12a
0851
(K12a
0851
)
A knot diagram
1
Linearized knot diagam
4 5 9 10 3 11 12 1 2 6 8 7
Solving Sequence
8,11
12 7
1,4
2 9 3 6 5 10
c
11
c
7
c
12
c
1
c
8
c
3
c
6
c
5
c
10
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h1.96995 × 10
27
u
83
+ 4.01738 × 10
27
u
82
+ ··· + 7.57665 × 10
26
b + 2.90943 × 10
27
,
3.36403 × 10
27
u
83
+ 4.75814 × 10
27
u
82
+ ··· + 7.57665 × 10
26
a + 8.28011 × 10
27
, u
84
+ 2u
83
+ ··· + u + 1i
I
u
2
= hu
2
+ b + 1, u
2
+ a + 2, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 87 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
= h1.97 × 10
27
u
83
+ 4.02 × 10
27
u
82
+ · · · + 7.58 × 10
26
b + 2.91 × 10
27
, 3.36 ×
10
27
u
83
+4.76×10
27
u
82
+· · ·+7.58×10
26
a+8.28×10
27
, u
84
+2u
83
+· · ·+u+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
4
=
−4.44000u
83
− 6.28001u
82
+ ··· + 13.8326u − 10.9285
−2.60003u
83
− 5.30231u
82
+ ··· + 4.48846u − 3.84000
a
2
=
1.40000u
83
+ 2.40001u
82
+ ··· − 3.79679u + 2.51127
0.399993u
83
− 0.0251959u
82
+ ··· − 1.11127u + 1.40000
a
9
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
3
=
−3.44000u
83
− 4.07994u
82
+ ··· + 11.1855u − 8.67404
−2.60000u
83
− 5.37729u
82
+ ··· + 6.07404u − 5.44000
a
6
=
u
3
+ 2u
u
3
+ u
a
5
=
−2.84000u
83
− 3.08004u
82
+ ··· + 10.4753u − 7.72107
−2.40000u
83
− 5.07341u
82
+ ··· + 6.32107u − 4.84000
a
10
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
166187238455782666427778654
757665015935352679203803969
u
83
−
2271996256671495868603397318
757665015935352679203803969
u
82
+
··· −
32408113859025183515073838511
757665015935352679203803969
u +
15078033543060885625800068293
757665015935352679203803969
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
84
− 13u
83
+ ··· − 36u + 8
c
2
, c
5
u
84
+ 4u
83
+ ··· − 24u − 1
c
3
u
84
− u
83
+ ··· + 3950u + 817
c
4
u
84
+ u
83
+ ··· + 6u − 1
c
6
, c
8
, c
10
u
84
− 2u
83
+ ··· + 69u + 17
c
7
, c
11
, c
12
u
84
+ 2u
83
+ ··· + u + 1
c
9
u
84
+ 2u
83
+ ··· − u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
84
− 21y
83
+ ··· − 1232y + 64
c
2
, c
5
y
84
− 48y
83
+ ··· − 224y + 1
c
3
y
84
+ 85y
83
+ ··· + 14128130y + 667489
c
4
y
84
+ 69y
83
+ ··· − 234y + 1
c
6
, c
8
, c
10
y
84
− 86y
83
+ ··· − 8807y + 289
c
7
, c
11
, c
12
y
84
+ 66y
83
+ ··· − 7y + 1
c
9
y
84
− 14y
83
+ ··· − 7y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.110255 + 0.992663I
a = −0.879103 + 0.499853I
b = −0.471696 + 0.417164I
−0.328989 + 0.984245I 0
u = 0.110255 − 0.992663I
a = −0.879103 − 0.499853I
b = −0.471696 − 0.417164I
−0.328989 − 0.984245I 0
u = −0.299517 + 0.904327I
a = 0.653627 − 0.164395I
b = 0.784926 − 0.330738I
1.86898 + 5.21583I 0
u = −0.299517 − 0.904327I
a = 0.653627 + 0.164395I
b = 0.784926 + 0.330738I
1.86898 − 5.21583I 0
u = 0.897832 + 0.024492I
a = −2.22947 + 0.33832I
b = −2.32818 + 0.38487I
−9.21782 − 0.46343I −10.71619 − 2.01137I
u = 0.897832 − 0.024492I
a = −2.22947 − 0.33832I
b = −2.32818 − 0.38487I
−9.21782 + 0.46343I −10.71619 + 2.01137I
u = −0.891180 + 0.064032I
a = −3.33093 + 0.18661I
b = −3.33496 + 0.25333I
−6.55361 + 12.23930I −7.27322 − 7.00359I
u = −0.891180 − 0.064032I
a = −3.33093 − 0.18661I
b = −3.33496 − 0.25333I
−6.55361 − 12.23930I −7.27322 + 7.00359I
u = −0.883835 + 0.038054I
a = 3.58891 − 0.56439I
b = 3.46927 − 0.50941I
−9.90913 + 5.72369I −9.95611 − 4.95013I
u = −0.883835 − 0.038054I
a = 3.58891 + 0.56439I
b = 3.46927 + 0.50941I
−9.90913 − 5.72369I −9.95611 + 4.95013I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.878158 + 0.070546I
a = 1.87000 − 0.49822I
b = 1.77275 − 0.49012I
−7.77096 − 4.58033I −9.29631 + 5.11031I
u = 0.878158 − 0.070546I
a = 1.87000 + 0.49822I
b = 1.77275 + 0.49012I
−7.77096 + 4.58033I −9.29631 − 5.11031I
u = 0.866332 + 0.010497I
a = 1.40515 + 2.06321I
b = 1.41089 + 2.80011I
−5.88867 − 0.76646I −10.18802 − 8.51416I
u = 0.866332 − 0.010497I
a = 1.40515 − 2.06321I
b = 1.41089 − 2.80011I
−5.88867 + 0.76646I −10.18802 + 8.51416I
u = −0.858093 + 0.025331I
a = −0.95600 − 2.32630I
b = −0.93847 − 1.67793I
−4.28941 + 3.74724I −4.52184 − 5.72401I
u = −0.858093 − 0.025331I
a = −0.95600 + 2.32630I
b = −0.93847 + 1.67793I
−4.28941 − 3.74724I −4.52184 + 5.72401I
u = −0.845503
a = −4.49525
b = −4.08680
−2.58884 −1.97210
u = −0.087999 + 1.205740I
a = 0.851450 + 0.079273I
b = 0.72162 − 1.91902I
3.95448 + 1.52948I 0
u = −0.087999 − 1.205740I
a = 0.851450 − 0.079273I
b = 0.72162 + 1.91902I
3.95448 − 1.52948I 0
u = 0.317565 + 0.699615I
a = 0.591065 − 0.420702I
b = 0.719031 − 0.569754I
2.21608 + 5.47917I −2.75820 − 3.97142I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.317565 − 0.699615I
a = 0.591065 + 0.420702I
b = 0.719031 + 0.569754I
2.21608 − 5.47917I −2.75820 + 3.97142I
u = 0.021059 + 1.251870I
a = 0.240736 + 0.329853I
b = 0.90373 − 1.23188I
4.24491 + 1.49996I 0
u = 0.021059 − 1.251870I
a = 0.240736 − 0.329853I
b = 0.90373 + 1.23188I
4.24491 − 1.49996I 0
u = −0.181517 + 1.243580I
a = 0.515986 + 0.448015I
b = 0.940034 + 0.122735I
2.79988 + 2.45137I 0
u = −0.181517 − 1.243580I
a = 0.515986 − 0.448015I
b = 0.940034 − 0.122735I
2.79988 − 2.45137I 0
u = −0.138549 + 1.255270I
a = −2.68245 + 0.16980I
b = −1.20511 + 3.23411I
4.81134 + 2.18966I 0
u = −0.138549 − 1.255270I
a = −2.68245 − 0.16980I
b = −1.20511 − 3.23411I
4.81134 − 2.18966I 0
u = −0.695609 + 0.224245I
a = −0.419849 − 0.863366I
b = −0.303627 − 0.104607I
−0.17080 − 1.42571I −7.44428 + 4.45956I
u = −0.695609 − 0.224245I
a = −0.419849 + 0.863366I
b = −0.303627 + 0.104607I
−0.17080 + 1.42571I −7.44428 − 4.45956I
u = 0.427785 + 1.209710I
a = −0.848570 + 0.981327I
b = −1.31346 − 0.86465I
−4.26362 − 0.09192I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.427785 − 1.209710I
a = −0.848570 − 0.981327I
b = −1.31346 + 0.86465I
−4.26362 + 0.09192I 0
u = 0.100979 + 1.283630I
a = 1.188160 − 0.660522I
b = 0.053190 − 1.124170I
7.14933 − 0.86028I 0
u = 0.100979 − 1.283630I
a = 1.188160 + 0.660522I
b = 0.053190 + 1.124170I
7.14933 + 0.86028I 0
u = −0.441193 + 1.219920I
a = 0.69098 + 2.04564I
b = 2.85592 − 0.56509I
−2.99168 − 7.48111I 0
u = −0.441193 − 1.219920I
a = 0.69098 − 2.04564I
b = 2.85592 + 0.56509I
−2.99168 + 7.48111I 0
u = 0.145710 + 1.293010I
a = 0.359277 − 0.661475I
b = −1.037240 + 0.805161I
6.61179 − 4.37101I 0
u = 0.145710 − 1.293010I
a = 0.359277 + 0.661475I
b = −1.037240 − 0.805161I
6.61179 + 4.37101I 0
u = 0.196690 + 1.297810I
a = −0.747120 + 0.507520I
b = −0.657638 + 0.978220I
2.21675 − 6.47162I 0
u = 0.196690 − 1.297810I
a = −0.747120 − 0.507520I
b = −0.657638 − 0.978220I
2.21675 + 6.47162I 0
u = −0.398240 + 1.251920I
a = 1.53701 − 0.13733I
b = 0.90130 − 1.85883I
−0.492846 + 0.757796I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.398240 − 1.251920I
a = 1.53701 + 0.13733I
b = 0.90130 + 1.85883I
−0.492846 − 0.757796I 0
u = −0.426210 + 1.243960I
a = −0.59680 − 2.27508I
b = −3.11386 + 0.31059I
−6.18171 − 1.03877I 0
u = −0.426210 − 1.243960I
a = −0.59680 + 2.27508I
b = −3.11386 − 0.31059I
−6.18171 + 1.03877I 0
u = 0.607131 + 0.307088I
a = −1.58483 + 0.55700I
b = −0.266796 − 0.243230I
0.95192 − 9.05338I −4.97546 + 9.13072I
u = 0.607131 − 0.307088I
a = −1.58483 − 0.55700I
b = −0.266796 + 0.243230I
0.95192 + 9.05338I −4.97546 − 9.13072I
u = 0.404594 + 1.266270I
a = 0.90231 + 1.11982I
b = −2.53793 + 2.04408I
−1.99365 − 3.78777I 0
u = 0.404594 − 1.266270I
a = 0.90231 − 1.11982I
b = −2.53793 − 2.04408I
−1.99365 + 3.78777I 0
u = −0.386566 + 1.274330I
a = 1.25324 + 2.56499I
b = 3.95929 − 1.00569I
1.37013 + 4.42279I 0
u = −0.386566 − 1.274330I
a = 1.25324 − 2.56499I
b = 3.95929 + 1.00569I
1.37013 − 4.42279I 0
u = 0.437176 + 1.259540I
a = 0.82445 − 1.23687I
b = 1.91181 + 1.01797I
−5.39444 − 4.30513I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.437176 − 1.259540I
a = 0.82445 + 1.23687I
b = 1.91181 − 1.01797I
−5.39444 + 4.30513I 0
u = 0.402005 + 1.283320I
a = −1.73173 + 0.46694I
b = −0.08092 − 3.11543I
−1.86506 − 5.31277I 0
u = 0.402005 − 1.283320I
a = −1.73173 − 0.46694I
b = −0.08092 + 3.11543I
−1.86506 + 5.31277I 0
u = −0.394513 + 1.293300I
a = −1.04471 + 1.30566I
b = 0.91800 + 1.57492I
−0.18144 + 8.24146I 0
u = −0.394513 − 1.293300I
a = −1.04471 − 1.30566I
b = 0.91800 − 1.57492I
−0.18144 − 8.24146I 0
u = 0.422095 + 1.298160I
a = 0.45105 − 1.40256I
b = 2.40460 + 0.35338I
−5.09893 − 5.18369I 0
u = 0.422095 − 1.298160I
a = 0.45105 + 1.40256I
b = 2.40460 − 0.35338I
−5.09893 + 5.18369I 0
u = −0.409997 + 1.305830I
a = −1.37607 − 1.93579I
b = −3.48232 + 1.34187I
−5.71744 + 10.35760I 0
u = −0.409997 − 1.305830I
a = −1.37607 + 1.93579I
b = −3.48232 − 1.34187I
−5.71744 − 10.35760I 0
u = 0.203910 + 1.353850I
a = 0.723619 − 0.741474I
b = 0.573766 − 0.626084I
6.17343 − 11.87550I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.203910 − 1.353850I
a = 0.723619 + 0.741474I
b = 0.573766 + 0.626084I
6.17343 + 11.87550I 0
u = 0.022685 + 1.376350I
a = 0.153766 − 0.103161I
b = −0.846260 + 0.832839I
8.47907 + 4.89675I 0
u = 0.022685 − 1.376350I
a = 0.153766 + 0.103161I
b = −0.846260 − 0.832839I
8.47907 − 4.89675I 0
u = −0.156955 + 1.372250I
a = −0.036947 − 0.278434I
b = 0.214434 − 0.433302I
4.30450 + 3.68208I 0
u = −0.156955 − 1.372250I
a = −0.036947 + 0.278434I
b = 0.214434 + 0.433302I
4.30450 − 3.68208I 0
u = −0.269039 + 1.357850I
a = −0.169242 + 0.428987I
b = −0.022494 + 0.257815I
4.81001 + 2.03195I 0
u = −0.269039 − 1.357850I
a = −0.169242 − 0.428987I
b = −0.022494 − 0.257815I
4.81001 − 2.03195I 0
u = 0.400272 + 1.326730I
a = −0.325967 + 1.177850I
b = −1.98872 + 0.03276I
−3.39774 − 9.16489I 0
u = 0.400272 − 1.326730I
a = −0.325967 − 1.177850I
b = −1.98872 − 0.03276I
−3.39774 + 9.16489I 0
u = −0.409526 + 1.324500I
a = 1.10413 + 1.91526I
b = 3.42129 − 1.08649I
−2.2113 + 16.8981I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.409526 − 1.324500I
a = 1.10413 − 1.91526I
b = 3.42129 + 1.08649I
−2.2113 − 16.8981I 0
u = −0.499272 + 0.346825I
a = 0.004363 + 0.910218I
b = −0.060032 + 0.136687I
−1.12877 + 1.42262I −10.39314 − 6.10068I
u = −0.499272 − 0.346825I
a = 0.004363 − 0.910218I
b = −0.060032 − 0.136687I
−1.12877 − 1.42262I −10.39314 + 6.10068I
u = 0.557733 + 0.195919I
a = 1.72195 − 0.43966I
b = 0.111216 − 0.097143I
−2.38342 − 3.80840I −9.51840 + 7.64998I
u = 0.557733 − 0.195919I
a = 1.72195 + 0.43966I
b = 0.111216 + 0.097143I
−2.38342 + 3.80840I −9.51840 − 7.64998I
u = −0.499502
a = −1.24909
b = −0.365987
−0.990570 −10.1890
u = 0.413247 + 0.213468I
a = −0.048963 + 0.431755I
b = 0.301016 − 0.915224I
2.03282 − 2.37496I −1.25289 + 9.23814I
u = 0.413247 − 0.213468I
a = −0.048963 − 0.431755I
b = 0.301016 + 0.915224I
2.03282 + 2.37496I −1.25289 − 9.23814I
u = −0.131361 + 0.439258I
a = −0.910942 + 0.658284I
b = −0.363524 + 0.420591I
−0.414341 + 1.299990I −5.47130 − 4.04671I
u = −0.131361 − 0.439258I
a = −0.910942 − 0.658284I
b = −0.363524 − 0.420591I
−0.414341 − 1.299990I −5.47130 + 4.04671I
12
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.413765 + 0.073813I
a = −0.52748 − 5.74207I
b = −0.83044 − 2.00086I
0.809864 + 0.226391I 16.7645 + 17.2989I
u = −0.413765 − 0.073813I
a = −0.52748 + 5.74207I
b = −0.83044 + 2.00086I
0.809864 − 0.226391I 16.7645 − 17.2989I
u = 0.212225 + 0.259248I
a = −2.31187 + 1.75601I
b = 0.562007 − 0.127169I
2.62354 + 0.38861I 2.48144 + 3.79646I
u = 0.212225 − 0.259248I
a = −2.31187 − 1.75601I
b = 0.562007 + 0.127169I
2.62354 − 0.38861I 2.48144 − 3.79646I
13
II. I
u
2
= hu
2
+ b + 1, u
2
+ a + 2, 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
4
=
−u
2
− 2
−u
2
− 1
a
2
=
u
2
+ 1
u
2
+ u + 1
a
9
=
1
0
a
3
=
−1
−u
2
− 1
a
6
=
−u
2
− 1
−u
2
− u − 1
a
5
=
−u
2
− 2
−2u
2
− u − 2
a
10
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
− 3u − 3
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
c
2
(u + 1)
3
c
3
, c
4
u
3
− u − 1
c
5
(u − 1)
3
c
6
, c
8
, c
9
u
3
+ u
2
− 1
c
7
u
3
− u
2
+ 2u − 1
c
10
u
3
− u
2
+ 1
c
11
, c
12
u
3
+ u
2
+ 2u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
3
c
2
, c
5
(y −1)
3
c
3
, c
4
y
3
− 2y
2
+ y −1
c
6
, c
8
, c
9
c
10
y
3
− y
2
+ 2y −1
c
7
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.337641 + 0.562280I
b = 0.662359 + 0.562280I
4.66906 + 2.82812I −0.69240 − 3.35914I
u = −0.215080 − 1.307140I
a = −0.337641 − 0.562280I
b = 0.662359 − 0.562280I
4.66906 − 2.82812I −0.69240 + 3.35914I
u = −0.569840
a = −2.32472
b = −1.32472
0.531480 −1.61520
17
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
3
(u
84
− 13u
83
+ ··· − 36u + 8)
c
2
((u + 1)
3
)(u
84
+ 4u
83
+ ··· − 24u − 1)
c
3
(u
3
− u − 1)(u
84
− u
83
+ ··· + 3950u + 817)
c
4
(u
3
− u − 1)(u
84
+ u
83
+ ··· + 6u − 1)
c
5
((u − 1)
3
)(u
84
+ 4u
83
+ ··· − 24u − 1)
c
6
, c
8
(u
3
+ u
2
− 1)(u
84
− 2u
83
+ ··· + 69u + 17)
c
7
(u
3
− u
2
+ 2u − 1)(u
84
+ 2u
83
+ ··· + u + 1)
c
9
(u
3
+ u
2
− 1)(u
84
+ 2u
83
+ ··· − u − 1)
c
10
(u
3
− u
2
+ 1)(u
84
− 2u
83
+ ··· + 69u + 17)
c
11
, c
12
(u
3
+ u
2
+ 2u + 1)(u
84
+ 2u
83
+ ··· + u + 1)
18
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
3
(y
84
− 21y
83
+ ··· − 1232y + 64)
c
2
, c
5
((y −1)
3
)(y
84
− 48y
83
+ ··· − 224y + 1)
c
3
(y
3
− 2y
2
+ y −1)(y
84
+ 85y
83
+ ··· + 1.41281 × 10
7
y + 667489)
c
4
(y
3
− 2y
2
+ y −1)(y
84
+ 69y
83
+ ··· − 234y + 1)
c
6
, c
8
, c
10
(y
3
− y
2
+ 2y −1)(y
84
− 86y
83
+ ··· − 8807y + 289)
c
7
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)(y
84
+ 66y
83
+ ··· − 7y + 1)
c
9
(y
3
− y
2
+ 2y −1)(y
84
− 14y
83
+ ··· − 7y + 1)
19
