12a
0234
(K12a
0234
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 5 11 12 4 1 8 9
Solving Sequence
2,5
6 3 7
1,10
11 4 9 12 8
c
5
c
2
c
6
c
1
c
10
c
4
c
9
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
74
+ u
73
+ ··· + b − u, −10u
74
− 43u
73
+ ··· + 2a − 15, u
75
+ 4u
74
+ ··· + 4u + 1i
I
u
2
= hb, a
2
− au − u
2
+ a + 2u − 1, u
3
− u
2
+ 1i
I
u
3
= hb − 1, a − 2, u − 1i
* 3 irreducible components of dim
C
= 0, with total 82 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
=
hu
74
+u
73
+· · ·+b−u, −10u
74
−43u
73
+· · ·+2a−15, u
75
+4u
74
+· · ·+4u+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
u
2
a
1
=
u
3
u
5
− u
3
+ u
a
10
=
5u
74
+
43
2
u
73
+ ··· +
39
2
u +
15
2
−u
74
− u
73
+ ··· + 4u
2
+ u
a
11
=
5
2
u
74
+ 11u
73
+ ··· +
13
2
u + 3
1
2
u
74
+
5
2
u
73
+ ··· + 4u + 1
a
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
2u
74
+
5
2
u
73
+ ··· −
19
2
u −
5
2
6u
74
+ 23u
73
+ ··· + 27u + 8
a
12
=
−
1
2
u
73
− u
72
+ ··· +
5
2
u +
1
2
u
26
− 4u
24
+ ··· − 4u
3
− 3u
2
a
8
=
1
2
u
74
+ u
73
+ ··· −
13
2
u
2
−
7
2
u
1
2
u
74
+
5
2
u
73
+ ··· + 4u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −31u
74
−
187
2
u
73
+ ··· −
159
2
u − 23
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
75
+ 24u
74
+ ··· − 16u + 1
c
2
, c
5
u
75
+ 4u
74
+ ··· + 4u + 1
c
3
u
75
− 2u
74
+ ··· − 2768u + 1009
c
4
, c
9
u
75
− 2u
74
+ ··· + 32u − 64
c
7
, c
8
, c
11
c
12
u
75
− 3u
74
+ ··· − 7u − 1
c
10
u
75
+ 21u
74
+ ··· − 4045u + 239
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
75
+ 56y
74
+ ··· + 160y −1
c
2
, c
5
y
75
− 24y
74
+ ··· − 16y −1
c
3
y
75
− 4y
74
+ ··· − 1197196y −1018081
c
4
, c
9
y
75
− 34y
74
+ ··· + 82944y −4096
c
7
, c
8
, c
11
c
12
y
75
− 87y
74
+ ··· + 53y −1
c
10
y
75
− 3y
74
+ ··· + 27550093y −57121
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.711207 + 0.731942I
a = 0.073783 + 0.958389I
b = −0.923873 − 0.439880I
3.44957 + 0.11709I 0
u = −0.711207 − 0.731942I
a = 0.073783 − 0.958389I
b = −0.923873 + 0.439880I
3.44957 − 0.11709I 0
u = −1.016490 + 0.108693I
a = 0.1180520 − 0.0250300I
b = −0.383356 − 1.026800I
5.05953 + 3.13748I 0
u = −1.016490 − 0.108693I
a = 0.1180520 + 0.0250300I
b = −0.383356 + 1.026800I
5.05953 − 3.13748I 0
u = −0.660509 + 0.785762I
a = −0.663561 − 0.985879I
b = 1.107580 + 0.521055I
−0.12076 − 2.74550I 0
u = −0.660509 − 0.785762I
a = −0.663561 + 0.985879I
b = 1.107580 − 0.521055I
−0.12076 + 2.74550I 0
u = −0.965638 + 0.051778I
a = −0.0394728 + 0.0390828I
b = 0.172922 + 0.877447I
−1.86895 + 1.44234I 0
u = −0.965638 − 0.051778I
a = −0.0394728 − 0.0390828I
b = 0.172922 − 0.877447I
−1.86895 − 1.44234I 0
u = 0.722226 + 0.744545I
a = 0.18997 + 1.79205I
b = −0.477443 − 0.900673I
3.49030 + 0.98913I 0
u = 0.722226 − 0.744545I
a = 0.18997 − 1.79205I
b = −0.477443 + 0.900673I
3.49030 − 0.98913I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.697965 + 0.783588I
a = −0.35474 − 1.96967I
b = 0.569105 + 1.003110I
11.08350 + 2.94021I 0
u = 0.697965 − 0.783588I
a = −0.35474 + 1.96967I
b = 0.569105 − 1.003110I
11.08350 − 2.94021I 0
u = 0.785691 + 0.697977I
a = 0.09437 − 1.44165I
b = 0.324972 + 0.756336I
2.20022 − 1.98773I 0
u = 0.785691 − 0.697977I
a = 0.09437 + 1.44165I
b = 0.324972 − 0.756336I
2.20022 + 1.98773I 0
u = −0.671002 + 0.820090I
a = 0.83769 + 1.20399I
b = −1.140810 − 0.631940I
1.38833 − 6.65529I 0
u = −0.671002 − 0.820090I
a = 0.83769 − 1.20399I
b = −1.140810 + 0.631940I
1.38833 + 6.65529I 0
u = 1.06726
a = 1.51914
b = 1.24765
−1.46846 0
u = 1.063370 + 0.091921I
a = −1.48543 + 0.57951I
b = −1.204920 − 0.351467I
−6.19357 − 2.44894I 0
u = 1.063370 − 0.091921I
a = −1.48543 − 0.57951I
b = −1.204920 + 0.351467I
−6.19357 + 2.44894I 0
u = −0.766146 + 0.749616I
a = 0.30396 − 1.44485I
b = 0.764353 + 0.496302I
12.27310 + 1.21172I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.766146 − 0.749616I
a = 0.30396 + 1.44485I
b = 0.764353 − 0.496302I
12.27310 − 1.21172I 0
u = 0.908509 + 0.580630I
a = −0.94613 + 1.40143I
b = −0.107565 − 0.998481I
7.57381 − 2.23045I 0
u = 0.908509 − 0.580630I
a = −0.94613 − 1.40143I
b = −0.107565 + 0.998481I
7.57381 + 2.23045I 0
u = 1.072730 + 0.131160I
a = 1.38515 − 0.78395I
b = 1.210390 + 0.505142I
−5.10143 − 6.45888I 0
u = 1.072730 − 0.131160I
a = 1.38515 + 0.78395I
b = 1.210390 − 0.505142I
−5.10143 + 6.45888I 0
u = −0.682230 + 0.841524I
a = −0.94322 − 1.37024I
b = 1.151480 + 0.720103I
9.20789 − 9.23314I 0
u = −0.682230 − 0.841524I
a = −0.94322 + 1.37024I
b = 1.151480 − 0.720103I
9.20789 + 9.23314I 0
u = −0.993937 + 0.434871I
a = 0.133903 − 0.616443I
b = −1.174540 − 0.467698I
4.04103 − 2.59869I 0
u = −0.993937 − 0.434871I
a = 0.133903 + 0.616443I
b = −1.174540 + 0.467698I
4.04103 + 2.59869I 0
u = −0.971580 + 0.501941I
a = 0.063789 + 0.774786I
b = 1.174790 + 0.276036I
−2.94814 − 0.18097I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.971580 − 0.501941I
a = 0.063789 − 0.774786I
b = 1.174790 − 0.276036I
−2.94814 + 0.18097I 0
u = 1.081740 + 0.163201I
a = −1.30157 + 0.93027I
b = −1.210650 − 0.632851I
2.40082 − 9.12230I 0
u = 1.081740 − 0.163201I
a = −1.30157 − 0.93027I
b = −1.210650 + 0.632851I
2.40082 + 9.12230I 0
u = 0.888282
a = −3.24544
b = −0.629954
7.31928 −10.6220
u = −0.982759 + 0.562329I
a = −0.216181 − 1.041120I
b = −1.209770 − 0.100303I
−3.43339 + 3.68758I 0
u = −0.982759 − 0.562329I
a = −0.216181 + 1.041120I
b = −1.209770 + 0.100303I
−3.43339 − 3.68758I 0
u = −0.499811 + 0.691420I
a = 0.744111 − 0.062254I
b = −1.178410 − 0.102990I
3.70237 − 1.18565I 0.324301 + 0.368937I
u = −0.499811 − 0.691420I
a = 0.744111 + 0.062254I
b = −1.178410 + 0.102990I
3.70237 + 1.18565I 0.324301 − 0.368937I
u = 0.836198 + 0.792398I
a = 0.716749 + 1.015140I
b = −0.683410 − 0.394873I
4.26586 − 3.75101I 0
u = 0.836198 − 0.792398I
a = 0.716749 − 1.015140I
b = −0.683410 + 0.394873I
4.26586 + 3.75101I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.937039 + 0.676923I
a = 1.012900 − 0.915123I
b = −0.211047 + 0.866238I
1.72641 − 3.30860I 0
u = 0.937039 − 0.676923I
a = 1.012900 + 0.915123I
b = −0.211047 − 0.866238I
1.72641 + 3.30860I 0
u = 0.826326 + 0.828687I
a = −1.00165 − 1.24716I
b = 0.907683 + 0.488606I
11.80470 − 5.23133I 0
u = 0.826326 − 0.828687I
a = −1.00165 + 1.24716I
b = 0.907683 − 0.488606I
11.80470 + 5.23133I 0
u = −1.010410 + 0.619026I
a = 0.27560 + 1.47068I
b = 1.274680 − 0.127579I
2.28478 + 6.17362I 0
u = −1.010410 − 0.619026I
a = 0.27560 − 1.47068I
b = 1.274680 + 0.127579I
2.28478 − 6.17362I 0
u = −0.952521 + 0.709229I
a = −1.25985 − 2.13172I
b = −0.819855 + 0.405800I
11.70100 + 4.33992I 0
u = −0.952521 − 0.709229I
a = −1.25985 + 2.13172I
b = −0.819855 − 0.405800I
11.70100 − 4.33992I 0
u = 0.914778 + 0.769195I
a = −1.046570 − 0.021063I
b = 0.622847 − 0.319420I
4.02488 − 2.11438I 0
u = 0.914778 − 0.769195I
a = −1.046570 + 0.021063I
b = 0.622847 + 0.319420I
4.02488 + 2.11438I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.982465 + 0.687726I
a = 0.78718 + 1.95441I
b = 1.032380 − 0.389930I
2.62497 + 5.32088I 0
u = −0.982465 − 0.687726I
a = 0.78718 − 1.95441I
b = 1.032380 + 0.389930I
2.62497 − 5.32088I 0
u = 0.976735 + 0.698504I
a = −1.30467 + 0.82935I
b = 0.428493 − 0.975607I
2.71841 − 6.49505I 0
u = 0.976735 − 0.698504I
a = −1.30467 − 0.82935I
b = 0.428493 + 0.975607I
2.71841 + 6.49505I 0
u = 0.999165 + 0.711748I
a = 1.48010 − 0.81442I
b = −0.557335 + 1.061150I
10.17200 − 8.59556I 0
u = 0.999165 − 0.711748I
a = 1.48010 + 0.81442I
b = −0.557335 − 1.061150I
10.17200 + 8.59556I 0
u = 0.940369 + 0.791453I
a = 1.42868 + 0.04316I
b = −0.886134 + 0.434484I
11.45350 − 0.81358I 0
u = 0.940369 − 0.791453I
a = 1.42868 − 0.04316I
b = −0.886134 − 0.434484I
11.45350 + 0.81358I 0
u = −1.015930 + 0.701587I
a = −0.47859 − 2.18042I
b = −1.177240 + 0.537485I
−1.18776 + 8.36921I 0
u = −1.015930 − 0.701587I
a = −0.47859 + 2.18042I
b = −1.177240 − 0.537485I
−1.18776 − 8.36921I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.023410 + 0.718352I
a = 0.43055 + 2.34903I
b = 1.186660 − 0.649196I
0.31813 + 12.42890I 0
u = −1.023410 − 0.718352I
a = 0.43055 − 2.34903I
b = 1.186660 + 0.649196I
0.31813 − 12.42890I 0
u = −1.026990 + 0.731679I
a = −0.40791 − 2.47441I
b = −1.181080 + 0.737477I
8.1534 + 15.1124I 0
u = −1.026990 − 0.731679I
a = −0.40791 + 2.47441I
b = −1.181080 − 0.737477I
8.1534 − 15.1124I 0
u = −0.170137 + 0.694294I
a = −0.93682 + 1.35426I
b = 1.121250 − 0.579634I
6.51786 + 6.48932I 3.20143 − 4.96610I
u = −0.170137 − 0.694294I
a = −0.93682 − 1.35426I
b = 1.121250 + 0.579634I
6.51786 − 6.48932I 3.20143 + 4.96610I
u = −0.222380 + 0.645831I
a = 0.772263 − 1.148320I
b = −1.070320 + 0.435701I
−0.91107 + 4.17004I −0.03505 − 6.87095I
u = −0.222380 − 0.645831I
a = 0.772263 + 1.148320I
b = −1.070320 − 0.435701I
−0.91107 − 4.17004I −0.03505 + 6.87095I
u = −0.330794 + 0.589043I
a = −0.558254 + 0.771750I
b = 1.021100 − 0.213253I
−1.84200 + 0.63905I −3.61059 − 0.19351I
u = −0.330794 − 0.589043I
a = −0.558254 − 0.771750I
b = 1.021100 + 0.213253I
−1.84200 − 0.63905I −3.61059 + 0.19351I
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.671696
a = 0.135845
b = 0.375009
−0.909794 −11.9330
u = 0.185398 + 0.482510I
a = −0.25875 + 2.28529I
b = 0.376277 − 0.783708I
8.74623 − 1.35684I 6.78331 + 0.85499I
u = 0.185398 − 0.482510I
a = −0.25875 − 2.28529I
b = 0.376277 + 0.783708I
8.74623 + 1.35684I 6.78331 − 0.85499I
u = 0.066176 + 0.284258I
a = −0.35019 − 2.23595I
b = −0.345577 + 0.468199I
1.171050 − 0.405728I 7.00634 + 1.53169I
u = 0.066176 − 0.284258I
a = −0.35019 + 2.23595I
b = −0.345577 − 0.468199I
1.171050 + 0.405728I 7.00634 − 1.53169I
12
II. I
u
2
= hb, a
2
− au − u
2
+ a + 2u − 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
1
=
u
2
− 1
−u
2
a
10
=
a
0
a
11
=
−au
−u
2
a + au + a
a
4
=
1
0
a
9
=
a
0
a
12
=
u
2
− a − u
−u
2
a
8
=
au − u + 1
u
2
a − au − a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
a + 3u
2
− a + 1
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
, c
10
(u
2
− u − 1)
3
c
11
, c
12
(u
2
+ u − 1)
3
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.198308 + 1.205210I
b = 0
11.90680 − 2.82812I 3.46158 + 2.71621I
u = 0.877439 + 0.744862I
a = 0.075747 − 0.460350I
b = 0
4.01109 − 2.82812I 0.95146 + 4.38177I
u = 0.877439 − 0.744862I
a = −0.198308 − 1.205210I
b = 0
11.90680 + 2.82812I 3.46158 − 2.71621I
u = 0.877439 − 0.744862I
a = 0.075747 + 0.460350I
b = 0
4.01109 + 2.82812I 0.95146 − 4.38177I
u = −0.754878
a = 1.08457
b = 0
−0.126494 1.00690
u = −0.754878
a = −2.83945
b = 0
7.76919 7.16700
16
III. I
u
3
= hb − 1, a − 2, u − 1i
(i) Arc colorings
a
2
=
0
1
a
5
=
1
0
a
6
=
1
1
a
3
=
−1
0
a
7
=
0
1
a
1
=
1
1
a
10
=
2
1
a
11
=
1
0
a
4
=
−1
−1
a
9
=
1
0
a
12
=
0
1
a
8
=
1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
9
u + 1
c
2
, c
3
, c
5
c
7
, c
8
, c
10
c
11
, c
12
u − 1
18
(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
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 2.00000
b = 1.00000
−1.64493 −6.00000
20
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
3
− u
2
+ 2u − 1)
2
(u
75
+ 24u
74
+ ··· − 16u + 1)
c
2
(u − 1)(u
3
+ u
2
− 1)
2
(u
75
+ 4u
74
+ ··· + 4u + 1)
c
3
(u − 1)(u
3
− u
2
+ 2u − 1)
2
(u
75
− 2u
74
+ ··· − 2768u + 1009)
c
4
, c
9
u
6
(u + 1)(u
75
− 2u
74
+ ··· + 32u − 64)
c
5
(u − 1)(u
3
− u
2
+ 1)
2
(u
75
+ 4u
74
+ ··· + 4u + 1)
c
6
(u + 1)(u
3
+ u
2
+ 2u + 1)
2
(u
75
+ 24u
74
+ ··· − 16u + 1)
c
7
, c
8
(u − 1)(u
2
− u − 1)
3
(u
75
− 3u
74
+ ··· − 7u − 1)
c
10
(u − 1)(u
2
− u − 1)
3
(u
75
+ 21u
74
+ ··· − 4045u + 239)
c
11
, c
12
(u − 1)(u
2
+ u − 1)
3
(u
75
− 3u
74
+ ··· − 7u − 1)
21
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y − 1)(y
3
+ 3y
2
+ 2y − 1)
2
(y
75
+ 56y
74
+ ··· + 160y − 1)
c
2
, c
5
(y − 1)(y
3
− y
2
+ 2y − 1)
2
(y
75
− 24y
74
+ ··· − 16y − 1)
c
3
(y − 1)(y
3
+ 3y
2
+ 2y − 1)
2
(y
75
− 4y
74
+ ··· − 1197196y − 1018081)
c
4
, c
9
y
6
(y − 1)(y
75
− 34y
74
+ ··· + 82944y − 4096)
c
7
, c
8
, c
11
c
12
(y − 1)(y
2
− 3y + 1)
3
(y
75
− 87y
74
+ ··· + 53y − 1)
c
10
(y − 1)(y
2
− 3y + 1)
3
(y
75
− 3y
74
+ ··· + 2.75501 × 10
7
y − 57121)
22
