12n
0437
(K12n
0437
)
A knot diagram
1
Linearized knot diagam
3 6 9 7 2 5 11 4 12 7 9 10
Solving Sequence
4,7 5,11
8 9 12 3 6 2 1 10
c
4
c
7
c
8
c
11
c
3
c
6
c
2
c
1
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h9.19324 × 10
24
u
24
− 9.37535 × 10
25
u
23
+ ··· + 7.01855 × 10
25
b + 8.63495 × 10
25
,
− 4.13470 × 10
25
u
24
+ 4.13261 × 10
26
u
23
+ ··· + 7.01855 × 10
25
a − 2.79968 × 10
27
,
u
25
− 10u
24
+ ··· + 72u − 1i
I
u
2
= h−u
4
+ u
3
− 4u
2
+ b + 3u − 3, a, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
I
u
3
= hu
2
a + b + u, u
2
a + a
2
− au − 2u
2
+ 2a + u − 3, u
3
− u
2
+ 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 36 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
= h9.19 × 10
24
u
24
− 9.38 × 10
25
u
23
+ · · · + 7.02 × 10
25
b + 8.63 ×
10
25
, −4.13 × 10
25
u
24
+ 4.13 × 10
26
u
23
+ · · · + 7.02 × 10
25
a − 2.80 ×
10
27
, u
25
− 10u
24
+ · · · + 72u − 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
11
=
0.589110u
24
− 5.88812u
23
+ ··· − 335.813u + 39.8897
−0.130985u
24
+ 1.33580u
23
+ ··· + 39.4166u − 1.23030
a
8
=
0.452645u
24
− 4.39464u
23
+ ··· − 195.754u + 22.1379
−0.146274u
24
+ 1.40258u
23
+ ··· + 20.5478u − 0.642297
a
9
=
0.598919u
24
− 5.79722u
23
+ ··· − 216.301u + 22.7802
−0.146274u
24
+ 1.40258u
23
+ ··· + 20.5478u − 0.642297
a
12
=
0.598919u
24
− 5.79722u
23
+ ··· − 216.301u + 22.7802
0.0345873u
24
− 0.219643u
23
+ ··· + 21.9603u − 0.710241
a
3
=
−0.0157966u
24
+ 0.0835882u
23
+ ··· − 49.3795u + 8.57675
−0.0421281u
24
+ 0.395025u
23
+ ··· + 1.19935u − 0.137223
a
6
=
u
u
3
+ u
a
2
=
0.0628449u
24
− 0.637985u
23
+ ··· − 55.7029u + 8.66493
0.0743783u
24
− 0.692119u
23
+ ··· − 9.71411u + 0.0157966
a
1
=
0.137223u
24
− 1.33010u
23
+ ··· − 65.4171u + 8.68072
0.100634u
24
− 0.951779u
23
+ ··· − 12.6101u + 0.0579248
a
10
=
0.589110u
24
− 5.88812u
23
+ ··· − 335.813u + 39.8897
−0.137742u
24
+ 1.39067u
23
+ ··· + 39.7915u − 1.22733
(ii) Obstruction class = −1
(iii) Cusp Shapes =
25106355762453601367160863
23395155619836929736818172
u
24
−
62945700046644877607050528
5848788904959232434204543
u
23
+
··· −
511603421605188008732954519
1376185624696289984518716
u +
351161482484964948702456565
23395155619836929736818172
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
25
+ 10u
24
+ ··· + 72u + 1
c
2
, c
5
u
25
+ 4u
24
+ ··· − 12u − 1
c
3
, c
8
u
25
− 2u
24
+ ··· − 32u − 64
c
7
, c
10
u
25
− 4u
24
+ ··· − 192u − 32
c
9
, c
11
, c
12
u
25
+ 9u
24
+ ··· − 41u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
y
25
+ 14y
24
+ ··· + 4016y −1
c
2
, c
5
y
25
− 10y
24
+ ··· + 72y −1
c
3
, c
8
y
25
− 28y
24
+ ··· + 29696y −4096
c
7
, c
10
y
25
+ 24y
24
+ ··· + 51712y −1024
c
9
, c
11
, c
12
y
25
− 9y
24
+ ··· + 1947y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.799543 + 0.627415I
a = −0.93941 + 1.56370I
b = −0.01758 + 1.51331I
−4.36096 − 1.13139I 0.75657 + 1.52598I
u = −0.799543 − 0.627415I
a = −0.93941 − 1.56370I
b = −0.01758 − 1.51331I
−4.36096 + 1.13139I 0.75657 − 1.52598I
u = −0.034202 + 0.923614I
a = −0.686187 − 0.237402I
b = 0.108494 + 0.547445I
1.97950 − 1.66008I −0.69040 + 2.96263I
u = −0.034202 − 0.923614I
a = −0.686187 + 0.237402I
b = 0.108494 − 0.547445I
1.97950 + 1.66008I −0.69040 − 2.96263I
u = 0.856820 + 0.829058I
a = 1.063360 − 0.668986I
b = 0.022399 − 0.950691I
−0.38972 − 2.81828I −1.13877 + 3.80627I
u = 0.856820 − 0.829058I
a = 1.063360 + 0.668986I
b = 0.022399 + 0.950691I
−0.38972 + 2.81828I −1.13877 − 3.80627I
u = 0.755262
a = −2.29591
b = 0.108132
7.52575 −13.1500
u = 0.240993 + 1.276750I
a = −0.034747 + 0.379118I
b = 0.13663 + 3.40519I
4.24524 − 2.77554I 34.2342 + 3.0457I
u = 0.240993 − 1.276750I
a = −0.034747 − 0.379118I
b = 0.13663 − 3.40519I
4.24524 + 2.77554I 34.2342 − 3.0457I
u = −0.563738 + 1.206990I
a = 0.839591 − 1.130370I
b = −0.40514 − 1.62951I
−2.40018 + 6.37988I 2.18378 − 2.52933I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.563738 − 1.206990I
a = 0.839591 + 1.130370I
b = −0.40514 + 1.62951I
−2.40018 − 6.37988I 2.18378 + 2.52933I
u = 0.75670 + 1.19685I
a = 0.860055 + 0.001764I
b = −0.149981 + 0.594534I
0.73025 − 3.27384I −1.24326 + 3.27643I
u = 0.75670 − 1.19685I
a = 0.860055 − 0.001764I
b = −0.149981 − 0.594534I
0.73025 + 3.27384I −1.24326 − 3.27643I
u = 0.462757 + 0.342084I
a = −0.172877 − 1.143600I
b = −0.66811 − 1.47469I
0.902564 − 0.255949I −3.12150 + 6.64716I
u = 0.462757 − 0.342084I
a = −0.172877 + 1.143600I
b = −0.66811 + 1.47469I
0.902564 + 0.255949I −3.12150 − 6.64716I
u = 0.447970
a = −0.336060
b = 0.456804
−0.908338 −11.6550
u = 0.11903 + 1.59221I
a = −0.113155 + 0.609313I
b = 0.018929 + 0.249452I
13.29460 − 3.19957I 9.89238 + 1.87771I
u = 0.11903 − 1.59221I
a = −0.113155 − 0.609313I
b = 0.018929 − 0.249452I
13.29460 + 3.19957I 9.89238 − 1.87771I
u = 1.63956 + 0.34345I
a = −0.019703 − 1.413500I
b = 0.15371 − 1.68285I
−10.18570 − 4.57384I 0. + 2.62009I
u = 1.63956 − 0.34345I
a = −0.019703 + 1.413500I
b = 0.15371 + 1.68285I
−10.18570 + 4.57384I 0. − 2.62009I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.04678 + 1.44405I
a = 0.689190 + 1.046230I
b = −0.09880 + 1.59912I
−7.03225 − 4.52109I 0
u = 1.04678 − 1.44405I
a = 0.689190 − 1.046230I
b = −0.09880 − 1.59912I
−7.03225 + 4.52109I 0
u = 0.66532 + 1.68962I
a = −0.626781 − 0.927153I
b = 0.44778 − 1.73312I
−3.94520 − 12.71920I 0
u = 0.66532 − 1.68962I
a = −0.626781 + 0.927153I
b = 0.44778 + 1.73312I
−3.94520 + 12.71920I 0
u = 0.0157871
a = 34.9133
b = −0.661595
1.12664 9.59670
7
II. I
u
2
= h−u
4
+ u
3
− 4u
2
+ b + 3u − 3, a, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
11
=
0
u
4
− u
3
+ 4u
2
− 3u + 3
a
8
=
0
u
a
9
=
−u
u
a
12
=
u
u
4
− u
3
+ 4u
2
− 4u + 3
a
3
=
u
2
+ 1
−u
2
a
6
=
u
u
3
+ u
a
2
=
u
3
+ 2u
−u
3
− u
a
1
=
u
−u
a
10
=
0
u
4
− u
3
+ 4u
2
− 3u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −18u
4
+ 11u
3
− 65u
2
+ 29u − 38
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
2
u
5
− u
4
+ u
2
+ u − 1
c
5
u
5
+ u
4
− u
2
+ u + 1
c
6
, c
8
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
7
, c
10
u
5
c
9
(u + 1)
5
c
11
, c
12
(u − 1)
5
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
8
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
2
, c
5
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
7
, c
10
y
5
c
9
, c
11
, c
12
(y −1)
5
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = 0
b = 0.278580 − 1.055720I
3.46474 − 2.21397I 3.79538 + 3.60694I
u = 0.233677 − 0.885557I
a = 0
b = 0.278580 + 1.055720I
3.46474 + 2.21397I 3.79538 − 3.60694I
u = 0.416284
a = 0
b = 2.40221
0.762751 −36.9390
u = 0.05818 + 1.69128I
a = 0
b = 0.020316 − 0.590570I
12.60320 − 3.33174I −2.32599 + 3.47010I
u = 0.05818 − 1.69128I
a = 0
b = 0.020316 + 0.590570I
12.60320 + 3.33174I −2.32599 − 3.47010I
11
III. I
u
3
= hu
2
a + b + u, u
2
a + a
2
− au − 2u
2
+ 2a + u − 3, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
11
=
a
−u
2
a − u
a
8
=
u
2
− a − u + 2
0
a
9
=
u
2
− a − u + 2
0
a
12
=
u
2
− a − u + 2
−u
2
a − u
a
3
=
1
0
a
6
=
u
u
2
− u + 1
a
2
=
u
−u
a
1
=
0
−u
a
10
=
a
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3au + 5u
2
− 3a + 2u + 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
3
, c
8
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
9
(u
2
− u − 1)
3
c
10
, c
11
, c
12
(u
2
+ u − 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
(y
3
+ 3y
2
+ 2y − 1)
2
c
2
, c
5
(y
3
− y
2
+ 2y − 1)
2
c
3
, c
8
y
6
c
7
, c
9
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.198308 + 1.205210I
b = 0.132927 + 0.807858I
11.90680 − 2.82812I 1.56739 + 1.81005I
u = 0.215080 + 1.307140I
a = 0.075747 − 0.460350I
b = −0.34801 − 2.11500I
4.01109 − 2.82812I −5.96298 + 6.80673I
u = 0.215080 − 1.307140I
a = −0.198308 − 1.205210I
b = 0.132927 − 0.807858I
11.90680 + 2.82812I 1.56739 − 1.81005I
u = 0.215080 − 1.307140I
a = 0.075747 + 0.460350I
b = −0.34801 + 2.11500I
4.01109 + 2.82812I −5.96298 − 6.80673I
u = 0.569840
a = 1.08457
b = −0.922021
−0.126494 1.65540
u = 0.569840
a = −2.83945
b = 0.352181
7.76919 20.1360
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
3
− u
2
+ 2u − 1)
2
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
· (u
25
+ 10u
24
+ ··· + 72u + 1)
c
2
((u
3
+ u
2
− 1)
2
)(u
5
− u
4
+ u
2
+ u − 1)(u
25
+ 4u
24
+ ··· − 12u − 1)
c
3
u
6
(u
5
− u
4
+ ··· + 3u − 1)(u
25
− 2u
24
+ ··· − 32u − 64)
c
5
((u
3
− u
2
+ 1)
2
)(u
5
+ u
4
− u
2
+ u + 1)(u
25
+ 4u
24
+ ··· − 12u − 1)
c
6
(u
3
+ u
2
+ 2u + 1)
2
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
· (u
25
+ 10u
24
+ ··· + 72u + 1)
c
7
u
5
(u
2
− u − 1)
3
(u
25
− 4u
24
+ ··· − 192u − 32)
c
8
u
6
(u
5
+ u
4
+ ··· + 3u + 1)(u
25
− 2u
24
+ ··· − 32u − 64)
c
9
((u + 1)
5
)(u
2
− u − 1)
3
(u
25
+ 9u
24
+ ··· − 41u + 1)
c
10
u
5
(u
2
+ u − 1)
3
(u
25
− 4u
24
+ ··· − 192u − 32)
c
11
, c
12
((u − 1)
5
)(u
2
+ u − 1)
3
(u
25
+ 9u
24
+ ··· − 41u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
(y
3
+ 3y
2
+ 2y − 1)
2
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
· (y
25
+ 14y
24
+ ··· + 4016y − 1)
c
2
, c
5
(y
3
− y
2
+ 2y − 1)
2
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)
· (y
25
− 10y
24
+ ··· + 72y − 1)
c
3
, c
8
y
6
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
· (y
25
− 28y
24
+ ··· + 29696y − 4096)
c
7
, c
10
y
5
(y
2
− 3y + 1)
3
(y
25
+ 24y
24
+ ··· + 51712y − 1024)
c
9
, c
11
, c
12
((y − 1)
5
)(y
2
− 3y + 1)
3
(y
25
− 9y
24
+ ··· + 1947y − 1)
17
