12n
0473
(K12n
0473
)
A knot diagram
1
Linearized knot diagam
3 6 9 10 2 12 11 3 4 6 7 8
Solving Sequence
3,8
9 4
6,10
11 2 1 5 7 12
c
8
c
3
c
9
c
10
c
2
c
1
c
5
c
7
c
12
c
4
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h5447u
8
− 4753u
7
+ ··· + 46532b + 23300, −461u
8
− 3128u
7
+ ··· + 93064a − 27096,
u
9
− 18u
7
− 36u
6
+ 12u
5
+ 68u
4
+ 36u
3
+ 8u
2
+ 24u + 8i
I
u
2
= hb
3
− b
2
+ 1, 2a − u, u
2
− 2i
I
v
1
= ha, v
2
+ b − 1, v
3
− v + 1i
* 3 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
= h5447u
8
− 4753u
7
+ · · · + 46532b + 23300, −461u
8
− 3128u
7
+ · · · +
93064a − 27096, u
9
− 18u
7
+ · · · + 24u + 8i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
4
=
−u
−u
3
+ u
a
6
=
0.00495358u
8
+ 0.0336113u
7
+ ··· − 1.27001u + 0.291154
−0.117059u
8
+ 0.102145u
7
+ ··· − 0.727843u − 0.500731
a
10
=
−u
2
+ 1
−u
4
+ 2u
2
a
11
=
0.0289801u
8
− 0.0392633u
7
+ ··· − 0.182670u + 0.813977
0.0609903u
8
− 0.0503739u
7
+ ··· + 1.76017u + 0.788705
a
2
=
0.0267235u
8
− 0.0442706u
7
+ ··· + 0.191954u − 0.199347
0.0998238u
8
− 0.159439u
7
+ ··· + 1.80830u + 0.494198
a
1
=
0.0267235u
8
− 0.0442706u
7
+ ··· + 0.191954u − 0.199347
0.00758618u
8
− 0.00221353u
7
+ ··· + 0.959598u + 0.140033
a
5
=
u
3
− 2u
u
5
− 3u
3
+ u
a
7
=
0.0326442u
8
− 0.0442276u
7
+ ··· − 0.952893u + 0.526090
−0.0316341u
8
− 0.0297215u
7
+ ··· − 0.338606u − 0.139173
a
12
=
0.0343097u
8
− 0.0464841u
7
+ ··· + 1.15155u − 0.0593140
0.00758618u
8
− 0.00221353u
7
+ ··· + 0.959598u + 0.140033
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
20957
46532
u
8
+
5941
23266
u
7
+
185067
23266
u
6
+
274717
23266
u
5
−
139019
11633
u
4
−
299261
11633
u
3
−
57101
11633
u
2
+
28694
11633
u−
223158
11633
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
+ 73u
8
+ ··· + 729u + 49
c
2
, c
5
u
9
+ 13u
8
+ 48u
7
+ 72u
6
+ 18u
5
− 14u
4
− 40u
3
− 27u − 7
c
3
, c
4
, c
8
c
9
u
9
− 18u
7
− 36u
6
+ 12u
5
+ 68u
4
+ 36u
3
+ 8u
2
+ 24u + 8
c
6
, c
7
, c
11
u
9
+ 3u
8
+ 7u
7
+ 14u
6
+ 14u
5
+ 25u
4
+ 11u
3
+ 15u
2
− 1
c
10
, c
12
u
9
+ 9u
8
− 73u
7
+ 56u
6
+ 430u
5
− 75u
4
− 417u
3
+ 173u
2
+ 36u − 13
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
− 4393y
8
+ ··· + 723913y −2401
c
2
, c
5
y
9
− 73y
8
+ ··· + 729y −49
c
3
, c
4
, c
8
c
9
y
9
− 36y
8
+ ··· + 448y −64
c
6
, c
7
, c
11
y
9
+ 5y
8
− 7y
7
− 128y
6
− 440y
5
− 731y
4
− 601y
3
− 175y
2
+ 30y −1
c
10
, c
12
y
9
− 227y
8
+ ··· + 5794y −169
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.186450 + 0.424170I
a = 0.251161 + 0.846300I
b = 1.76218 + 0.93575I
−1.02910 − 2.89378I −13.46616 + 2.37667I
u = −1.186450 − 0.424170I
a = 0.251161 − 0.846300I
b = 1.76218 − 0.93575I
−1.02910 + 2.89378I −13.46616 − 2.37667I
u = 0.273011 + 0.580428I
a = −0.750166 − 0.744053I
b = 1.201410 + 0.238186I
2.44926 − 1.76826I −8.73509 + 3.07632I
u = 0.273011 − 0.580428I
a = −0.750166 + 0.744053I
b = 1.201410 − 0.238186I
2.44926 + 1.76826I −8.73509 − 3.07632I
u = 1.44806
a = 0.605279
b = −0.107770
−6.65928 −13.3420
u = −0.347212
a = 0.692622
b = −0.151972
−0.501693 −19.7740
u = −2.09595 + 0.74847I
a = 1.180400 − 0.226493I
b = 1.57561 − 4.87250I
13.5998 + 8.3082I −13.79311 − 2.86755I
u = −2.09595 − 0.74847I
a = 1.180400 + 0.226493I
b = 1.57561 + 4.87250I
13.5998 − 8.3082I −13.79311 + 2.86755I
u = 4.91794
a = −1.66068
b = −37.8186
6.72995 −14.8960
5
II. I
u
2
= hb
3
− b
2
+ 1, 2a − u, u
2
− 2i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
2
a
4
=
−u
−u
a
6
=
1
2
u
b
a
10
=
−1
0
a
11
=
−
1
2
bu − 1
−b
2
a
2
=
1
2
u
b + u
a
1
=
1
2
u
b
a
5
=
0
−u
a
7
=
−
1
2
b
2
u − b
2
+
1
2
u + 1
−b
2
+ b + 1
a
12
=
b +
1
2
u
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b − 20
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
8
c
9
(u
2
− 2)
3
c
6
, c
7
(u
3
+ u
2
+ 2u + 1)
2
c
10
, c
12
(u
3
+ u
2
− 1)
2
c
11
(u
3
− u
2
+ 2u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
8
c
9
(y −2)
6
c
6
, c
7
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
10
, c
12
(y
3
− y
2
+ 2y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = 0.707107
b = 0.877439 + 0.744862I
−3.55561 − 2.82812I −16.4902 + 2.9794I
u = 1.41421
a = 0.707107
b = 0.877439 − 0.744862I
−3.55561 + 2.82812I −16.4902 − 2.9794I
u = 1.41421
a = 0.707107
b = −0.754878
−7.69319 −23.0200
u = −1.41421
a = −0.707107
b = 0.877439 + 0.744862I
−3.55561 − 2.82812I −16.4902 + 2.9794I
u = −1.41421
a = −0.707107
b = 0.877439 − 0.744862I
−3.55561 + 2.82812I −16.4902 − 2.9794I
u = −1.41421
a = −0.707107
b = −0.754878
−7.69319 −23.0200
9
III. I
v
1
= ha, v
2
+ b − 1, v
3
− v + 1i
(i) Arc colorings
a
3
=
v
0
a
8
=
1
0
a
9
=
1
0
a
4
=
v
0
a
6
=
0
−v
2
+ 1
a
10
=
1
0
a
11
=
1
−v
2
− v + 1
a
2
=
v
v
2
− 1
a
1
=
0
v
2
− 1
a
5
=
v
0
a
7
=
v
2
+ v
v + 1
a
12
=
v
2
− 1
v
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4v
2
− 2v − 10
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
8
c
9
u
3
c
5
(u + 1)
3
c
6
, c
7
u
3
− u
2
+ 2u − 1
c
10
, c
12
u
3
− u
2
+ 1
c
11
u
3
+ u
2
+ 2u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
3
c
3
, c
4
, c
8
c
9
y
3
c
6
, c
7
, c
11
y
3
+ 3y
2
+ 2y − 1
c
10
, c
12
y
3
− y
2
+ 2y − 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.662359 + 0.562280I
a = 0
b = 0.877439 − 0.744862I
1.37919 + 2.82812I −11.81496 − 4.10401I
v = 0.662359 − 0.562280I
a = 0
b = 0.877439 + 0.744862I
1.37919 − 2.82812I −11.81496 + 4.10401I
v = −1.32472
a = 0
b = −0.754878
−2.75839 −14.3700
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
9
+ 73u
8
+ ··· + 729u + 49)
c
2
(u − 1)
3
(u + 1)
6
· (u
9
+ 13u
8
+ 48u
7
+ 72u
6
+ 18u
5
− 14u
4
− 40u
3
− 27u − 7)
c
3
, c
4
, c
8
c
9
u
3
(u
2
− 2)
3
(u
9
− 18u
7
+ ··· + 24u + 8)
c
5
(u − 1)
6
(u + 1)
3
· (u
9
+ 13u
8
+ 48u
7
+ 72u
6
+ 18u
5
− 14u
4
− 40u
3
− 27u − 7)
c
6
, c
7
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
· (u
9
+ 3u
8
+ 7u
7
+ 14u
6
+ 14u
5
+ 25u
4
+ 11u
3
+ 15u
2
− 1)
c
10
, c
12
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
· (u
9
+ 9u
8
− 73u
7
+ 56u
6
+ 430u
5
− 75u
4
− 417u
3
+ 173u
2
+ 36u − 13)
c
11
(u
3
− u
2
+ 2u − 1)
2
(u
3
+ u
2
+ 2u + 1)
· (u
9
+ 3u
8
+ 7u
7
+ 14u
6
+ 14u
5
+ 25u
4
+ 11u
3
+ 15u
2
− 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
9
)(y
9
− 4393y
8
+ ··· + 723913y − 2401)
c
2
, c
5
((y − 1)
9
)(y
9
− 73y
8
+ ··· + 729y − 49)
c
3
, c
4
, c
8
c
9
y
3
(y − 2)
6
(y
9
− 36y
8
+ ··· + 448y − 64)
c
6
, c
7
, c
11
(y
3
+ 3y
2
+ 2y − 1)
3
· (y
9
+ 5y
8
− 7y
7
− 128y
6
− 440y
5
− 731y
4
− 601y
3
− 175y
2
+ 30y − 1)
c
10
, c
12
((y
3
− y
2
+ 2y − 1)
3
)(y
9
− 227y
8
+ ··· + 5794y − 169)
15
