12n
0346
(K12n
0346
)
A knot diagram
1
Linearized knot diagam
3 6 8 9 2 11 5 4 12 6 9 10
Solving Sequence
5,9
4 8 3
7,12
11 6 2 1 10
c
4
c
8
c
3
c
7
c
11
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
= h−8563315867395u
19
+ 12169590007513u
18
+ ··· + 13082431761068b + 64802000815324,
52899877847623u
19
− 79722292100065u
18
+ ··· + 13082431761068a − 413103210378158,
u
20
− 2u
19
+ ··· − 12u + 4i
I
u
2
= hu
7
− u
6
− 2u
5
+ 3u
4
− 2u
2
+ b + 4u − 2, −u
7
− u
6
+ 3u
5
+ 2u
4
− 3u
3
+ a − 2,
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
I
u
3
= h−au + b − u − 1, 2a
2
+ au − 1, u
2
− 2i
I
v
1
= ha, b − v + 2, v
2
− 3v + 1i
* 4 irreducible components of dim
C
= 0, with total 34 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−8.56 × 10
12
u
19
+ 1.22 × 10
13
u
18
+ · · · + 1.31 × 10
13
b + 6.48 ×
10
13
, 5.29 × 10
13
u
19
− 7.97 × 10
13
u
18
+ · · · + 1.31 × 10
13
a − 4.13 ×
10
14
, u
20
− 2u
19
+ · · · − 12u + 4i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
8
=
u
−u
3
+ u
a
3
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
−u
3
+ 2u
−u
3
+ u
a
12
=
−4.04358u
19
+ 6.09384u
18
+ ··· − 25.5560u + 31.5769
0.654566u
19
− 0.930224u
18
+ ··· + 6.84847u − 4.95336
a
11
=
−4.04358u
19
+ 6.09384u
18
+ ··· − 25.5560u + 31.5769
−0.367222u
19
+ 0.611916u
18
+ ··· − 0.897037u + 3.01992
a
6
=
−6.21544u
19
+ 9.28273u
18
+ ··· − 48.2193u + 51.0252
0.348398u
19
− 0.661202u
18
+ ··· + 3.63448u − 3.80249
a
2
=
−4.90774u
19
+ 7.58167u
18
+ ··· − 38.9378u + 42.2351
2.23801u
19
− 3.26288u
18
+ ··· + 16.4563u − 17.7254
a
1
=
−5.35239u
19
+ 8.30016u
18
+ ··· − 43.5388u + 45.8952
2.16712u
19
− 3.30945u
18
+ ··· + 15.7609u − 18.5509
a
10
=
5.60383u
19
− 8.48450u
18
+ ··· + 48.3882u − 47.1067
−2.25347u
19
+ 3.54770u
18
+ ··· − 15.9192u + 19.1670
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
8651471282093
3270607940267
u
19
−
12603254258761
3270607940267
u
18
+ ··· −
78299168927818
3270607940267
u −
83998088848016
3270607940267
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 24u
18
+ ··· + 3807u + 81
c
2
, c
5
u
20
+ 4u
19
+ ··· − 93u − 9
c
3
, c
4
, c
8
u
20
+ 2u
19
+ ··· + 12u + 4
c
6
, c
10
u
20
+ 2u
19
+ ··· + 3200u + 256
c
7
u
20
− 6u
19
+ ··· − 6212u − 964
c
9
, c
11
, c
12
u
20
+ 12u
19
+ ··· − 58u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
+ 48y
19
+ ··· − 10684791y + 6561
c
2
, c
5
y
20
+ 24y
18
+ ··· − 3807y + 81
c
3
, c
4
, c
8
y
20
− 14y
19
+ ··· − 432y + 16
c
6
, c
10
y
20
− 60y
19
+ ··· − 5160960y + 65536
c
7
y
20
+ 58y
19
+ ··· − 44905072y + 929296
c
9
, c
11
, c
12
y
20
− 44y
19
+ ··· − 2922y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.311540 + 0.820503I
a = −1.183960 + 0.104977I
b = −0.936205 − 0.261735I
1.94026 − 0.82547I 0.912486 + 0.761117I
u = −0.311540 − 0.820503I
a = −1.183960 − 0.104977I
b = −0.936205 + 0.261735I
1.94026 + 0.82547I 0.912486 − 0.761117I
u = 0.806269 + 0.099052I
a = −0.029427 − 0.611307I
b = 0.383913 − 0.245700I
−1.290640 + 0.060456I −5.65835 + 0.55419I
u = 0.806269 − 0.099052I
a = −0.029427 + 0.611307I
b = 0.383913 + 0.245700I
−1.290640 − 0.060456I −5.65835 − 0.55419I
u = 1.33638
a = −1.16154
b = 1.16836
3.62783 3.71420
u = −1.301610 + 0.401831I
a = 0.508418 − 0.295516I
b = −0.439577 + 0.851076I
−1.51005 + 5.52302I −5.54282 − 3.24717I
u = −1.301610 − 0.401831I
a = 0.508418 + 0.295516I
b = −0.439577 − 0.851076I
−1.51005 − 5.52302I −5.54282 + 3.24717I
u = 1.037800 + 0.883330I
a = 0.77758 − 1.27910I
b = −2.72069 − 2.90147I
4.14315 − 3.36874I 0.04800 + 2.58345I
u = 1.037800 − 0.883330I
a = 0.77758 + 1.27910I
b = −2.72069 + 2.90147I
4.14315 + 3.36874I 0.04800 − 2.58345I
u = −1.38998
a = −0.0133698
b = −1.23390
−6.53389 −14.0580
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.307583 + 1.370650I
a = 0.05460 + 1.81689I
b = −1.41200 + 5.33955I
−16.9450 + 5.1999I 0.48479 − 2.11695I
u = −0.307583 − 1.370650I
a = 0.05460 − 1.81689I
b = −1.41200 − 5.33955I
−16.9450 − 5.1999I 0.48479 + 2.11695I
u = 1.43456
a = 0.671677
b = 7.72000
−4.98064 60.9990
u = 0.486279
a = −3.47218
b = 1.37914
6.88313 −9.17840
u = 0.388073
a = 0.457271
b = 0.718784
−1.01688 −12.8430
u = 1.58364 + 0.49918I
a = −1.064100 + 0.688577I
b = 3.34498 + 2.38478I
16.4234 − 11.7755I −1.93237 + 4.54638I
u = 1.58364 − 0.49918I
a = −1.064100 − 0.688577I
b = 3.34498 − 2.38478I
16.4234 + 11.7755I −1.93237 − 4.54638I
u = −0.283054
a = −3.06452
b = −0.370034
1.22670 10.7000
u = −1.49310 + 0.84801I
a = 1.22823 + 0.86872I
b = −3.91160 + 3.62679I
19.0199 + 2.7480I −0.478669 − 0.983748I
u = −1.49310 − 0.84801I
a = 1.22823 − 0.86872I
b = −3.91160 − 3.62679I
19.0199 − 2.7480I −0.478669 + 0.983748I
6
II. I
u
2
= hu
7
− u
6
− 2u
5
+ 3u
4
− 2u
2
+ b + 4u − 2, −u
7
− u
6
+ 3u
5
+ 2u
4
−
3u
3
+ a − 2, u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
8
=
u
−u
3
+ u
a
3
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
−u
3
+ 2u
−u
3
+ u
a
12
=
u
7
+ u
6
− 3u
5
− 2u
4
+ 3u
3
+ 2
−u
7
+ u
6
+ 2u
5
− 3u
4
+ 2u
2
− 4u + 2
a
11
=
u
7
+ u
6
− 3u
5
− 2u
4
+ 3u
3
+ 2
−u
7
+ u
6
+ 2u
5
− 3u
4
+ 2u
2
− 3u + 2
a
6
=
−u
3
+ 2u
−u
3
+ u
a
2
=
u
5
− 2u
3
+ u
−u
7
+ 3u
5
− 2u
3
− u
a
1
=
0
−u
a
10
=
u
7
+ u
6
− 3u
5
− 2u
4
+ 3u
3
+ 2
−u
7
+ u
6
+ 2u
5
− 3u
4
+ 2u
2
− 3u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
7
− 9u
6
− 10u
5
+ 27u
4
− 2u
3
− 18u
2
+ 20u − 17
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
2
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
3
, c
4
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
5
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
6
, c
10
u
8
c
7
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
8
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
9
(u + 1)
8
c
11
, c
12
(u − 1)
8
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
2
, c
5
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
3
, c
4
, c
8
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
6
, c
10
y
8
c
9
, c
11
, c
12
(y −1)
8
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.180120 + 0.268597I
a = 0.805639 − 0.183365I
b = −1.14297 − 0.89911I
0.604279 − 1.131230I −1.38132 + 1.25921I
u = 1.180120 − 0.268597I
a = 0.805639 + 0.183365I
b = −1.14297 + 0.89911I
0.604279 + 1.131230I −1.38132 − 1.25921I
u = 0.108090 + 0.747508I
a = 0.189481 − 1.310380I
b = −0.02521 − 1.55019I
3.80435 − 2.57849I 1.74277 + 4.63100I
u = 0.108090 − 0.747508I
a = 0.189481 + 1.310380I
b = −0.02521 + 1.55019I
3.80435 + 2.57849I 1.74277 − 4.63100I
u = −1.37100
a = −0.729394
b = 6.70204
−4.85780 −25.4550
u = −1.334530 + 0.318930I
a = −0.708845 − 0.169402I
b = 1.07471 − 1.15185I
−0.73474 + 6.44354I −1.71699 − 7.87618I
u = −1.334530 − 0.318930I
a = −0.708845 + 0.169402I
b = 1.07471 + 1.15185I
−0.73474 − 6.44354I −1.71699 + 7.87618I
u = 0.463640
a = 2.15684
b = 0.484913
0.799899 −10.8330
10
III. I
u
3
= h−au + b − u − 1, 2a
2
+ au − 1, u
2
− 2i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
4
=
1
−2
a
8
=
u
−u
a
3
=
−1
0
a
7
=
0
−u
a
12
=
a
au + u + 1
a
11
=
a
au + 2a + u + 1
a
6
=
−a +
1
2
u
1
a
2
=
−a +
1
2
u − 1
1
a
1
=
−a +
1
2
u
1
a
10
=
−a +
1
2
u
2a + u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
7
c
8
(u
2
− 2)
2
c
6
, c
11
, c
12
(u
2
+ u − 1)
2
c
9
, c
10
(u
2
− u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
7
c
8
(y −2)
4
c
6
, c
9
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −1.14412
b = 0.796180
2.30291 −4.00000
u = 1.41421
a = 0.437016
b = 3.03225
−5.59278 −4.00000
u = −1.41421
a = 1.14412
b = −2.03225
2.30291 −4.00000
u = −1.41421
a = −0.437016
b = 0.203820
−5.59278 −4.00000
14
IV. I
v
1
= ha, b − v + 2, v
2
− 3v + 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
v
0
a
4
=
1
0
a
8
=
v
0
a
3
=
1
0
a
7
=
v
0
a
12
=
0
v − 2
a
11
=
−2v + 1
v − 2
a
6
=
−2v + 1
1
a
2
=
2v
−1
a
1
=
2v − 1
−1
a
10
=
v
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 14
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
7
c
8
u
2
c
5
(u + 1)
2
c
6
, c
9
u
2
− u − 1
c
10
, c
11
, c
12
u
2
+ u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
2
c
3
, c
4
, c
7
c
8
y
2
c
6
, c
9
, c
10
c
11
, c
12
y
2
− 3y + 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.381966
a = 0
b = −1.61803
−0.657974 14.0000
v = 2.61803
a = 0
b = 0.618034
7.23771 14.0000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
20
+ 24u
18
+ ··· + 3807u + 81)
c
2
(u − 1)
2
(u + 1)
4
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
· (u
20
+ 4u
19
+ ··· − 93u − 9)
c
3
, c
4
u
2
(u
2
− 2)
2
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
· (u
20
+ 2u
19
+ ··· + 12u + 4)
c
5
(u − 1)
4
(u + 1)
2
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
· (u
20
+ 4u
19
+ ··· − 93u − 9)
c
6
u
8
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
20
+ 2u
19
+ ··· + 3200u + 256)
c
7
u
2
(u
2
− 2)
2
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
20
− 6u
19
+ ··· − 6212u − 964)
c
8
u
2
(u
2
− 2)
2
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)
· (u
20
+ 2u
19
+ ··· + 12u + 4)
c
9
((u + 1)
8
)(u
2
− u − 1)
3
(u
20
+ 12u
19
+ ··· − 58u + 1)
c
10
u
8
(u
2
− u − 1)
2
(u
2
+ u − 1)(u
20
+ 2u
19
+ ··· + 3200u + 256)
c
11
, c
12
((u − 1)
8
)(u
2
+ u − 1)
3
(u
20
+ 12u
19
+ ··· − 58u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y − 1)
6
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
20
+ 48y
19
+ ··· − 10684791y + 6561)
c
2
, c
5
(y − 1)
6
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
20
+ 24y
18
+ ··· − 3807y + 81)
c
3
, c
4
, c
8
y
2
(y − 2)
4
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
20
− 14y
19
+ ··· − 432y + 16)
c
6
, c
10
y
8
(y
2
− 3y + 1)
3
(y
20
− 60y
19
+ ··· − 5160960y + 65536)
c
7
y
2
(y − 2)
4
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
20
+ 58y
19
+ ··· − 44905072y + 929296)
c
9
, c
11
, c
12
((y − 1)
8
)(y
2
− 3y + 1)
3
(y
20
− 44y
19
+ ··· − 2922y + 1)
20
