12n
0309
(K12n
0309
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 5 11 12 5 7 8 9
Solving Sequence
7,11
8
5,12
6 10 4 3 2 1 9
c
7
c
11
c
6
c
10
c
4
c
3
c
2
c
1
c
9
c
5
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
4
+ 5u
2
+ 4b + 6u + 1, 2u
5
+ 5u
4
− 6u
3
− 29u
2
+ 4a − 24u − 9, u
6
+ 3u
5
− 2u
4
− 17u
3
− 18u
2
− 5u + 1i
I
u
2
= hb + a, a
3
− a
2
+ 2a − 1, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 12 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
4
+ 5u
2
+ 4b + 6u + 1, 2u
5
+ 5u
4
− 6u
3
− 29u
2
+ 4a − 24u −
9, u
6
+ 3u
5
− 2u
4
− 17u
3
− 18u
2
− 5u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
5
=
−
1
2
u
5
−
5
4
u
4
+ ··· + 6u +
9
4
1
4
u
4
−
5
4
u
2
−
3
2
u −
1
4
a
12
=
u
−u
3
+ u
a
6
=
−
1
4
u
5
−
1
2
u
4
+ ··· +
13
4
u + 2
1
4
u
5
+
1
2
u
4
−
7
4
u
3
− 3u
2
−
5
4
u
a
10
=
−u
u
a
4
=
−
3
2
u
5
−
11
4
u
4
+ ··· + 9u +
7
4
u
5
+
7
4
u
4
+ ··· −
9
2
u +
1
4
a
3
=
−
1
2
u
5
− u
4
+
3
2
u
3
+ 6u
2
+
9
2
u + 2
u
5
+
7
4
u
4
+ ··· −
9
2
u +
1
4
a
2
=
1
1
4
u
5
+
1
2
u
4
−
3
4
u
3
− 4u
2
−
9
4
u
a
1
=
u
3
− 2u
−u
5
+ 3u
3
− u
a
9
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
5
+ 6u
4
− 4u
3
−
69
2
u
2
−
71
2
u −
13
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
6
+ 6u
5
+ 17u
4
+ 22u
3
− 6u
2
− 4u + 1
c
2
, c
5
u
6
+ 2u
5
− u
4
− 6u
3
− 4u
2
− 2u − 1
c
3
u
6
− 32u
5
+ 357u
4
− 1330u
3
− 670u
2
− 786u − 433
c
4
, c
9
u
6
+ 4u
5
− 40u
4
− 160u
3
+ 192u
2
− 96u − 64
c
7
, c
8
, c
10
c
11
, c
12
u
6
− 3u
5
− 2u
4
+ 17u
3
− 18u
2
+ 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
6
− 2y
5
+ 13y
4
− 638y
3
+ 246y
2
− 28y + 1
c
2
, c
5
y
6
− 6y
5
+ 17y
4
− 22y
3
− 6y
2
+ 4y + 1
c
3
y
6
− 310y
5
+ ··· − 37576y + 187489
c
4
, c
9
y
6
− 96y
5
+ 3264y
4
− 40320y
3
+ 11264y
2
− 33792y + 4096
c
7
, c
8
, c
10
c
11
, c
12
y
6
− 13y
5
+ 70y
4
− 185y
3
+ 150y
2
− 61y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.769155 + 0.318981I
a = 0.780154 − 0.426955I
b = 0.291213 + 0.014710I
1.40855 − 0.49854I 5.08917 + 1.32495I
u = −0.769155 − 0.318981I
a = 0.780154 + 0.426955I
b = 0.291213 − 0.014710I
1.40855 + 0.49854I 5.08917 − 1.32495I
u = 0.130756
a = 3.16146
b = −0.467433
−0.927213 −11.7390
u = −1.98103 + 0.85464I
a = −0.545050 + 0.888077I
b = −1.58699 − 2.45706I
−4.00977 − 6.34376I 1.76361 + 2.48758I
u = −1.98103 − 0.85464I
a = −0.545050 − 0.888077I
b = −1.58699 + 2.45706I
−4.00977 + 6.34376I 1.76361 − 2.48758I
u = 2.36961
a = 0.368332
b = −2.94101
6.12964 1.03330
5
II. I
u
2
= hb + a, a
3
− a
2
+ 2a − 1, u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u − 1
a
5
=
a
−a
a
12
=
u
−u − 1
a
6
=
a
2
+ 1
−a
2
a
10
=
−u
u
a
4
=
a
−a
a
3
=
0
−a
a
2
=
−1
−a
2
a
1
=
−1
0
a
9
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = a
2
u − au + 3a + 3u + 1
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
(u
2
− u − 1)
3
c
10
, c
11
, c
12
(u
2
+ u − 1)
3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.215080 + 1.307140I
b = −0.215080 − 1.307140I
4.01109 − 2.82812I 0.95146 + 4.38177I
u = −0.618034
a = 0.215080 − 1.307140I
b = −0.215080 + 1.307140I
4.01109 + 2.82812I 0.95146 − 4.38177I
u = −0.618034
a = 0.569840
b = −0.569840
−0.126494 1.00690
u = 1.61803
a = 0.215080 + 1.307140I
b = −0.215080 − 1.307140I
11.90680 − 2.82812I 3.46158 + 2.71621I
u = 1.61803
a = 0.215080 − 1.307140I
b = −0.215080 + 1.307140I
11.90680 + 2.82812I 3.46158 − 2.71621I
u = 1.61803
a = 0.569840
b = −0.569840
7.76919 7.16700
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 2u − 1)
2
(u
6
+ 6u
5
+ 17u
4
+ 22u
3
− 6u
2
− 4u + 1)
c
2
(u
3
+ u
2
− 1)
2
(u
6
+ 2u
5
− u
4
− 6u
3
− 4u
2
− 2u − 1)
c
3
(u
3
− u
2
+ 2u − 1)
2
· (u
6
− 32u
5
+ 357u
4
− 1330u
3
− 670u
2
− 786u − 433)
c
4
, c
9
u
6
(u
6
+ 4u
5
− 40u
4
− 160u
3
+ 192u
2
− 96u − 64)
c
5
(u
3
− u
2
+ 1)
2
(u
6
+ 2u
5
− u
4
− 6u
3
− 4u
2
− 2u − 1)
c
6
(u
3
+ u
2
+ 2u + 1)
2
(u
6
+ 6u
5
+ 17u
4
+ 22u
3
− 6u
2
− 4u + 1)
c
7
, c
8
(u
2
− u − 1)
3
(u
6
− 3u
5
− 2u
4
+ 17u
3
− 18u
2
+ 5u + 1)
c
10
, c
11
, c
12
(u
2
+ u − 1)
3
(u
6
− 3u
5
− 2u
4
+ 17u
3
− 18u
2
+ 5u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
(y
6
− 2y
5
+ 13y
4
− 638y
3
+ 246y
2
− 28y + 1)
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
(y
6
− 6y
5
+ 17y
4
− 22y
3
− 6y
2
+ 4y + 1)
c
3
((y
3
+ 3y
2
+ 2y −1)
2
)(y
6
− 310y
5
+ ··· − 37576y + 187489)
c
4
, c
9
y
6
(y
6
− 96y
5
+ 3264y
4
− 40320y
3
+ 11264y
2
− 33792y + 4096)
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
(y
6
− 13y
5
+ 70y
4
− 185y
3
+ 150y
2
− 61y + 1)
11
