12n
0051
(K12n
0051
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 10 12 4 7 6 8 11
Solving Sequence
7,9
10 6
4,11
3 8 5 12 2 1
c
9
c
6
c
10
c
3
c
8
c
4
c
11
c
2
c
1
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.54023 × 10
25
u
21
− 4.36397 × 10
25
u
20
+ ··· + 2.35281 × 10
27
b − 1.19412 × 10
26
,
4.19615 × 10
28
u
21
+ 1.33730 × 10
29
u
20
+ ··· + 1.37404 × 10
30
a + 5.83977 × 10
30
,
u
22
+ 3u
21
+ ··· − 160u + 73i
I
u
2
= hb, 6u
3
a − 4u
2
a − 3u
3
+ 4a
2
+ 14au − 2u
2
− 6a − 7u − 7, u
4
− u
3
+ 3u
2
− 2u + 1i
I
u
3
= h−a
4
u + a
3
u + a
3
− 2a
2
+ 4au + 4b − 4a − 4u, a
5
+ a
4
u − a
4
− 2a
3
u − 4a
2
u − 4a
2
+ 4a − 4u + 4, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 40 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−1.54 × 10
25
u
21
− 4.36 × 10
25
u
20
+ · · · + 2.35 × 10
27
b − 1.19 ×
10
26
, 4.20 × 10
28
u
21
+ 1.34 × 10
29
u
20
+ · · · + 1.37 × 10
30
a + 5.84 ×
10
30
, u
22
+ 3u
21
+ · · · − 160u + 73i
(i) Arc colorings
a
7
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
−u
u
3
+ u
a
4
=
−0.0305388u
21
− 0.0973265u
20
+ ··· − 6.20155u − 4.25007
0.00654633u
21
+ 0.0185480u
20
+ ··· + 4.03131u + 0.0507530
a
11
=
u
2
+ 1
−u
4
− 2u
2
a
3
=
−0.0269488u
21
− 0.0862650u
20
+ ··· − 5.94433u − 3.75869
0.00399982u
21
+ 0.0101440u
20
+ ··· + 3.98951u − 0.419348
a
8
=
−0.0111557u
21
− 0.0331545u
20
+ ··· − 11.4392u + 1.82405
0.00209278u
21
+ 0.00628667u
20
+ ··· + 2.38388u − 0.320958
a
5
=
−0.0325563u
21
− 0.0995247u
20
+ ··· − 12.3684u − 1.68756
0.00662300u
21
+ 0.0176776u
20
+ ··· + 5.55048u − 0.614116
a
12
=
0.00503747u
21
+ 0.0176554u
20
+ ··· + 0.745076u + 3.13839
−0.000640784u
21
− 0.00237253u
20
+ ··· − 0.108742u − 0.457982
a
2
=
−0.0116342u
21
− 0.0404516u
20
+ ··· + 5.25864u − 5.57795
0.000768008u
21
+ 0.00322339u
20
+ ··· + 0.588075u + 0.675650
a
1
=
−0.00471751u
21
− 0.0164802u
20
+ ··· − 0.689358u − 2.34200
0.000443931u
21
+ 0.00178731u
20
+ ··· + 0.0810747u + 0.491670
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.0299967u
21
+ 0.106465u
20
+ ··· − 19.9627u + 12.9465
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
22
+ 19u
21
+ ··· + 79u + 16
c
2
, c
5
u
22
+ 7u
21
+ ··· + 35u + 4
c
3
u
22
− 16u
21
+ ··· + 25000u + 3104
c
4
, c
8
u
22
− u
21
+ ··· + 1536u + 2048
c
6
, c
9
, c
10
u
22
+ 3u
21
+ ··· − 160u + 73
c
7
, c
11
u
22
+ 3u
21
+ ··· + 182u + 73
c
12
u
22
+ 7u
21
+ ··· − 67032u + 5329
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
22
− 25y
21
+ ··· + 179903y + 256
c
2
, c
5
y
22
+ 19y
21
+ ··· + 79y + 16
c
3
y
22
− 78y
21
+ ··· + 78714048y + 9634816
c
4
, c
8
y
22
+ 91y
21
+ ··· + 30670848y + 4194304
c
6
, c
9
, c
10
y
22
+ 45y
21
+ ··· + 149016y + 5329
c
7
, c
11
y
22
− 7y
21
+ ··· + 67032y + 5329
c
12
y
22
+ 85y
21
+ ··· + 2794246372y + 28398241
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.166885 + 0.855784I
a = 0.884204 − 0.564449I
b = −0.685307 − 0.431142I
−1.86083 + 1.88410I −4.29628 − 4.39442I
u = 0.166885 − 0.855784I
a = 0.884204 + 0.564449I
b = −0.685307 + 0.431142I
−1.86083 − 1.88410I −4.29628 + 4.39442I
u = 1.245170 + 0.161308I
a = −0.584895 + 0.469137I
b = 0.88430 − 1.76284I
−3.01876 + 2.75600I 1.05384 − 1.99167I
u = 1.245170 − 0.161308I
a = −0.584895 − 0.469137I
b = 0.88430 + 1.76284I
−3.01876 − 2.75600I 1.05384 + 1.99167I
u = 0.065911 + 1.393150I
a = −0.167886 + 0.219714I
b = 0.208154 + 0.992360I
−7.41484 + 5.99413I −4.98068 − 7.65331I
u = 0.065911 − 1.393150I
a = −0.167886 − 0.219714I
b = 0.208154 − 0.992360I
−7.41484 − 5.99413I −4.98068 + 7.65331I
u = −0.147428 + 0.530014I
a = 2.56324 − 1.19415I
b = −0.391902 − 0.411319I
0.18307 − 2.82080I 2.85537 − 1.68871I
u = −0.147428 − 0.530014I
a = 2.56324 + 1.19415I
b = −0.391902 + 0.411319I
0.18307 + 2.82080I 2.85537 + 1.68871I
u = 0.309359 + 0.401971I
a = −0.508079 − 0.818004I
b = 0.193284 + 0.440196I
0.445026 + 1.231770I 4.87220 − 5.67709I
u = 0.309359 − 0.401971I
a = −0.508079 + 0.818004I
b = 0.193284 − 0.440196I
0.445026 − 1.231770I 4.87220 + 5.67709I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.14190 + 1.59059I
a = −0.104582 − 0.310705I
b = −0.841835 − 0.832729I
−5.61868 + 1.54212I 1.66178 − 2.03716I
u = −0.14190 − 1.59059I
a = −0.104582 + 0.310705I
b = −0.841835 + 0.832729I
−5.61868 − 1.54212I 1.66178 + 2.03716I
u = −0.009993 + 0.350116I
a = −3.01267 − 0.71642I
b = 0.469397 + 0.461238I
0.96093 + 1.37462I 8.72525 − 4.65494I
u = −0.009993 − 0.350116I
a = −3.01267 + 0.71642I
b = 0.469397 − 0.461238I
0.96093 − 1.37462I 8.72525 + 4.65494I
u = −0.69231 + 1.86047I
a = −0.585229 − 1.015730I
b = 1.41273 − 1.99234I
15.9166 − 13.0727I 0.81219 + 5.19676I
u = −0.69231 − 1.86047I
a = −0.585229 + 1.015730I
b = 1.41273 + 1.99234I
15.9166 + 13.0727I 0.81219 − 5.19676I
u = −0.61577 + 2.17742I
a = 0.394124 + 0.984688I
b = −1.03067 + 2.69186I
19.2619 − 5.9056I 2.16347 + 1.69823I
u = −0.61577 − 2.17742I
a = 0.394124 − 0.984688I
b = −1.03067 − 2.69186I
19.2619 + 5.9056I 2.16347 − 1.69823I
u = −1.57086 + 2.18266I
a = 0.515590 + 0.201211I
b = 1.02473 + 4.03132I
−13.72130 − 3.06559I 0
u = −1.57086 − 2.18266I
a = 0.515590 − 0.201211I
b = 1.02473 − 4.03132I
−13.72130 + 3.06559I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.10906 + 2.96605I
a = −0.102719 − 0.793534I
b = −0.74289 − 4.42273I
12.96110 + 1.49730I 0
u = −0.10906 − 2.96605I
a = −0.102719 + 0.793534I
b = −0.74289 + 4.42273I
12.96110 − 1.49730I 0
7
II. I
u
2
= hb, 6u
3
a − 3u
3
+ · · · − 6a − 7, u
4
− u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
−u
u
3
+ u
a
4
=
a
0
a
11
=
u
2
+ 1
−u
3
+ u
2
− 2u + 1
a
3
=
u
3
a − 2u
2
a + 2au
u
3
a − 2u
2
a + au
a
8
=
1
0
a
5
=
a
0
a
12
=
u
3
+ 2u
−u
3
+ u
2
− 2u + 1
a
2
=
u
3
a − 2u
2
a +
3
2
u
3
+ 2au − u
2
+
7
2
u −
3
2
u
3
a − 2u
2
a + au
a
1
=
u
−u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes =
9
2
u
3
a − 5u
2
a −
19
4
u
3
+
21
2
au +
7
2
u
2
−
7
2
a −
55
4
u +
17
4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
8
u
8
c
6
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
7
(u
4
+ u
3
+ u
2
+ 1)
2
c
9
, c
10
, c
12
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
11
(u
4
− u
3
+ u
2
+ 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
4
c
4
, c
8
y
8
c
6
, c
9
, c
10
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
7
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = −1.13839 − 1.09665I
b = 0
0.21101 + 3.44499I 3.71851 − 10.46973I
u = 0.395123 + 0.506844I
a = 1.51892 − 0.43755I
b = 0
0.211005 − 0.614778I 1.372162 − 0.328352I
u = 0.395123 − 0.506844I
a = −1.13839 + 1.09665I
b = 0
0.21101 − 3.44499I 3.71851 + 10.46973I
u = 0.395123 − 0.506844I
a = 1.51892 + 0.43755I
b = 0
0.211005 + 0.614778I 1.372162 + 0.328352I
u = 0.10488 + 1.55249I
a = −0.435815 + 0.100890I
b = 0
−6.79074 + 5.19385I 0.529613 − 1.243149I
u = 0.10488 + 1.55249I
a = 0.305281 + 0.326982I
b = 0
−6.79074 + 1.13408I −4.49529 − 1.20873I
u = 0.10488 − 1.55249I
a = −0.435815 − 0.100890I
b = 0
−6.79074 − 5.19385I 0.529613 + 1.243149I
u = 0.10488 − 1.55249I
a = 0.305281 − 0.326982I
b = 0
−6.79074 − 1.13408I −4.49529 + 1.20873I
11
III. I
u
3
= h−a
4
u + a
3
u + · · · − 2a
2
− 4a, a
4
u − 2a
3
u + · · · + 4a + 4, u
2
+ 1i
(i) Arc colorings
a
7
=
0
u
a
9
=
1
0
a
10
=
1
1
a
6
=
−u
0
a
4
=
a
1
4
a
4
u −
1
4
a
3
u + ··· +
1
2
a
2
+ a
a
11
=
0
1
a
3
=
−
1
4
a
4
u +
1
4
a
3
u + ··· +
1
4
a
3
−
1
2
a
2
1
4
a
4
u −
1
4
a
3
u + ··· +
1
2
a
2
+ a
a
8
=
−u
−
1
2
a
3
u + a
2
u + ··· +
1
2
a − 3
a
5
=
1
4
a
3
u −
1
2
a
2
u + ··· −
1
4
a
3
+ 1
−
1
2
a
4
u +
3
4
a
3
u + ··· − 2a
2
− a
a
12
=
1
−
1
4
a
4
u +
1
2
a
3
u + ··· −
1
2
a + 1
a
2
=
−
1
2
a
4
u +
1
4
a
3
u + ··· − 2a
2
− 1
1
2
a
4
u −
3
4
a
3
u + ··· + 2a
2
+
1
2
a
a
1
=
1
−
1
4
a
4
u +
1
2
a
3
u + ··· − a
2
−
1
2
a
(ii) Obstruction class = 1
(iii) Cusp Shapes = a
4
+ 2a
3
u − 2a
3
− 6a
2
u + 8
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
2
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
3
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
4
, c
8
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
c
5
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
6
, c
7
, c
9
c
10
, c
11
(u
2
+ 1)
5
c
12
(u − 1)
10
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
c
2
, c
5
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
c
3
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
c
4
, c
8
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
c
6
, c
7
, c
9
c
10
, c
11
(y + 1)
10
c
12
(y − 1)
10
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −0.794743 + 0.582062I
b = 0.21917 + 1.41878I
−5.87256 − 4.40083I −0.74431 + 3.49859I
u = 1.000000I
a = −0.582062 + 0.794743I
b = −0.21917 + 1.41878I
−5.87256 + 4.40083I −0.74431 − 3.49859I
u = 1.000000I
a = 0.821196 − 0.821196I
b = − 1.217740I
−2.40108 2.51886 + 0.I
u = 1.000000I
a = 2.15793 + 0.60232I
b = −0.549911 − 0.309916I
−0.32910 + 1.53058I 3.48489 − 4.43065I
u = 1.000000I
a = −0.60232 − 2.15793I
b = 0.549911 − 0.309916I
−0.32910 − 1.53058I 3.48489 + 4.43065I
u = − 1.000000I
a = −0.582062 − 0.794743I
b = −0.21917 − 1.41878I
−5.87256 + 4.40083I −0.74431 − 3.49859I
u = − 1.000000I
a = −0.794743 − 0.582062I
b = 0.21917 − 1.41878I
−5.87256 − 4.40083I −0.74431 + 3.49859I
u = − 1.000000I
a = 0.821196 + 0.821196I
b = 1.217740I
−2.40108 2.51886 + 0.I
u = − 1.000000I
a = 2.15793 − 0.60232I
b = −0.549911 + 0.309916I
−0.32910 − 1.53058I 3.48489 + 4.43065I
u = − 1.000000I
a = −0.60232 + 2.15793I
b = 0.549911 + 0.309916I
−0.32910 + 1.53058I 3.48489 − 4.43065I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
4
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
· (u
22
+ 19u
21
+ ··· + 79u + 16)
c
2
((u
2
+ u + 1)
4
)(u
5
− u
4
+ ··· + u − 1)
2
(u
22
+ 7u
21
+ ··· + 35u + 4)
c
3
(u
2
− u + 1)
4
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
· (u
22
− 16u
21
+ ··· + 25000u + 3104)
c
4
, c
8
u
8
(u
10
+ 5u
8
+ ··· − u
2
+ 1)(u
22
− u
21
+ ··· + 1536u + 2048)
c
5
((u
2
− u + 1)
4
)(u
5
+ u
4
+ ··· + u + 1)
2
(u
22
+ 7u
21
+ ··· + 35u + 4)
c
6
((u
2
+ 1)
5
)(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
(u
22
+ 3u
21
+ ··· − 160u + 73)
c
7
((u
2
+ 1)
5
)(u
4
+ u
3
+ u
2
+ 1)
2
(u
22
+ 3u
21
+ ··· + 182u + 73)
c
9
, c
10
((u
2
+ 1)
5
)(u
4
− u
3
+ 3u
2
− 2u + 1)
2
(u
22
+ 3u
21
+ ··· − 160u + 73)
c
11
((u
2
+ 1)
5
)(u
4
− u
3
+ u
2
+ 1)
2
(u
22
+ 3u
21
+ ··· + 182u + 73)
c
12
(u − 1)
10
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· (u
22
+ 7u
21
+ ··· − 67032u + 5329)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)
4
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
· (y
22
− 25y
21
+ ··· + 179903y + 256)
c
2
, c
5
(y
2
+ y + 1)
4
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
· (y
22
+ 19y
21
+ ··· + 79y + 16)
c
3
(y
2
+ y + 1)
4
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
· (y
22
− 78y
21
+ ··· + 78714048y + 9634816)
c
4
, c
8
y
8
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
· (y
22
+ 91y
21
+ ··· + 30670848y + 4194304)
c
6
, c
9
, c
10
(y + 1)
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
22
+ 45y
21
+ ··· + 149016y + 5329)
c
7
, c
11
(y + 1)
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
22
− 7y
21
+ ··· + 67032y + 5329)
c
12
(y − 1)
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
22
+ 85y
21
+ ··· + 2794246372y + 28398241)
17
