12a
1234
(K12a
1234
)
A knot diagram
1
Linearized knot diagam
5 6 9 10 2 11 12 1 4 3 8 7
Solving Sequence
8,11
12 7 1 9
2,6
3 5 10 4
c
11
c
7
c
12
c
8
c
6
c
2
c
5
c
10
c
4
c
1
, c
3
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h1.93075 × 10
22
u
71
− 9.36918 × 10
22
u
70
+ ··· + 2.61600 × 10
23
b − 2.51001 × 10
23
,
3.91390 × 10
23
u
71
− 8.79983 × 10
23
u
70
+ ··· + 2.61600 × 10
23
a − 2.96665 × 10
24
, u
72
− 2u
71
+ ··· − 8u − 1i
I
u
2
= h−au − u
2
+ b + u − 1, a
2
− 4u
2
+ 2u − 6, u
3
− u
2
+ 2u − 1i
I
u
3
= h−u
2
+ b − u − 1, a, u
3
+ u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 81 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
= h1.93 × 10
22
u
71
− 9.37 × 10
22
u
70
+ · · · + 2.62 × 10
23
b − 2.51 × 10
23
, 3.91 ×
10
23
u
71
−8.80×10
23
u
70
+· · ·+2.62×10
23
a−2.97×10
24
, u
72
−2u
71
+· · ·−8u−1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
2
=
−1.49614u
71
+ 3.36385u
70
+ ··· + 9.35049u + 11.3404
−0.0738055u
71
+ 0.358149u
70
+ ··· − 1.96687u + 0.959482
a
6
=
u
3
+ 2u
u
3
+ u
a
3
=
−1.70670u
71
+ 4.43356u
70
+ ··· + 11.3889u + 11.8788
−0.0934270u
71
+ 1.10929u
70
+ ··· − 1.29641u + 1.19703
a
5
=
0.959482u
71
− 1.99277u
70
+ ··· − 1.06519u − 9.64273
0.371572u
71
− 0.601476u
70
+ ··· − 0.628716u − 1.49614
a
10
=
−2.52657u
71
+ 5.46669u
70
+ ··· + 6.75726u + 15.3627
−0.515104u
71
+ 0.594756u
70
+ ··· + 2.97411u + 2.22254
a
4
=
−1.82146u
71
+ 4.39966u
70
+ ··· + 11.3717u + 12.1549
0.225358u
71
+ 0.517327u
70
+ ··· − 1.91838u + 0.946928
(ii) Obstruction class = −1
(iii) Cusp Shapes =
131451318810141352881632
130800077995111299411101
u
71
−
277192896356154580549909
130800077995111299411101
u
70
+ ··· +
732571967736991836052242
130800077995111299411101
u +
336112699848274877396314
130800077995111299411101
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
72
+ 4u
71
+ ··· + 51u − 7
c
3
, c
4
, c
9
u
72
− u
71
+ ··· + 8u − 8
c
6
, c
8
u
72
− 2u
71
+ ··· + 5260u − 481
c
7
, c
11
, c
12
u
72
+ 2u
71
+ ··· + 8u − 1
c
10
u
72
+ 3u
71
+ ··· − 2888u − 5768
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
72
− 70y
71
+ ··· + 423y + 49
c
3
, c
4
, c
9
y
72
− 65y
71
+ ··· − 448y + 64
c
6
, c
8
y
72
− 52y
71
+ ··· − 12759486y + 231361
c
7
, c
11
, c
12
y
72
+ 60y
71
+ ··· − 62y + 1
c
10
y
72
+ 19y
71
+ ··· − 312337216y + 33269824
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.855221 + 0.104555I
a = −3.51455 − 0.94548I
b = −2.90722 − 0.53414I
−11.19460 + 5.99079I −11.45914 − 4.34329I
u = −0.855221 − 0.104555I
a = −3.51455 + 0.94548I
b = −2.90722 + 0.53414I
−11.19460 − 5.99079I −11.45914 + 4.34329I
u = 0.846928 + 0.141420I
a = −3.30013 + 1.21203I
b = −2.78162 + 0.68434I
−5.95434 − 10.47780I −7.43345 + 6.18173I
u = 0.846928 − 0.141420I
a = −3.30013 − 1.21203I
b = −2.78162 − 0.68434I
−5.95434 + 10.47780I −7.43345 − 6.18173I
u = 0.849158 + 0.055790I
a = −3.83366 + 0.58255I
b = −3.09139 + 0.33000I
−8.96835 − 1.04646I −9.68773 − 0.28213I
u = 0.849158 − 0.055790I
a = −3.83366 − 0.58255I
b = −3.09139 − 0.33000I
−8.96835 + 1.04646I −9.68773 + 0.28213I
u = 0.413905 + 1.104350I
a = 1.63797 − 1.57519I
b = 2.11853 + 0.95743I
−3.00550 + 5.94009I 0
u = 0.413905 − 1.104350I
a = 1.63797 + 1.57519I
b = 2.11853 − 0.95743I
−3.00550 − 5.94009I 0
u = 0.805296 + 0.104399I
a = 1.033840 − 0.702962I
b = 0.912278 − 0.053298I
0.09644 − 6.17633I −4.65210 + 5.74472I
u = 0.805296 − 0.104399I
a = 1.033840 + 0.702962I
b = 0.912278 + 0.053298I
0.09644 + 6.17633I −4.65210 − 5.74472I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.340749 + 1.153980I
a = −0.553879 + 0.101561I
b = −0.951773 − 0.318003I
3.29054 + 2.00837I 0
u = 0.340749 − 1.153980I
a = −0.553879 − 0.101561I
b = −0.951773 + 0.318003I
3.29054 − 2.00837I 0
u = −0.789046 + 0.047967I
a = 0.926833 + 0.645835I
b = 0.870318 + 0.061520I
−4.46192 + 2.46018I −10.16635 − 4.00752I
u = −0.789046 − 0.047967I
a = 0.926833 − 0.645835I
b = 0.870318 − 0.061520I
−4.46192 − 2.46018I −10.16635 + 4.00752I
u = 0.782297 + 0.041244I
a = 0.717071 + 0.660038I
b = 0.814839 + 0.084621I
−1.27923 − 1.03150I −6.99400 + 0.63948I
u = 0.782297 − 0.041244I
a = 0.717071 − 0.660038I
b = 0.814839 − 0.084621I
−1.27923 + 1.03150I −6.99400 − 0.63948I
u = −0.414055 + 1.158140I
a = 1.60375 + 1.71714I
b = 2.38953 − 0.91844I
−7.96420 − 1.43495I 0
u = −0.414055 − 1.158140I
a = 1.60375 − 1.71714I
b = 2.38953 + 0.91844I
−7.96420 + 1.43495I 0
u = −0.481039 + 0.598411I
a = 0.375183 − 1.317490I
b = 0.464545 + 0.103622I
−1.22913 + 5.72696I −4.74473 − 6.15763I
u = −0.481039 − 0.598411I
a = 0.375183 + 1.317490I
b = 0.464545 − 0.103622I
−1.22913 − 5.72696I −4.74473 + 6.15763I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.767390
a = −5.25539
b = −3.90280
−0.107189 −8.73150
u = −0.089470 + 1.249190I
a = 0.634254 + 0.081148I
b = −0.714517 + 1.152590I
1.49723 + 1.57721I 0
u = −0.089470 − 1.249190I
a = 0.634254 − 0.081148I
b = −0.714517 − 1.152590I
1.49723 − 1.57721I 0
u = 0.324593 + 1.231220I
a = 0.095849 + 0.773188I
b = −0.573454 − 0.363220I
2.37868 − 2.96753I 0
u = 0.324593 − 1.231220I
a = 0.095849 − 0.773188I
b = −0.573454 + 0.363220I
2.37868 + 2.96753I 0
u = −0.332624 + 1.229470I
a = −0.494776 − 0.054404I
b = −0.964484 + 0.433638I
−0.83246 + 1.58738I 0
u = −0.332624 − 1.229470I
a = −0.494776 + 0.054404I
b = −0.964484 − 0.433638I
−0.83246 − 1.58738I 0
u = −0.621231 + 0.374947I
a = 0.245228 − 1.110140I
b = 0.612440 − 0.033140I
−1.94589 − 1.80647I −6.41560 − 0.05777I
u = −0.621231 − 0.374947I
a = 0.245228 + 1.110140I
b = 0.612440 + 0.033140I
−1.94589 + 1.80647I −6.41560 + 0.05777I
u = 0.401285 + 1.213440I
a = 1.52913 − 1.92577I
b = 2.74245 + 0.83179I
−5.40292 − 3.44004I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.401285 − 1.213440I
a = 1.52913 + 1.92577I
b = 2.74245 − 0.83179I
−5.40292 + 3.44004I 0
u = 0.526133 + 0.479268I
a = 0.264306 + 1.274860I
b = 0.551953 − 0.042979I
−5.48345 − 1.89351I −9.94754 + 3.92631I
u = 0.526133 − 0.479268I
a = 0.264306 − 1.274860I
b = 0.551953 + 0.042979I
−5.48345 + 1.89351I −9.94754 − 3.92631I
u = −0.021601 + 1.289030I
a = 0.982879 − 0.665839I
b = −1.11100 − 1.60954I
7.38621 + 0.45778I 0
u = −0.021601 − 1.289030I
a = 0.982879 + 0.665839I
b = −1.11100 + 1.60954I
7.38621 − 0.45778I 0
u = 0.260985 + 1.272720I
a = −0.153726 + 0.524674I
b = −0.776623 − 0.528444I
2.21876 − 3.34010I 0
u = 0.260985 − 1.272720I
a = −0.153726 − 0.524674I
b = −0.776623 + 0.528444I
2.21876 + 3.34010I 0
u = 0.047615 + 1.309970I
a = −0.701791 + 0.007598I
b = 0.040193 − 0.254286I
4.63025 − 1.60850I 0
u = 0.047615 − 1.309970I
a = −0.701791 − 0.007598I
b = 0.040193 + 0.254286I
4.63025 + 1.60850I 0
u = −0.327283 + 1.273420I
a = 1.52077 + 2.75302I
b = 3.91217 − 0.83641I
3.84964 + 3.94691I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.327283 − 1.273420I
a = 1.52077 − 2.75302I
b = 3.91217 + 0.83641I
3.84964 − 3.94691I 0
u = 0.663829
a = 0.875997
b = 0.804619
−1.74281 −3.26010
u = 0.340856 + 1.295450I
a = −0.439050 − 0.044410I
b = −1.021930 − 0.522230I
2.89162 − 5.08382I 0
u = 0.340856 − 1.295450I
a = −0.439050 + 0.044410I
b = −1.021930 + 0.522230I
2.89162 + 5.08382I 0
u = −0.345526 + 1.301720I
a = −0.055600 − 0.900464I
b = −0.736941 + 0.208894I
−0.24447 + 6.55260I 0
u = −0.345526 − 1.301720I
a = −0.055600 + 0.900464I
b = −0.736941 − 0.208894I
−0.24447 − 6.55260I 0
u = −0.240433 + 1.331850I
a = −0.521834 − 0.837953I
b = −1.098430 + 0.565000I
8.17221 + 2.72991I 0
u = −0.240433 − 1.331850I
a = −0.521834 + 0.837953I
b = −1.098430 − 0.565000I
8.17221 − 2.72991I 0
u = 0.381480 + 1.308120I
a = 1.02332 − 2.24320I
b = 3.24181 + 0.18133I
−4.70869 − 5.46372I 0
u = 0.381480 − 1.308120I
a = 1.02332 + 2.24320I
b = 3.24181 − 0.18133I
−4.70869 + 5.46372I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.070481 + 1.375000I
a = −0.710558 + 0.082467I
b = 0.434145 + 0.464159I
10.20230 + 4.09393I 0
u = −0.070481 − 1.375000I
a = −0.710558 − 0.082467I
b = 0.434145 − 0.464159I
10.20230 − 4.09393I 0
u = 0.351583 + 1.335210I
a = −0.094715 + 0.966773I
b = −0.805731 − 0.126163I
4.61739 − 10.34930I 0
u = 0.351583 − 1.335210I
a = −0.094715 − 0.966773I
b = −0.805731 + 0.126163I
4.61739 + 10.34930I 0
u = −0.380105 + 1.341130I
a = 0.77057 + 2.17800I
b = 3.17864 + 0.12703I
−6.65706 + 10.42950I 0
u = −0.380105 − 1.341130I
a = 0.77057 − 2.17800I
b = 3.17864 − 0.12703I
−6.65706 − 10.42950I 0
u = 0.142108 + 1.400830I
a = 0.073563 − 0.435223I
b = −1.082410 − 0.884379I
0.48745 − 4.08574I 0
u = 0.142108 − 1.400830I
a = 0.073563 + 0.435223I
b = −1.082410 + 0.884379I
0.48745 + 4.08574I 0
u = 0.369333 + 1.361460I
a = 0.58444 − 2.15508I
b = 3.15725 − 0.33932I
−1.2207 − 14.8549I 0
u = 0.369333 − 1.361460I
a = 0.58444 + 2.15508I
b = 3.15725 + 0.33932I
−1.2207 + 14.8549I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.317187 + 0.489927I
a = 0.090286 − 0.752772I
b = −0.635338 − 0.425944I
4.43235 + 2.91765I 1.16854 − 6.33914I
u = −0.317187 − 0.489927I
a = 0.090286 + 0.752772I
b = −0.635338 + 0.425944I
4.43235 − 2.91765I 1.16854 + 6.33914I
u = −0.22131 + 1.40591I
a = −0.133181 + 0.324090I
b = −1.068390 + 0.764012I
3.69568 + 1.17982I 0
u = −0.22131 − 1.40591I
a = −0.133181 − 0.324090I
b = −1.068390 − 0.764012I
3.69568 − 1.17982I 0
u = −0.09732 + 1.43277I
a = 0.098014 + 0.602807I
b = −1.16805 + 0.91404I
5.29029 + 7.47149I 0
u = −0.09732 − 1.43277I
a = 0.098014 − 0.602807I
b = −1.16805 − 0.91404I
5.29029 − 7.47149I 0
u = −0.492033 + 0.179262I
a = 1.73174 + 0.27386I
b = 0.789924 − 0.245115I
3.53660 − 0.14057I −0.86825 − 1.39722I
u = −0.492033 − 0.179262I
a = 1.73174 − 0.27386I
b = 0.789924 + 0.245115I
3.53660 + 0.14057I −0.86825 + 1.39722I
u = −0.365925
a = −1.20803
b = 0.774716
−2.24878 1.74900
u = 0.213495 + 0.279988I
a = 0.805720 + 0.480004I
b = −0.165356 + 0.323439I
−0.160994 − 0.786143I −4.51480 + 8.79169I
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.213495 − 0.279988I
a = 0.805720 − 0.480004I
b = −0.165356 − 0.323439I
−0.160994 + 0.786143I −4.51480 − 8.79169I
u = −0.134176
a = 9.11290
b = 1.17072
3.24436 2.13710
12
II. I
u
2
= h−au − u
2
+ b + u − 1, a
2
− 4u
2
+ 2u − 6, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
7
=
u
u
2
− u + 1
a
1
=
u
2
+ 1
u
2
− u + 1
a
9
=
−1
0
a
2
=
a
au + u
2
− u + 1
a
6
=
u
2
+ 1
u
2
− u + 1
a
3
=
−u
2
+ a − 1
au
a
5
=
u
2
− a + 1
−au
a
10
=
−u
2
a + au + 2u
2
− a − 2u + 5
2
a
4
=
au − u
2
+ a − 1
au
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 4u − 8
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u + 1)
6
c
3
, c
4
, c
9
c
10
(u
2
− 2)
3
c
5
(u − 1)
6
c
6
, c
8
(u
3
− u
2
+ 1)
2
c
7
(u
3
+ u
2
+ 2u + 1)
2
c
11
, c
12
(u
3
− u
2
+ 2u − 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
9
c
10
(y −2)
6
c
6
, c
8
(y
3
− y
2
+ 2y −1)
2
c
7
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.173328 + 1.053390I
b = −2.29165 − 0.74486I
6.31400 − 2.82812I −0.49024 + 2.97945I
u = 0.215080 + 1.307140I
a = 0.173328 − 1.053390I
b = 0.536775 − 0.744862I
6.31400 − 2.82812I −0.49024 + 2.97945I
u = 0.215080 − 1.307140I
a = −0.173328 − 1.053390I
b = −2.29165 + 0.74486I
6.31400 + 2.82812I −0.49024 − 2.97945I
u = 0.215080 − 1.307140I
a = 0.173328 + 1.053390I
b = 0.536775 + 0.744862I
6.31400 + 2.82812I −0.49024 − 2.97945I
u = 0.569840
a = −2.48177
b = −0.659336
2.17641 −7.01950
u = 0.569840
a = 2.48177
b = 2.16909
2.17641 −7.01950
16
III. I
u
3
= h−u
2
+ b − u − 1, a, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
12
=
1
u
2
a
7
=
u
−u
2
− u − 1
a
1
=
u
2
+ 1
u
2
+ u + 1
a
9
=
1
0
a
2
=
0
u
2
+ u + 1
a
6
=
−u
2
− 1
−u
2
− u − 1
a
3
=
−u
2
− 1
0
a
5
=
−u
2
− 1
0
a
10
=
1
0
a
4
=
−u
2
− 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6u
2
− 4u − 16
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
9
c
10
u
3
c
5
(u + 1)
3
c
6
, c
8
u
3
+ u
2
− 1
c
7
u
3
− u
2
+ 2u − 1
c
11
, c
12
u
3
+ u
2
+ 2u + 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
9
c
10
y
3
c
6
, c
8
y
3
− y
2
+ 2y −1
c
7
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0
b = −0.877439 + 0.744862I
1.37919 + 2.82812I −5.16553 − 1.85489I
u = −0.215080 − 1.307140I
a = 0
b = −0.877439 − 0.744862I
1.37919 − 2.82812I −5.16553 + 1.85489I
u = −0.569840
a = 0
b = 0.754878
−2.75839 −15.6690
20
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
3
)(u + 1)
6
(u
72
+ 4u
71
+ ··· + 51u − 7)
c
3
, c
4
, c
9
u
3
(u
2
− 2)
3
(u
72
− u
71
+ ··· + 8u − 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
72
+ 4u
71
+ ··· + 51u − 7)
c
6
, c
8
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
72
− 2u
71
+ ··· + 5260u − 481)
c
7
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
(u
72
+ 2u
71
+ ··· + 8u − 1)
c
10
u
3
(u
2
− 2)
3
(u
72
+ 3u
71
+ ··· − 2888u − 5768)
c
11
, c
12
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
72
+ 2u
71
+ ··· + 8u − 1)
21
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
((y −1)
9
)(y
72
− 70y
71
+ ··· + 423y + 49)
c
3
, c
4
, c
9
y
3
(y −2)
6
(y
72
− 65y
71
+ ··· − 448y + 64)
c
6
, c
8
((y
3
− y
2
+ 2y −1)
3
)(y
72
− 52y
71
+ ··· − 1.27595 × 10
7
y + 231361)
c
7
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
72
+ 60y
71
+ ··· − 62y + 1)
c
10
y
3
(y −2)
6
(y
72
+ 19y
71
+ ··· − 3.12337 × 10
8
y + 3.32698 × 10
7
)
22
