12n
0402
(K12n
0402
)
A knot diagram
1
Linearized knot diagam
3 5 9 10 2 11 3 12 4 7 5 8
Solving Sequence
3,9
4 10 5 2
1,12
8 7 11 6
c
3
c
9
c
4
c
2
c
1
c
8
c
7
c
11
c
6
c
5
, c
10
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
9
+ 2u
8
+ 3u
7
− 4u
6
− 6u
5
− 2u
4
+ 9u
3
+ 4u
2
+ b − 1,
u
9
− u
8
− 3u
7
− u
6
+ 5u
5
+ 9u
4
− 4u
3
− 6u
2
+ 2a − 5u + 1,
u
10
− 3u
9
− u
8
+ 7u
7
+ 3u
6
− 5u
5
− 14u
4
+ 6u
3
+ 7u
2
+ 3u − 2i
I
u
2
= h−u
5
+ 3u
3
+ b − u + 1, u
6
− 4u
4
+ 4u
2
+ a, u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1i
I
u
3
= hb + 1, a
2
+ a + 2, u + 1i
* 3 irreducible components of dim
C
= 0, with total 20 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
9
+ 2u
8
+ · · · + b − 1, u
9
− u
8
+ · · · + 2a + 1, u
10
− 3u
9
+ · · · + 3u − 2i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
10
=
u
−u
3
+ u
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
−u
6
+ 3u
4
− 2u
2
+ 1
u
8
− 4u
6
+ 4u
4
a
1
=
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
u
8
− 4u
6
+ 4u
4
a
12
=
−
1
2
u
9
+
1
2
u
8
+ ··· +
5
2
u −
1
2
u
9
− 2u
8
− 3u
7
+ 4u
6
+ 6u
5
+ 2u
4
− 9u
3
− 4u
2
+ 1
a
8
=
1
2
u
9
−
1
2
u
8
+ ··· −
1
2
u +
1
2
u
9
− u
8
− 4u
7
+ u
6
+ 7u
5
+ 4u
4
− 6u
3
− 4u
2
+ 1
a
7
=
−
1
2
u
9
+
1
2
u
8
+ ··· −
1
2
u −
1
2
u
9
− u
8
− 4u
7
+ u
6
+ 7u
5
+ 4u
4
− 6u
3
− 4u
2
+ 1
a
11
=
3
2
u
9
−
5
2
u
8
+ ··· −
3
2
u +
5
2
−u
9
+ 3u
8
+ 3u
7
− 8u
6
− 6u
5
+ 2u
4
+ 8u
3
+ 4u
2
+ u − 1
a
6
=
3u
9
− 4u
8
− 7u
7
+ 5u
6
+ 5u
5
+ 9u
4
− 6u
3
− 4u
2
− 3u + 1
−8u
9
+ 8u
8
+ 32u
7
− 9u
6
− 56u
5
− 32u
4
+ 48u
3
+ 37u
2
+ 6u − 8
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
9
+ 6u
8
+ 12u
7
−6u
6
−22u
5
−22u
4
+ 22u
3
+ 20u
2
+ 18u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
+ 49u
9
+ ··· − 2401u + 64
c
2
, c
5
u
10
+ u
9
+ ··· − 9u − 8
c
3
, c
4
, c
9
u
10
− 3u
9
− u
8
+ 7u
7
+ 3u
6
− 5u
5
− 14u
4
+ 6u
3
+ 7u
2
+ 3u − 2
c
6
, c
8
, c
10
c
12
u
10
+ 13u
8
+ 2u
7
+ 48u
6
+ 30u
5
+ 20u
4
+ 14u
3
− u
2
+ 2u − 1
c
7
, c
11
u
10
− 2u
9
+ ··· − 54u − 29
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 535y
9
+ ··· − 3422273y + 4096
c
2
, c
5
y
10
+ 49y
9
+ ··· − 2401y + 64
c
3
, c
4
, c
9
y
10
− 11y
9
+ ··· − 37y + 4
c
6
, c
8
, c
10
c
12
y
10
+ 26y
9
+ ··· − 2y + 1
c
7
, c
11
y
10
+ 110y
9
+ ··· − 23448y + 841
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.632414 + 0.947419I
a = 2.18175 − 1.13028I
b = 2.28217 − 0.07300I
15.8724 − 3.1297I 7.30319 + 2.05885I
u = −0.632414 − 0.947419I
a = 2.18175 + 1.13028I
b = 2.28217 + 0.07300I
15.8724 + 3.1297I 7.30319 − 2.05885I
u = −0.481550 + 0.474579I
a = −0.609360 + 0.497018I
b = −0.743164 − 0.230554I
−1.67643 − 1.67312I 8.34167 + 5.35276I
u = −0.481550 − 0.474579I
a = −0.609360 − 0.497018I
b = −0.743164 + 0.230554I
−1.67643 + 1.67312I 8.34167 − 5.35276I
u = −1.44882
a = 0.519822
b = 0.723841
6.49727 15.2060
u = 1.53180 + 0.11762I
a = −0.119445 − 0.373636I
b = −0.797864 + 0.675313I
5.05958 + 3.70571I 13.2497 − 5.2095I
u = 1.53180 − 0.11762I
a = −0.119445 + 0.373636I
b = −0.797864 − 0.675313I
5.05958 − 3.70571I 13.2497 + 5.2095I
u = 0.358246
a = 0.785999
b = 0.146927
0.511729 19.5320
u = 1.62745 + 0.32233I
a = 0.64414 + 1.30973I
b = 2.32347 + 0.23227I
−16.1803 + 7.8809I 9.73645 − 2.75764I
u = 1.62745 − 0.32233I
a = 0.64414 − 1.30973I
b = 2.32347 − 0.23227I
−16.1803 − 7.8809I 9.73645 + 2.75764I
5
II. I
u
2
= h−u
5
+ 3u
3
+ b − u + 1, u
6
− 4u
4
+ 4u
2
+ a, u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
10
=
u
−u
3
+ u
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
−u
6
+ 3u
4
− 2u
2
+ 1
u
6
− 3u
4
+ 2u
2
− 1
a
1
=
0
u
6
− 3u
4
+ 2u
2
− 1
a
12
=
−u
6
+ 4u
4
− 4u
2
u
5
− 3u
3
+ u − 1
a
8
=
−u
7
+ 5u
5
− 7u
3
+ 2u
−u
7
+ 4u
5
− u
4
− 4u
3
+ 2u
2
+ u
a
7
=
u
5
+ u
4
− 3u
3
− 2u
2
+ u
−u
7
+ 4u
5
− u
4
− 4u
3
+ 2u
2
+ u
a
11
=
−u
5
+ u
4
+ 3u
3
− 2u
2
−u
6
+ u
5
+ 3u
4
− 4u
3
− 2u
2
+ 2u
a
6
=
−u
6
+ 3u
4
− 2u
2
+ 1
u
6
− 2u
4
− u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
6
+ 16u
4
− 16u
2
+ 8
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
2
(u
4
− u
3
+ u
2
+ 1)
2
c
3
, c
4
, c
9
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
c
5
(u
4
+ u
3
+ u
2
+ 1)
2
c
6
, c
8
, c
10
c
12
(u
2
+ 1)
4
c
7
u
8
− 2u
7
− 10u
5
+ 5u
4
+ 14u
3
+ 19u
2
+ 48u + 29
c
11
u
8
+ 2u
7
+ 10u
5
+ 5u
4
− 14u
3
+ 19u
2
− 48u + 29
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
2
, c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
3
, c
4
, c
9
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
c
6
, c
8
, c
10
c
12
(y + 1)
8
c
7
, c
11
y
8
− 4y
7
− 30y
6
− 6y
5
+ 555y
4
+ 954y
3
− 693y
2
− 1202y + 841
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.506844 + 0.395123I
a = −0.95668 − 1.22719I
b = −0.279658 − 0.351808I
−3.50087 + 1.41510I 4.17326 − 4.90874I
u = 0.506844 − 0.395123I
a = −0.95668 + 1.22719I
b = −0.279658 + 0.351808I
−3.50087 − 1.41510I 4.17326 + 4.90874I
u = −0.506844 + 0.395123I
a = −0.95668 + 1.22719I
b = −1.72034 − 0.35181I
−3.50087 − 1.41510I 4.17326 + 4.90874I
u = −0.506844 − 0.395123I
a = −0.95668 − 1.22719I
b = −1.72034 + 0.35181I
−3.50087 + 1.41510I 4.17326 − 4.90874I
u = 1.55249 + 0.10488I
a = −0.043315 − 0.641200I
b = −1.91129 + 0.85181I
3.50087 + 3.16396I 7.82674 − 2.56480I
u = 1.55249 − 0.10488I
a = −0.043315 + 0.641200I
b = −1.91129 − 0.85181I
3.50087 − 3.16396I 7.82674 + 2.56480I
u = −1.55249 + 0.10488I
a = −0.043315 + 0.641200I
b = −0.088708 + 0.851808I
3.50087 − 3.16396I 7.82674 + 2.56480I
u = −1.55249 − 0.10488I
a = −0.043315 − 0.641200I
b = −0.088708 − 0.851808I
3.50087 + 3.16396I 7.82674 − 2.56480I
9
III. I
u
3
= hb + 1, a
2
+ a + 2, u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
−1
a
4
=
1
−1
a
10
=
−1
0
a
5
=
0
−1
a
2
=
1
1
a
1
=
2
1
a
12
=
a
−1
a
8
=
−a − 2
−a − 1
a
7
=
−1
−a − 1
a
11
=
a
a − 1
a
6
=
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u − 1)
2
c
3
, c
4
, c
9
(u + 1)
2
c
6
, c
8
, c
10
c
12
u
2
− u + 2
c
7
, c
11
u
2
+ u + 2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
9
(y −1)
2
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
2
+ 3y + 4
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.50000 + 1.32288I
b = −1.00000
−1.64493 10.0000
u = −1.00000
a = −0.50000 − 1.32288I
b = −1.00000
−1.64493 10.0000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
4
− u
3
+ 3u
2
− 2u + 1)
2
(u
10
+ 49u
9
+ ··· − 2401u + 64)
c
2
((u − 1)
2
)(u
4
− u
3
+ u
2
+ 1)
2
(u
10
+ u
9
+ ··· − 9u − 8)
c
3
, c
4
, c
9
(u + 1)
2
(u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1)
· (u
10
− 3u
9
− u
8
+ 7u
7
+ 3u
6
− 5u
5
− 14u
4
+ 6u
3
+ 7u
2
+ 3u − 2)
c
5
((u − 1)
2
)(u
4
+ u
3
+ u
2
+ 1)
2
(u
10
+ u
9
+ ··· − 9u − 8)
c
6
, c
8
, c
10
c
12
(u
2
+ 1)
4
(u
2
− u + 2)
· (u
10
+ 13u
8
+ 2u
7
+ 48u
6
+ 30u
5
+ 20u
4
+ 14u
3
− u
2
+ 2u − 1)
c
7
(u
2
+ u + 2)(u
8
− 2u
7
− 10u
5
+ 5u
4
+ 14u
3
+ 19u
2
+ 48u + 29)
· (u
10
− 2u
9
+ ··· − 54u − 29)
c
11
(u
2
+ u + 2)(u
8
+ 2u
7
+ 10u
5
+ 5u
4
− 14u
3
+ 19u
2
− 48u + 29)
· (u
10
− 2u
9
+ ··· − 54u − 29)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
2
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
10
− 535y
9
+ ··· − 3422273y + 4096)
c
2
, c
5
((y −1)
2
)(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
(y
10
+ 49y
9
+ ··· − 2401y + 64)
c
3
, c
4
, c
9
((y −1)
2
)(y
4
− 5y
3
+ ··· − 2y + 1)
2
(y
10
− 11y
9
+ ··· − 37y + 4)
c
6
, c
8
, c
10
c
12
((y + 1)
8
)(y
2
+ 3y + 4)(y
10
+ 26y
9
+ ··· − 2y + 1)
c
7
, c
11
(y
2
+ 3y + 4)
· (y
8
− 4y
7
− 30y
6
− 6y
5
+ 555y
4
+ 954y
3
− 693y
2
− 1202y + 841)
· (y
10
+ 110y
9
+ ··· − 23448y + 841)
15
