30u
5
+ 45u
4
+ 46u
3
+ 32u
2
+ 22u + 14
c
3
, c
4
, c
8
c
10
u
7
+ u
6
− u
4
+ 3u
3
+ u
2
− 1
c
7
, c
9
u
7
+ u
6
+ 8u
5
+ 3u
4
+ 13u
3
+ 3u
2
+ 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
y
7
− 9y
6
+ 30y
5
− 45y
4
+ 28y
3
+ 4y
2
− 12y −4
c
2
y
7
− 21y
6
+ 182y
5
+ 203y
4
+ 304y
3
− 260y
2
− 412y −196
c
3
, c
4
, c
8
c
10
y
7
− y
6
+ 8y
5
− 3y
4
+ 13y
3
− 3y
2
+ 2y −1
c
7
, c
9
y
7
+ 15y
6
+ 84y
5
+ 197y
4
+ 181y
3
+ 37y
2
− 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.050170 + 0.492398I
a = −1.369620 − 0.237150I
b = −0.828738 + 0.848640I
3.39904 + 5.13113I 0.70211 − 5.71003I
u = 1.050170 − 0.492398I
a = −1.369620 + 0.237150I
b = −0.828738 − 0.848640I
3.39904 − 5.13113I 0.70211 + 5.71003I
u = −1.33623
a = −0.889511
b = −0.610544
3.10278 2.54950
u = −0.122110 + 0.584395I
a = 1.254020 + 0.529753I
b = 0.441920 + 0.538118I
−0.192432 − 1.318890I −1.84900 + 4.97200I
u = −0.122110 − 0.584395I
a = 1.254020 − 0.529753I
b = 0.441920 − 0.538118I
−0.192432 + 1.318890I −1.84900 − 4.97200I
u = −1.75995 + 0.15485I
a = 1.060360 − 0.362677I
b = 1.19209 + 0.98985I
13.3363 − 7.9365I 0.87212 + 4.07397I
u = −1.75995 − 0.15485I
a = 1.060360 + 0.362677I
b = 1.19209 − 0.98985I
13.3363 + 7.9365I 0.87212 − 4.07397I
5
II. I
u
2
= 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
5
=
1
0
a
6
=
1
−u
2
a
2
=
u
−u
2
− u + 1
a
3
=
−u
2
+ 1
−u
2
− u + 1
a
8
=
−u
2
+ u + 2
b
a
9
=
−u
2
− b + u + 2
b
a
4
=
u
2
b − bu − 2b + 1
−bu + u
2
− 1
a
7
=
−u
2
+ 1
u
2
+ u − 1
a
10
=
−u
2
− b + u + 2
−u
2
b + b + u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
(u
3
+ u
2
− 2u − 1)
2
c
2
(u
3
− 3u
2
− 4u − 1)
2
c
3
, c
4
, c
8
c
10
u
6
+ u
5
− 2u
3
+ 2u − 1
c
7
, c
9
u
6
+ u
5
+ 4u
4
+ 10u
3
+ 8u
2
+ 4u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
(y
3
− 5y
2
+ 6y −1)
2
c
2
(y
3
− 17y
2
+ 10y −1)
2
c
3
, c
4
, c
8
c
10
y
6
− y
5
+ 4y
4
− 10y
3
+ 8y
2
− 4y + 1
c
7
, c
9
y
6
+ 7y
5
+ 12y
4
− 42y
3
− 8y
2
+ 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = −0.801938
b = −0.623490 + 0.407699I
3.05488 2.00000
u = −1.24698
a = −0.801938
b = −0.623490 − 0.407699I
3.05488 2.00000
u = 0.445042
a = 2.24698
b = 1.14526
−2.58490 2.00000
u = 0.445042
a = 2.24698
b = −0.700221
−2.58490 2.00000
u = 1.80194
a = 0.554958
b = 0.90097 + 1.19801I
14.3344 2.00000
u = 1.80194
a = 0.554958
b = 0.90097 − 1.19801I
14.3344 2.00000
9
III. I
u
3
= hb + 1, 2a − u + 2, u
2
− 2i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
−2
a
2
=
u
−u
a
3
=
0
−u
a
8
=
1
2
u − 1
−1
a
9
=
1
2
u
−1
a
4
=
1
2
u
−1
a
7
=
−1
0
a
10
=
1
2
u
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
2
− 2
c
3
, c
7
, c
8
c
9
(u − 1)
2
c
4
, c
10
(u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
(y −2)
2
c
3
, c
4
, c
7
c
8
, c
9
, c
10
(y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −0.292893
b = −1.00000
1.64493 −4.00000
u = −1.41421
a = −1.70711
b = −1.00000
1.64493 −4.00000
13
IV. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
1
=
−1
0
a
5
=
1
0
a
6
=
1
0
a
2
=
−1
0
a
3
=
−1
0
a
8
=
0
1
a
9
=
−1
1
a
4
=
1
−1
a
7
=
1
0
a
10
=
−2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
c
3
, c
8
u + 1
c
4
, c
7
, c
9
c
10
u − 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
c
3
, c
4
, c
7
c
8
, c
9
, c
10
y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
u(u
2
− 2)(u
3
+ u
2
− 2u − 1)
2
(u
7
− 3u
6
+ 5u
4
− 4u
2
+ 2u − 2)
c
2
u(u
2
− 2)(u
3
− 3u
2
− 4u − 1)
2
· (u
7
+ 9u
6
+ 30u
5
+ 45u
4
+ 46u
3
+ 32u
2
+ 22u + 14)
c
3
, c
8
((u − 1)
2
)(u + 1)(u
6
+ u
5
+ ··· + 2u − 1)(u
7
+ u
6
+ ··· + u
2
− 1)
c
4
, c
10
(u − 1)(u + 1)
2
(u
6
+ u
5
+ ··· + 2u − 1)(u
7
+ u
6
+ ··· + u
2
− 1)
c
7
, c
9
(u − 1)
3
(u
6
+ u
5
+ 4u
4
+ 10u
3
+ 8u
2
+ 4u + 1)
· (u
7
+ u
6
+ 8u
5
+ 3u
4
+ 13u
3
+ 3u
2
+ 2u + 1)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
y(y −2)
2
(y
3
− 5y
2
+ 6y −1)
2
· (y
7
− 9y
6
+ 30y
5
− 45y
4
+ 28y
3
+ 4y
2
− 12y −4)
c
2
y(y −2)
2
(y
3
− 17y
2
+ 10y −1)
2
· (y
7
− 21y
6
+ 182y
5
+ 203y
4
+ 304y
3
− 260y
2
− 412y −196)
c
3
, c
4
, c
8
c
10
(y −1)
3
(y
6
− y
5
+ 4y
4
− 10y
3
+ 8y
2
− 4y + 1)
· (y
7
− y
6
+ 8y
5
− 3y
4
+ 13y
3
− 3y
2
+ 2y −1)
c
7
, c
9
(y −1)
3
(y
6
+ 7y
5
+ 12y
4
− 42y
3
− 8y
2
+ 1)
· (y
7
+ 15y
6
+ 84y
5
+ 197y
4
+ 181y
3
+ 37y
2
− 2y −1)
19
