12n
0356
(K12n
0356
)
A knot diagram
1
Linearized knot diagam
3 5 12 10 2 10 3 4 7 4 9 8
Solving Sequence
4,12 3,9
8 1 7 11 10 5 2 6
c
3
c
8
c
12
c
7
c
11
c
10
c
4
c
2
c
5
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h7.95983 × 10
44
u
41
− 1.50155 × 10
45
u
40
+ ··· + 1.14909 × 10
45
b − 1.16386 × 10
46
,
6.40800 × 10
45
u
41
− 3.37761 × 10
44
u
40
+ ··· + 3.44727 × 10
45
a − 2.38350 × 10
46
, u
42
+ 6u
40
+ ··· − 4u + 3i
I
u
2
= hb − 1, a + 1, u
2
+ u + 1i
I
u
3
= h−u
9
− 3u
7
− 5u
5
− u
4
− 10u
3
− u
2
+ b − 6u,
− 126u
9
− 4u
8
− 339u
7
− 31u
6
− 519u
5
− 187u
4
− 1073u
3
− 183u
2
+ 85a − 564u − 111,
u
10
+ 3u
8
+ 5u
6
+ u
5
+ 10u
4
+ u
3
+ 7u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 54 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
= h7.96 × 10
44
u
41
− 1.50 × 10
45
u
40
+ · · · + 1.15 × 10
45
b − 1.16 × 10
46
, 6.41 ×
10
45
u
41
−3.38×10
44
u
40
+· · ·+3.45×10
45
a−2.38×10
46
, u
42
+6u
40
+· · ·−4u+3i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
3
=
1
−u
2
a
9
=
−1.85886u
41
+ 0.0979791u
40
+ ··· − 27.6246u + 6.91416
−0.692707u
41
+ 1.30673u
40
+ ··· − 11.4825u + 10.1285
a
8
=
−2.55157u
41
+ 1.40471u
40
+ ··· − 39.1071u + 17.0427
−0.692707u
41
+ 1.30673u
40
+ ··· − 11.4825u + 10.1285
a
1
=
0.832926u
41
− 0.346814u
40
+ ··· + 10.5076u + 6.09743
−0.155548u
41
+ 0.571430u
40
+ ··· − 4.77363u + 6.26450
a
7
=
−2.86555u
41
+ 0.463510u
40
+ ··· − 40.8981u + 11.1283
−0.435280u
41
+ 1.07479u
40
+ ··· − 8.65966u + 7.30494
a
11
=
0.0541962u
41
− 0.173110u
40
+ ··· + 5.05694u + 7.82335
0.934277u
41
− 0.745135u
40
+ ··· + 12.2243u − 7.99042
a
10
=
0.988473u
41
− 0.918245u
40
+ ··· + 17.2813u − 0.167070
0.934277u
41
− 0.745135u
40
+ ··· + 12.2243u − 7.99042
a
5
=
2.78167u
41
− 0.519929u
40
+ ··· + 10.7131u − 6.92196
0.763818u
41
− 0.555089u
40
+ ··· + 9.41655u − 4.17445
a
2
=
1.26821u
41
− 1.42161u
40
+ ··· + 19.1673u − 1.20751
0.308673u
41
+ 0.214746u
40
+ ··· + 0.831385u + 3.04011
a
6
=
4.26347u
41
− 3.57388u
40
+ ··· + 63.8196u − 31.1222
2.53749u
41
− 2.17142u
40
+ ··· + 37.6429u − 19.2215
(ii) Obstruction class = −1
(iii) Cusp Shapes = 11.6391u
41
− 2.74474u
40
+ ··· + 169.526u − 46.9504
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
+ 12u
41
+ ··· + 230u + 9
c
2
, c
5
u
42
+ 6u
40
+ ··· + 4u + 3
c
3
u
42
+ 6u
40
+ ··· − 4u + 3
c
4
, c
10
u
42
+ 14u
40
+ ··· − 792u + 172
c
6
, c
9
u
42
+ 3u
41
+ ··· + 1471u + 111
c
7
u
42
+ u
41
+ ··· + 47u + 99
c
8
u
42
+ u
41
+ ··· − 131u + 111
c
11
u
42
− 3u
41
+ ··· − 1471u + 111
c
12
u
42
+ 14u
40
+ ··· + 792u + 172
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
+ 44y
41
+ ··· + 254y + 81
c
2
, c
3
, c
5
y
42
+ 12y
41
+ ··· + 230y + 9
c
4
, c
10
, c
12
y
42
+ 28y
41
+ ··· + 8448y + 29584
c
6
, c
9
, c
11
y
42
− 27y
41
+ ··· + 55937y + 12321
c
7
y
42
+ 21y
41
+ ··· + 383891y + 9801
c
8
y
42
− 29y
41
+ ··· + 95171y + 12321
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.544383 + 0.863324I
a = −0.51188 + 1.32823I
b = −0.497540 + 0.008587I
2.42641 − 0.18863I 3.24251 − 0.93757I
u = −0.544383 − 0.863324I
a = −0.51188 − 1.32823I
b = −0.497540 − 0.008587I
2.42641 + 0.18863I 3.24251 + 0.93757I
u = −0.687915 + 0.682286I
a = 1.102770 − 0.213792I
b = −1.75734 + 0.28348I
1.70620 + 4.86322I 2.22045 − 7.40969I
u = −0.687915 − 0.682286I
a = 1.102770 + 0.213792I
b = −1.75734 − 0.28348I
1.70620 − 4.86322I 2.22045 + 7.40969I
u = 0.750317 + 0.769897I
a = 1.340840 − 0.353204I
b = −1.37469 − 1.08554I
−3.29363 − 5.13657I −2.33265 + 8.72582I
u = 0.750317 − 0.769897I
a = 1.340840 + 0.353204I
b = −1.37469 + 1.08554I
−3.29363 + 5.13657I −2.33265 − 8.72582I
u = −0.932214 + 0.569996I
a = −1.037210 − 0.591475I
b = 0.860453 − 0.527673I
−3.75518 + 1.50447I −4.91299 + 0.56780I
u = −0.932214 − 0.569996I
a = −1.037210 + 0.591475I
b = 0.860453 + 0.527673I
−3.75518 − 1.50447I −4.91299 − 0.56780I
u = −0.144161 + 0.853584I
a = −1.010980 − 0.281506I
b = 0.414095 + 0.757109I
0.95566 + 2.59332I 5.16066 − 5.00921I
u = −0.144161 − 0.853584I
a = −1.010980 + 0.281506I
b = 0.414095 − 0.757109I
0.95566 − 2.59332I 5.16066 + 5.00921I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.547178 + 0.657889I
a = 1.21496 + 1.75367I
b = 0.157769 − 0.348092I
3.29363 − 5.13657I 2.33265 + 8.72582I
u = 0.547178 − 0.657889I
a = 1.21496 − 1.75367I
b = 0.157769 + 0.348092I
3.29363 + 5.13657I 2.33265 − 8.72582I
u = 0.402523 + 0.735634I
a = −1.213870 − 0.554358I
b = 1.46993 + 0.46496I
3.75518 + 1.50447I 4.91299 + 0.56780I
u = 0.402523 − 0.735634I
a = −1.213870 + 0.554358I
b = 1.46993 − 0.46496I
3.75518 − 1.50447I 4.91299 − 0.56780I
u = 0.618713 + 1.043480I
a = 0.248973 − 0.791074I
b = −1.309910 + 0.260236I
−2.42641 − 0.18863I −3.24251 − 0.93757I
u = 0.618713 − 1.043480I
a = 0.248973 + 0.791074I
b = −1.309910 − 0.260236I
−2.42641 + 0.18863I −3.24251 + 0.93757I
u = 0.831964 + 0.909186I
a = 1.28883 − 0.94404I
b = −1.80526 − 0.30079I
−5.39497 − 3.10535I −10.05216 + 0.I
u = 0.831964 − 0.909186I
a = 1.28883 + 0.94404I
b = −1.80526 + 0.30079I
−5.39497 + 3.10535I −10.05216 + 0.I
u = −1.042840 + 0.697476I
a = 0.494099 + 0.769179I
b = −1.144430 − 0.016547I
− 0.206317I 0
u = −1.042840 − 0.697476I
a = 0.494099 − 0.769179I
b = −1.144430 + 0.016547I
0.206317I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.033933 + 0.665190I
a = −0.60540 + 2.41235I
b = −0.141969 + 1.363470I
5.39497 + 3.10535I 10.05216 − 0.84923I
u = 0.033933 − 0.665190I
a = −0.60540 − 2.41235I
b = −0.141969 − 1.363470I
5.39497 − 3.10535I 10.05216 + 0.84923I
u = 1.122240 + 0.748923I
a = −0.720260 + 0.883365I
b = 1.377720 − 0.195992I
−1.23530 + 7.11550I 0
u = 1.122240 − 0.748923I
a = −0.720260 − 0.883365I
b = 1.377720 + 0.195992I
−1.23530 − 7.11550I 0
u = 0.984944 + 0.968446I
a = −0.621355 + 0.424387I
b = 1.41482 + 0.57562I
−8.49949 − 3.58541I 0
u = 0.984944 − 0.968446I
a = −0.621355 − 0.424387I
b = 1.41482 − 0.57562I
−8.49949 + 3.58541I 0
u = −0.360297 + 0.499277I
a = −0.145220 + 0.355309I
b = −0.293588 + 0.434809I
1.15367I 0. − 5.79196I
u = −0.360297 − 0.499277I
a = −0.145220 − 0.355309I
b = −0.293588 − 0.434809I
− 1.15367I 0. + 5.79196I
u = 0.129278 + 0.581586I
a = 0.221386 + 0.378940I
b = 0.01026 − 2.64515I
5.05194 − 3.77462I 11.6382 + 10.3724I
u = 0.129278 − 0.581586I
a = 0.221386 − 0.378940I
b = 0.01026 + 2.64515I
5.05194 + 3.77462I 11.6382 − 10.3724I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.865461 + 1.109310I
a = 0.989347 + 0.290574I
b = −1.56337 + 0.89130I
1.23530 + 7.11550I 0
u = −0.865461 − 1.109310I
a = 0.989347 − 0.290574I
b = −1.56337 − 0.89130I
1.23530 − 7.11550I 0
u = −1.05197 + 0.94991I
a = −1.180020 − 0.609379I
b = 1.382870 − 0.244233I
−5.05194 + 3.77462I 0
u = −1.05197 − 0.94991I
a = −1.180020 + 0.609379I
b = 1.382870 + 0.244233I
−5.05194 − 3.77462I 0
u = 0.89172 + 1.13300I
a = −1.161700 + 0.418276I
b = 1.72271 + 0.83595I
− 14.3453I 0
u = 0.89172 − 1.13300I
a = −1.161700 − 0.418276I
b = 1.72271 − 0.83595I
14.3453I 0
u = −0.76555 + 1.23008I
a = −0.662962 − 0.381871I
b = 1.242360 − 0.087577I
−1.70620 + 4.86322I 0
u = −0.76555 − 1.23008I
a = −0.662962 + 0.381871I
b = 1.242360 + 0.087577I
−1.70620 − 4.86322I 0
u = 0.157972 + 0.382405I
a = 3.18352 − 0.59179I
b = −0.795555 − 0.090965I
−0.95566 − 2.59332I −5.16066 + 5.00921I
u = 0.157972 − 0.382405I
a = 3.18352 + 0.59179I
b = −0.795555 + 0.090965I
−0.95566 + 2.59332I −5.16066 − 5.00921I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.07599 + 1.62893I
a = −0.047185 + 0.419190I
b = 0.130648 − 0.204139I
8.49949 + 3.58541I 0
u = −0.07599 − 1.62893I
a = −0.047185 − 0.419190I
b = 0.130648 + 0.204139I
8.49949 − 3.58541I 0
9
II. I
u
2
= hb − 1, a + 1, u
2
+ u + 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
3
=
1
u + 1
a
9
=
−1
1
a
8
=
0
1
a
1
=
0
u
a
7
=
−1
−u
a
11
=
u
0
a
10
=
u
0
a
5
=
1
0
a
2
=
−u
u + 1
a
6
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u − 4
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
9
u
2
− u + 1
c
2
, c
3
, c
6
c
11
u
2
+ u + 1
c
4
, c
10
, c
12
u
2
c
7
, c
8
(u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
9
c
11
y
2
+ y + 1
c
4
, c
10
, c
12
y
2
c
7
, c
8
(y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −1.00000
b = 1.00000
4.05977I 0. − 6.92820I
u = −0.500000 − 0.866025I
a = −1.00000
b = 1.00000
− 4.05977I 0. + 6.92820I
13
III. I
u
3
= h−u
9
− 3u
7
− 5u
5
− u
4
− 10u
3
− u
2
+ b − 6u, −126u
9
− 4u
8
+ · · · +
85a − 111, u
10
+ 3u
8
+ 5u
6
+ u
5
+ 10u
4
+ u
3
+ 7u
2
+ 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
3
=
1
−u
2
a
9
=
1.48235u
9
+ 0.0470588u
8
+ ··· + 6.63529u + 1.30588
u
9
+ 3u
7
+ 5u
5
+ u
4
+ 10u
3
+ u
2
+ 6u
a
8
=
2.48235u
9
+ 0.0470588u
8
+ ··· + 12.6353u + 1.30588
u
9
+ 3u
7
+ 5u
5
+ u
4
+ 10u
3
+ u
2
+ 6u
a
1
=
−3.69412u
9
− 1.48235u
8
+ ··· − 19.0118u − 5.63529
−2.02353u
9
− 0.270588u
8
+ ··· − 9.15294u − 1.25882
a
7
=
1.94118u
9
− 0.176471u
8
+ ··· + 9.11765u + 1.35294
1.07059u
9
− 0.188235u
8
+ ··· + 5.45882u − 0.223529
a
11
=
−0.647059u
9
− 0.941176u
8
+ ··· − 3.70588u − 3.11765
−1.02353u
9
− 0.270588u
8
+ ··· − 4.15294u − 1.25882
a
10
=
−1.67059u
9
− 1.21176u
8
+ ··· − 7.85882u − 4.37647
−1.02353u
9
− 0.270588u
8
+ ··· − 4.15294u − 1.25882
a
5
=
2.56471u
9
+ 0.494118u
8
+ ··· + 14.6706u + 1.21176
1.22353u
9
+ 0.0705882u
8
+ ··· + 5.95294u − 0.541176
a
2
=
−2.62353u
9
− 1.67059u
8
+ ··· − 13.5529u − 5.85882
−1.85882u
9
− 0.376471u
8
+ ··· − 8.08235u − 1.44706
a
6
=
3.05882u
9
+ 2.17647u
8
+ ··· + 17.8824u + 10.6471
2.10588u
9
+ 0.717647u
8
+ ··· + 11.1882u + 3.16471
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
9
85
u
9
−
24
85
u
8
+
91
85
u
7
−
101
85
u
6
+
201
85
u
5
−
1
5
u
4
+
362
85
u
3
−
163
85
u
2
+
356
85
u −
156
85
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− 6u
9
+ ··· − 14u + 1
c
2
, c
3
u
10
+ 3u
8
+ 5u
6
+ u
5
+ 10u
4
+ u
3
+ 7u
2
+ 1
c
4
, c
12
u
10
+ u
9
+ 2u
8
− 4u
7
− 9u
6
− 11u
5
+ 5u
4
+ 33u
3
+ 45u
2
+ 40u + 16
c
5
u
10
+ 3u
8
+ 5u
6
− u
5
+ 10u
4
− u
3
+ 7u
2
+ 1
c
6
, c
11
u
10
+ 5u
9
+ 8u
8
+ 2u
7
− 6u
6
− 4u
5
+ 5u
4
+ 9u
3
+ u
2
− 3u + 1
c
7
u
10
− 2u
9
+ 7u
8
− 13u
7
+ 21u
6
− 24u
5
+ 21u
4
− 13u
3
+ 7u
2
− 2u + 1
c
8
u
10
+ u
7
− 9u
6
− 2u
5
+ 14u
4
+ 5u
3
+ 15u
2
+ 2u + 1
c
9
u
10
− 5u
9
+ 8u
8
− 2u
7
− 6u
6
+ 4u
5
+ 5u
4
− 9u
3
+ u
2
+ 3u + 1
c
10
u
10
− u
9
+ 2u
8
+ 4u
7
− 9u
6
+ 11u
5
+ 5u
4
− 33u
3
+ 45u
2
− 40u + 16
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
+ 2y
9
+ ··· − 58y + 1
c
2
, c
3
, c
5
y
10
+ 6y
9
+ ··· + 14y + 1
c
4
, c
10
, c
12
y
10
+ 3y
9
+ ··· − 160y + 256
c
6
, c
9
, c
11
y
10
− 9y
9
+ ··· − 7y + 1
c
7
y
10
+ 10y
9
+ ··· + 10y + 1
c
8
y
10
− 18y
8
+ ··· + 26y + 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.028467 + 0.876055I
a = 1.150800 − 0.608136I
b = 0.065521 + 0.264223I
− 1.89767I 0. + 1.79792I
u = −0.028467 − 0.876055I
a = 1.150800 + 0.608136I
b = 0.065521 − 0.264223I
1.89767I 0. − 1.79792I
u = −0.930058 + 0.884199I
a = −1.28389 − 0.71814I
b = 1.49482 − 0.34728I
−4.66287 + 3.40367I 1.43158 − 1.46813I
u = −0.930058 − 0.884199I
a = −1.28389 + 0.71814I
b = 1.49482 + 0.34728I
−4.66287 − 3.40367I 1.43158 + 1.46813I
u = 0.993286 + 0.964746I
a = 0.781058 − 0.547640I
b = −1.51134 − 0.46158I
−9.09852 − 3.60546I −8.68376 + 2.68056I
u = 0.993286 − 0.964746I
a = 0.781058 + 0.547640I
b = −1.51134 + 0.46158I
−9.09852 + 3.60546I −8.68376 − 2.68056I
u = −0.05526 + 1.47925I
a = −0.133797 + 0.210607I
b = 0.080483 − 0.804181I
9.09852 + 3.60546I 8.68376 − 2.68056I
u = −0.05526 − 1.47925I
a = −0.133797 − 0.210607I
b = 0.080483 + 0.804181I
9.09852 − 3.60546I 8.68376 + 2.68056I
u = 0.020502 + 0.433246I
a = 0.98582 + 1.96028I
b = −0.12949 + 1.86975I
4.66287 − 3.40367I −1.43158 + 1.46813I
u = 0.020502 − 0.433246I
a = 0.98582 − 1.96028I
b = −0.12949 − 1.86975I
4.66287 + 3.40367I −1.43158 − 1.46813I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
10
− 6u
9
+ ··· − 14u + 1)(u
42
+ 12u
41
+ ··· + 230u + 9)
c
2
(u
2
+ u + 1)(u
10
+ 3u
8
+ 5u
6
+ u
5
+ 10u
4
+ u
3
+ 7u
2
+ 1)
· (u
42
+ 6u
40
+ ··· + 4u + 3)
c
3
(u
2
+ u + 1)(u
10
+ 3u
8
+ 5u
6
+ u
5
+ 10u
4
+ u
3
+ 7u
2
+ 1)
· (u
42
+ 6u
40
+ ··· − 4u + 3)
c
4
u
2
(u
10
+ u
9
+ ··· + 40u + 16)
· (u
42
+ 14u
40
+ ··· − 792u + 172)
c
5
(u
2
− u + 1)(u
10
+ 3u
8
+ 5u
6
− u
5
+ 10u
4
− u
3
+ 7u
2
+ 1)
· (u
42
+ 6u
40
+ ··· + 4u + 3)
c
6
(u
2
+ u + 1)(u
10
+ 5u
9
+ ··· − 3u + 1)
· (u
42
+ 3u
41
+ ··· + 1471u + 111)
c
7
(u + 1)
2
· (u
10
− 2u
9
+ 7u
8
− 13u
7
+ 21u
6
− 24u
5
+ 21u
4
− 13u
3
+ 7u
2
− 2u + 1)
· (u
42
+ u
41
+ ··· + 47u + 99)
c
8
(u + 1)
2
(u
10
+ u
7
− 9u
6
− 2u
5
+ 14u
4
+ 5u
3
+ 15u
2
+ 2u + 1)
· (u
42
+ u
41
+ ··· − 131u + 111)
c
9
(u
2
− u + 1)(u
10
− 5u
9
+ ··· + 3u + 1)
· (u
42
+ 3u
41
+ ··· + 1471u + 111)
c
10
u
2
(u
10
− u
9
+ ··· − 40u + 16)
· (u
42
+ 14u
40
+ ··· − 792u + 172)
c
11
(u
2
+ u + 1)(u
10
+ 5u
9
+ ··· − 3u + 1)
· (u
42
− 3u
41
+ ··· − 1471u + 111)
c
12
u
2
(u
10
+ u
9
+ ··· + 40u + 16)
· (u
42
+ 14u
40
+ ··· + 792u + 172)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
10
+ 2y
9
+ ··· − 58y + 1)(y
42
+ 44y
41
+ ··· + 254y + 81)
c
2
, c
3
, c
5
(y
2
+ y + 1)(y
10
+ 6y
9
+ ··· + 14y + 1)(y
42
+ 12y
41
+ ··· + 230y + 9)
c
4
, c
10
, c
12
y
2
(y
10
+ 3y
9
+ ··· − 160y + 256)(y
42
+ 28y
41
+ ··· + 8448y + 29584)
c
6
, c
9
, c
11
(y
2
+ y + 1)(y
10
− 9y
9
+ ··· − 7y + 1)
· (y
42
− 27y
41
+ ··· + 55937y + 12321)
c
7
((y − 1)
2
)(y
10
+ 10y
9
+ ··· + 10y + 1)
· (y
42
+ 21y
41
+ ··· + 383891y + 9801)
c
8
((y − 1)
2
)(y
10
− 18y
8
+ ··· + 26y + 1)
· (y
42
− 29y
41
+ ··· + 95171y + 12321)
19
