12n
0570
(K12n
0570
)
A knot diagram
1
Linearized knot diagam
3 7 9 10 8 2 12 3 4 5 7 11
Solving Sequence
4,9
10 5
7,11
12 3 2 1 6 8
c
9
c
4
c
10
c
11
c
3
c
2
c
1
c
6
c
8
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h6u
18
− 3u
17
+ ··· + 4b − 8, −5u
18
+ 3u
17
+ ··· + 4a + 2, u
19
− 2u
18
+ ··· − 2u + 2i
I
u
2
= hb + u − 1, 3a − 2u + 3, u
2
− 3i
I
u
3
= ha
2
u + 2a
2
+ 2au + b + 5a − 2u − 2, a
3
+ 2a
2
− 3au − u + 1, u
2
+ u − 1i
I
u
4
= hb − 1, a + 1, u + 1i
I
u
5
= hb, a + 1, u + 1i
I
u
6
= hb − 2, a + 1, u − 1i
I
u
7
= hb − 1, a, u − 1i
I
v
1
= ha, b − 1, v −1i
* 8 irreducible components of dim
C
= 0, with total 32 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
=
h6u
18
−3u
17
+· · ·+4b−8, −5u
18
+3u
17
+· · ·+4a+2, u
19
−2u
18
+· · ·−2u+2i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
5
=
u
−u
3
+ u
a
7
=
5
4
u
18
−
3
4
u
17
+ ··· + 2u −
1
2
−
3
2
u
18
+
3
4
u
17
+ ··· + 2u
2
+ 2
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
12
=
5
4
u
18
−
3
4
u
17
+ ··· + 2u −
1
2
−
5
4
u
18
+
1
4
u
17
+ ··· − u +
3
2
a
3
=
−u
u
a
2
=
−
1
4
u
18
+
11
4
u
16
+ ··· −
1
2
u −
1
2
1
4
u
18
−
5
2
u
16
+ ··· + u
2
+ u
a
1
=
1
4
u
16
−
5
2
u
14
+ ··· −
1
2
u −
1
2
1
2
u
11
−
7
2
u
9
+ ··· +
3
2
u
2
+ u
a
6
=
−u
7
+ 4u
5
− 4u
3
+ 2u
u
7
− 3u
5
+ u
a
8
=
−u
2
+ 1
u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
18
−20u
16
−4u
15
+ 74u
14
+ 38u
13
−118u
12
−130u
11
+ 54u
10
+
186u
9
+ 68u
8
− 90u
7
− 112u
6
− 20u
5
+ 38u
4
+ 62u
3
+ 30u
2
+ 4u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
19
+ 27u
18
+ ··· + 3437u + 121
c
2
, c
6
u
19
− u
18
+ ··· − 37u − 11
c
3
, c
4
, c
8
c
9
, c
10
u
19
− 2u
18
+ ··· − 2u + 2
c
5
u
19
+ 5u
18
+ ··· − 2958u + 842
c
7
, c
11
u
19
+ u
18
+ ··· + 11u − 5
c
12
u
19
− 3u
18
+ ··· + 261u − 25
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
19
− 63y
18
+ ··· + 5592117y −14641
c
2
, c
6
y
19
− 27y
18
+ ··· + 3437y −121
c
3
, c
4
, c
8
c
9
, c
10
y
19
− 24y
18
+ ··· + 195y
2
− 4
c
5
y
19
+ 39y
18
+ ··· + 8404544y −708964
c
7
, c
11
y
19
− 3y
18
+ ··· + 261y −25
c
12
y
19
+ 33y
18
+ ··· + 24021y −625
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.794152 + 0.543465I
a = 0.482447 + 0.204059I
b = 0.924151 + 0.772270I
−7.34662 − 0.67943I 5.65616 + 1.50255I
u = −0.794152 − 0.543465I
a = 0.482447 − 0.204059I
b = 0.924151 − 0.772270I
−7.34662 + 0.67943I 5.65616 − 1.50255I
u = 0.935136 + 0.498632I
a = −0.009698 − 0.441743I
b = −1.24024 + 1.35546I
−6.45737 + 7.68487I 7.03505 − 5.77603I
u = 0.935136 − 0.498632I
a = −0.009698 + 0.441743I
b = −1.24024 − 1.35546I
−6.45737 − 7.68487I 7.03505 + 5.77603I
u = −0.787128 + 0.325506I
a = 0.150228 + 0.339895I
b = −0.48300 − 1.36935I
1.07697 − 4.13087I 9.45859 + 7.55506I
u = −0.787128 − 0.325506I
a = 0.150228 − 0.339895I
b = −0.48300 + 1.36935I
1.07697 + 4.13087I 9.45859 − 7.55506I
u = −0.071777 + 0.733673I
a = 0.15515 − 1.56739I
b = −0.214541 − 0.685780I
−9.51900 − 3.56143I 3.03301 + 2.71216I
u = −0.071777 − 0.733673I
a = 0.15515 + 1.56739I
b = −0.214541 + 0.685780I
−9.51900 + 3.56143I 3.03301 − 2.71216I
u = −0.039838 + 0.447508I
a = 1.21035 + 1.34300I
b = −0.102434 + 0.177489I
−1.13039 + 1.43084I 1.52232 − 3.59827I
u = −0.039838 − 0.447508I
a = 1.21035 − 1.34300I
b = −0.102434 − 0.177489I
−1.13039 − 1.43084I 1.52232 + 3.59827I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.382837
a = 0.501969
b = 0.660647
0.831777 13.6610
u = 1.62442 + 0.15410I
a = −0.86935 + 1.16752I
b = 1.85650 − 1.36466I
0.84264 + 3.32574I 7.36493 − 0.96716I
u = 1.62442 − 0.15410I
a = −0.86935 − 1.16752I
b = 1.85650 + 1.36466I
0.84264 − 3.32574I 7.36493 + 0.96716I
u = 1.64903 + 0.08490I
a = 0.26722 − 2.31947I
b = −0.55282 + 2.88445I
9.55424 + 5.66378I 11.29825 − 4.88655I
u = 1.64903 − 0.08490I
a = 0.26722 + 2.31947I
b = −0.55282 − 2.88445I
9.55424 − 5.66378I 11.29825 + 4.88655I
u = −1.69418 + 0.14515I
a = 1.07205 + 2.00699I
b = −1.98554 − 2.27964I
2.65549 − 10.25360I 8.99721 + 4.95875I
u = −1.69418 − 0.14515I
a = 1.07205 − 2.00699I
b = −1.98554 + 2.27964I
2.65549 + 10.25360I 8.99721 − 4.95875I
u = 1.72122
a = −0.936889
b = 0.803538
14.6839 18.0050
u = −1.74710
a = −1.48185
b = 2.13166
11.7122 5.60290
6
II. I
u
2
= hb + u − 1, 3a − 2u + 3, u
2
− 3i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−3
a
5
=
u
−2u
a
7
=
2
3
u − 1
−u + 1
a
11
=
−2
3
a
12
=
−
2
3
u − 1
u + 2
a
3
=
−u
u
a
2
=
−
5
3
u + 1
2u − 1
a
1
=
−
2
3
u + 1
u − 1
a
6
=
−u
u
a
8
=
−2
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
11
c
12
(u − 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
− 3
c
6
, c
7
(u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y −3)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.73205
a = 0.154701
b = −0.732051
13.1595 12.0000
u = −1.73205
a = −2.15470
b = 2.73205
13.1595 12.0000
10
III.
I
u
3
= ha
2
u + 2a
2
+ 2au + b + 5a − 2u − 2, a
3
+ 2a
2
− 3au − u + 1, u
2
+ u − 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
u − 1
a
5
=
u
−u + 1
a
7
=
a
−a
2
u − 2a
2
− 2au − 5a + 2u + 2
a
11
=
u
−u
a
12
=
a
a
2
u + 2a
2
+ 3au + 4a − 2u − 2
a
3
=
−u
u
a
2
=
−a
−a
2
u − 2a
2
− 3au − 4a + 2u + 2
a
1
=
−a
2
− au − 3a + 2u
−a
2
u − a
2
− 2au − 2a + 2
a
6
=
u
−u + 1
a
8
=
u
−u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 4u
5
+ 6u
4
+ 7u
3
+ 7u
2
+ 3u + 1
c
2
, c
6
, c
7
c
11
u
6
− 2u
4
− u
3
+ u
2
+ u − 1
c
3
, c
4
, c
8
c
9
, c
10
(u
2
+ u − 1)
3
c
5
u
6
c
12
u
6
− 4u
5
+ 6u
4
− 7u
3
+ 7u
2
− 3u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
6
− 4y
5
− 6y
4
+ 13y
3
+ 19y
2
+ 5y + 1
c
2
, c
6
, c
7
c
11
y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1
c
3
, c
4
, c
8
c
9
, c
10
(y
2
− 3y + 1)
3
c
5
y
6
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0.365159 + 0.080975I
b = 0.626987 − 0.659789I
0.986960 10.0000
u = 0.618034
a = 0.365159 − 0.080975I
b = 0.626987 + 0.659789I
0.986960 10.0000
u = 0.618034
a = −2.73032
b = 0.746026
0.986960 10.0000
u = −1.61803
a = −0.659441
b = −0.238962
8.88264 10.0000
u = −1.61803
a = −0.67028 + 1.87638I
b = 1.11948 − 2.34901I
8.88264 10.0000
u = −1.61803
a = −0.67028 − 1.87638I
b = 1.11948 + 2.34901I
8.88264 10.0000
14
IV. I
u
4
= hb − 1, a + 1, u + 1i
(i) Arc colorings
a
4
=
0
−1
a
9
=
1
0
a
10
=
1
−1
a
5
=
−1
0
a
7
=
−1
1
a
11
=
0
−1
a
12
=
−1
0
a
3
=
1
−1
a
2
=
1
−1
a
1
=
1
−1
a
6
=
−1
1
a
8
=
0
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 18
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
c
3
, c
4
, c
7
c
8
, c
9
, c
10
c
11
u + 1
c
5
, c
12
u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
c
3
, c
4
, c
5
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y − 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 1.00000
4.93480 18.0000
18
V. I
u
5
= hb, a + 1, u + 1i
(i) Arc colorings
a
4
=
0
−1
a
9
=
1
0
a
10
=
1
−1
a
5
=
−1
0
a
7
=
−1
0
a
11
=
0
−1
a
12
=
−1
−1
a
3
=
1
−1
a
2
=
0
−1
a
1
=
−1
0
a
6
=
−1
1
a
8
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
, c
12
u − 1
c
2
, c
5
, c
8
c
9
, c
10
, c
11
u + 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y − 1
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
3.28987 12.0000
22
VI. I
u
6
= hb − 2, a + 1, u − 1i
(i) Arc colorings
a
4
=
0
1
a
9
=
1
0
a
10
=
1
−1
a
5
=
1
0
a
7
=
−1
2
a
11
=
0
−1
a
12
=
−1
1
a
3
=
−1
1
a
2
=
−2
3
a
1
=
−1
2
a
6
=
1
−1
a
8
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
12
u − 1
c
2
, c
3
, c
4
c
11
u + 1
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y − 1
25
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 2.00000
3.28987 12.0000
26
VII. I
u
7
= hb − 1, a, u − 1i
(i) Arc colorings
a
4
=
0
1
a
9
=
1
0
a
10
=
1
−1
a
5
=
1
0
a
7
=
0
1
a
11
=
0
−1
a
12
=
0
−1
a
3
=
−1
1
a
2
=
−1
2
a
1
=
0
1
a
6
=
1
−1
a
8
=
0
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
27
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u + 1
c
2
, c
3
, c
4
c
6
, c
8
, c
9
c
10
u − 1
c
7
, c
11
, c
12
u
28
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
y − 1
c
7
, c
11
, c
12
y
29
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = 1.00000
1.64493 6.00000
30
VIII. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
1
0
a
10
=
1
0
a
5
=
1
0
a
7
=
0
1
a
11
=
1
0
a
12
=
1
−1
a
3
=
1
0
a
2
=
1
−1
a
1
=
0
−1
a
6
=
1
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
31
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
11
c
12
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
7
u + 1
32
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
y − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
33
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
34
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u − 1)
5
(u + 1)(u
6
+ 4u
5
+ 6u
4
+ 7u
3
+ 7u
2
+ 3u + 1)
· (u
19
+ 27u
18
+ ··· + 3437u + 121)
c
2
u(u − 1)
4
(u + 1)
2
(u
6
− 2u
4
− u
3
+ u
2
+ u − 1)
· (u
19
− u
18
+ ··· − 37u − 11)
c
3
, c
4
, c
8
c
9
, c
10
u(u − 1)
2
(u + 1)
2
(u
2
− 3)(u
2
+ u − 1)
3
(u
19
− 2u
18
+ ··· − 2u + 2)
c
5
u
7
(u − 1)
2
(u + 1)
2
(u
2
− 3)(u
19
+ 5u
18
+ ··· − 2958u + 842)
c
6
u(u − 1)
3
(u + 1)
3
(u
6
− 2u
4
− u
3
+ u
2
+ u − 1)
· (u
19
− u
18
+ ··· − 37u − 11)
c
7
u(u − 1)
2
(u + 1)
4
(u
6
− 2u
4
− u
3
+ u
2
+ u − 1)
· (u
19
+ u
18
+ ··· + 11u − 5)
c
11
u(u − 1)
3
(u + 1)
3
(u
6
− 2u
4
− u
3
+ u
2
+ u − 1)
· (u
19
+ u
18
+ ··· + 11u − 5)
c
12
u(u − 1)
6
(u
6
− 4u
5
+ 6u
4
− 7u
3
+ 7u
2
− 3u + 1)
· (u
19
− 3u
18
+ ··· + 261u − 25)
35
X. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y(y − 1)
6
(y
6
− 4y
5
− 6y
4
+ 13y
3
+ 19y
2
+ 5y + 1)
· (y
19
− 63y
18
+ ··· + 5592117y − 14641)
c
2
, c
6
y(y − 1)
6
(y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1)
· (y
19
− 27y
18
+ ··· + 3437y − 121)
c
3
, c
4
, c
8
c
9
, c
10
y(y − 3)
2
(y − 1)
4
(y
2
− 3y + 1)
3
(y
19
− 24y
18
+ ··· + 195y
2
− 4)
c
5
y
7
(y − 3)
2
(y − 1)
4
(y
19
+ 39y
18
+ ··· + 8404544y − 708964)
c
7
, c
11
y(y − 1)
6
(y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1)
· (y
19
− 3y
18
+ ··· + 261y − 25)
c
12
y(y − 1)
6
(y
6
− 4y
5
− 6y
4
+ 13y
3
+ 19y
2
+ 5y + 1)
· (y
19
+ 33y
18
+ ··· + 24021y − 625)
36
