12a
0420
(K12a
0420
)
A knot diagram
1
Linearized knot diagam
3 6 10 7 2 5 11 12 4 1 8 9
Solving Sequence
2,6
3 1 5
7,11
8 12 4 10 9
c
2
c
1
c
5
c
6
c
7
c
11
c
4
c
10
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
60
+ u
59
+ ··· + 2b − u, 2u
60
− 7u
59
+ ··· + 2a + 6, u
61
− 3u
60
+ ··· + 6u − 1i
I
u
2
= hb − a, −u
2
a + a
2
− au − u − 1, u
3
+ u
2
− 1i
* 2 irreducible components of dim
C
= 0, with total 67 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−2u
60
+u
59
+· · ·+2b −u, 2u
60
−7u
59
+· · ·+2a +6, u
61
−3u
60
+· · ·+6u −1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
7
=
−u
3
−u
3
+ u
a
11
=
−u
60
+
7
2
u
59
+ ··· +
5
2
u − 3
u
60
−
1
2
u
59
+ ··· + 8u
2
+
1
2
u
a
8
=
1
2
u
59
− u
58
+ ··· −
1
2
u − 1
1
2
u
59
− u
58
+ ··· + 2u
2
+
3
2
u
a
12
=
−
1
2
u
58
+ u
57
+ ··· − 2u −
3
2
−
1
2
u
60
+ u
59
+ ··· +
1
2
u
2
+ u
a
4
=
u
5
+ u
u
5
− u
3
+ u
a
10
=
−2u
60
+ 8u
59
+ ··· + 12u −
9
2
7
2
u
60
− 3u
59
+ ··· +
31
2
u
2
− u
a
9
=
−u
59
+
1
2
u
58
+ ··· − 8u −
1
2
−
1
2
u
60
+
7
2
u
58
+ ··· − 3u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
13
2
u
60
+ 11u
59
+ ··· + 29u −
41
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
61
+ 15u
60
+ ··· + 32u + 1
c
2
, c
5
u
61
+ 3u
60
+ ··· + 6u + 1
c
3
, c
9
u
61
+ u
60
+ ··· + 288u + 64
c
7
, c
8
, c
11
c
12
u
61
+ 4u
60
+ ··· + u + 1
c
10
u
61
− 14u
60
+ ··· + 1883u + 233
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
y
61
+ 65y
60
+ ··· + 360y −1
c
2
, c
5
y
61
− 15y
60
+ ··· + 32y −1
c
3
, c
9
y
61
+ 35y
60
+ ··· − 48128y −4096
c
7
, c
8
, c
11
c
12
y
61
− 70y
60
+ ··· + 33y −1
c
10
y
61
+ 14y
60
+ ··· − 820731y −54289
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.903295 + 0.462746I
a = −0.695546 − 0.562869I
b = −0.271804 − 0.008793I
1.64912 + 3.33623I −8.90726 − 4.06280I
u = −0.903295 − 0.462746I
a = −0.695546 + 0.562869I
b = −0.271804 + 0.008793I
1.64912 − 3.33623I −8.90726 + 4.06280I
u = 0.965632 + 0.071466I
a = 0.255352 − 0.811455I
b = 0.583357 − 0.467416I
−1.45506 + 1.38293I −14.5746 − 4.7781I
u = 0.965632 − 0.071466I
a = 0.255352 + 0.811455I
b = 0.583357 + 0.467416I
−1.45506 − 1.38293I −14.5746 + 4.7781I
u = 0.891866 + 0.371875I
a = 0.025117 + 0.353693I
b = 0.48374 − 1.47821I
−9.20574 − 3.82983I −18.3699 + 4.7525I
u = 0.891866 − 0.371875I
a = 0.025117 − 0.353693I
b = 0.48374 + 1.47821I
−9.20574 + 3.82983I −18.3699 − 4.7525I
u = 1.027100 + 0.122511I
a = −0.577403 + 0.901803I
b = −1.53356 + 0.54056I
−8.53214 + 3.20048I −17.8758 − 2.0778I
u = 1.027100 − 0.122511I
a = −0.577403 − 0.901803I
b = −1.53356 − 0.54056I
−8.53214 − 3.20048I −17.8758 + 2.0778I
u = −0.973990 + 0.420575I
a = 0.605744 + 0.194442I
b = −0.223113 − 0.672957I
0.54572 + 6.96672I −12.0000 − 9.8175I
u = −0.973990 − 0.420575I
a = 0.605744 − 0.194442I
b = −0.223113 + 0.672957I
0.54572 − 6.96672I −12.0000 + 9.8175I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.015250 + 0.392641I
a = −0.579050 + 0.148607I
b = 0.52059 + 1.42782I
−6.92985 + 9.41761I −12.0000 − 7.9710I
u = −1.015250 − 0.392641I
a = −0.579050 − 0.148607I
b = 0.52059 − 1.42782I
−6.92985 − 9.41761I −12.0000 + 7.9710I
u = −0.558526 + 0.717048I
a = 1.30900 + 0.57411I
b = 0.615035 + 0.129276I
−2.84419 + 2.86741I −10.16234 − 3.32231I
u = −0.558526 − 0.717048I
a = 1.30900 − 0.57411I
b = 0.615035 − 0.129276I
−2.84419 − 2.86741I −10.16234 + 3.32231I
u = −0.877865 + 0.718484I
a = −0.507669 − 0.573262I
b = −0.710801 − 0.319209I
2.45258 + 2.74784I 0
u = −0.877865 − 0.718484I
a = −0.507669 + 0.573262I
b = −0.710801 + 0.319209I
2.45258 − 2.74784I 0
u = 0.804516 + 0.318031I
a = 0.196739 − 0.606198I
b = −0.308799 + 0.790106I
−1.74945 − 2.32779I −16.2262 + 7.6362I
u = 0.804516 − 0.318031I
a = 0.196739 + 0.606198I
b = −0.308799 − 0.790106I
−1.74945 + 2.32779I −16.2262 − 7.6362I
u = −0.958834 + 0.651543I
a = 0.741491 + 1.069760I
b = 1.03234 + 1.20463I
−3.94098 + 2.19391I 0
u = −0.958834 − 0.651543I
a = 0.741491 − 1.069760I
b = 1.03234 − 1.20463I
−3.94098 − 2.19391I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.808720 + 0.190086I
a = −1.90972 − 1.06246I
b = −2.26441 − 0.31552I
−10.29000 + 0.53665I −18.8353 − 9.0691I
u = −0.808720 − 0.190086I
a = −1.90972 + 1.06246I
b = −2.26441 + 0.31552I
−10.29000 − 0.53665I −18.8353 + 9.0691I
u = −0.761071 + 0.320943I
a = 1.15383 + 0.94285I
b = 1.234600 + 0.239084I
−1.85024 + 1.25608I −15.1118 − 4.8976I
u = −0.761071 − 0.320943I
a = 1.15383 − 0.94285I
b = 1.234600 − 0.239084I
−1.85024 − 1.25608I −15.1118 + 4.8976I
u = 0.898416 + 0.813498I
a = 1.20213 − 0.99182I
b = 1.22884 − 0.86819I
−4.53737 − 3.04417I 0
u = 0.898416 − 0.813498I
a = 1.20213 + 0.99182I
b = 1.22884 + 0.86819I
−4.53737 + 3.04417I 0
u = −0.856068 + 0.862516I
a = −1.22457 + 2.63205I
b = 1.20834 + 2.93387I
−1.50415 − 0.89126I 0
u = −0.856068 − 0.862516I
a = −1.22457 − 2.63205I
b = 1.20834 − 2.93387I
−1.50415 + 0.89126I 0
u = 0.820898 + 0.899902I
a = −2.06541 − 1.89264I
b = −0.15813 − 2.95041I
1.58971 + 7.54690I 0
u = 0.820898 − 0.899902I
a = −2.06541 + 1.89264I
b = −0.15813 + 2.95041I
1.58971 − 7.54690I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.886180 + 0.839133I
a = 1.67113 − 1.82782I
b = 0.00358 − 2.86721I
5.10715 + 1.32327I 0
u = −0.886180 − 0.839133I
a = 1.67113 + 1.82782I
b = 0.00358 + 2.86721I
5.10715 − 1.32327I 0
u = 0.842192 + 0.896151I
a = 1.85971 + 1.34927I
b = 0.56016 + 2.65833I
9.11750 + 4.54405I 0
u = 0.842192 − 0.896151I
a = 1.85971 − 1.34927I
b = 0.56016 − 2.65833I
9.11750 − 4.54405I 0
u = −0.413830 + 0.644735I
a = −0.606570 − 0.053578I
b = −0.437841 − 0.476673I
3.19857 + 0.74045I −5.48051 − 3.33968I
u = −0.413830 − 0.644735I
a = −0.606570 + 0.053578I
b = −0.437841 + 0.476673I
3.19857 − 0.74045I −5.48051 + 3.33968I
u = −0.921108 + 0.827807I
a = −2.24258 + 1.12030I
b = −1.29799 + 2.86920I
4.99860 + 4.88810I 0
u = −0.921108 − 0.827807I
a = −2.24258 − 1.12030I
b = −1.29799 − 2.86920I
4.99860 − 4.88810I 0
u = 0.864695 + 0.889666I
a = −1.48550 − 0.77368I
b = −0.87847 − 2.12723I
10.20040 + 0.14626I 0
u = 0.864695 − 0.889666I
a = −1.48550 + 0.77368I
b = −0.87847 + 2.12723I
10.20040 − 0.14626I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.898542 + 0.870061I
a = 0.474229 + 0.038157I
b = 0.796656 + 0.869534I
5.34141 − 3.08631I 0
u = 0.898542 − 0.870061I
a = 0.474229 − 0.038157I
b = 0.796656 − 0.869534I
5.34141 + 3.08631I 0
u = −0.218197 + 0.712691I
a = 0.502777 + 0.882359I
b = −0.541108 − 0.948954I
−4.36643 − 5.46439I −10.79014 + 3.37482I
u = −0.218197 − 0.712691I
a = 0.502777 − 0.882359I
b = −0.541108 + 0.948954I
−4.36643 + 5.46439I −10.79014 − 3.37482I
u = 0.928575 + 0.852722I
a = −0.522895 − 0.567519I
b = 0.296827 − 0.457150I
5.24177 − 3.29859I 0
u = 0.928575 − 0.852722I
a = −0.522895 + 0.567519I
b = 0.296827 + 0.457150I
5.24177 + 3.29859I 0
u = −0.952556 + 0.826132I
a = 2.82339 − 0.54814I
b = 2.58806 − 2.93805I
−1.80666 + 7.16585I 0
u = −0.952556 − 0.826132I
a = 2.82339 + 0.54814I
b = 2.58806 + 2.93805I
−1.80666 − 7.16585I 0
u = −0.297713 + 0.668924I
a = 0.038251 − 0.431492I
b = 0.427836 + 0.712970I
2.69330 − 2.99488I −7.10794 + 4.32236I
u = −0.297713 − 0.668924I
a = 0.038251 + 0.431492I
b = 0.427836 − 0.712970I
2.69330 + 2.99488I −7.10794 − 4.32236I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.963248 + 0.846430I
a = 1.30769 + 1.41106I
b = 0.00611 + 2.16662I
9.88678 − 6.57001I 0
u = 0.963248 − 0.846430I
a = 1.30769 − 1.41106I
b = 0.00611 − 2.16662I
9.88678 + 6.57001I 0
u = 0.979698 + 0.836583I
a = −1.85878 − 1.61789I
b = −0.64482 − 2.99145I
8.68110 − 10.95380I 0
u = 0.979698 − 0.836583I
a = −1.85878 + 1.61789I
b = −0.64482 + 2.99145I
8.68110 + 10.95380I 0
u = 0.992523 + 0.826283I
a = 2.32375 + 1.68082I
b = 1.36651 + 3.55718I
1.04634 − 13.93150I 0
u = 0.992523 − 0.826283I
a = 2.32375 − 1.68082I
b = 1.36651 − 3.55718I
1.04634 + 13.93150I 0
u = 0.584427 + 0.143584I
a = −0.768708 + 0.414329I
b = 0.282890 − 0.137929I
−0.764972 − 0.051966I −11.69477 − 1.11358I
u = 0.584427 − 0.143584I
a = −0.768708 − 0.414329I
b = 0.282890 + 0.137929I
−0.764972 + 0.051966I −11.69477 + 1.11358I
u = 0.327187 + 0.491156I
a = 1.56928 − 1.31367I
b = −0.647027 + 0.145084I
−7.50998 + 0.52039I −13.78782 + 1.07094I
u = 0.327187 − 0.491156I
a = 1.56928 + 1.31367I
b = −0.647027 − 0.145084I
−7.50998 − 0.52039I −13.78782 − 1.07094I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.227354
a = −2.03044
b = 0.364783
−0.700986 −13.7360
11
II. I
u
2
= hb − a, −u
2
a + a
2
− au − u − 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
5
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
11
=
a
a
a
8
=
−a − u − 1
−a − 1
a
12
=
−u
2
a − au − u
2
− u
−au − u
2
− u
a
4
=
1
u
2
a
10
=
u
2
a + au
au
a
9
=
u
2
a + au
au
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
a + 2au + 3u
2
− 19
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
3
, 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
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
3
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −1.071720 − 0.909787I
b = −1.071720 − 0.909787I
−5.85852 + 2.82812I −16.5384 − 2.7162I
u = −0.877439 + 0.744862I
a = 0.409360 + 0.347508I
b = 0.409360 + 0.347508I
2.03717 + 2.82812I −19.0485 − 4.3818I
u = −0.877439 − 0.744862I
a = −1.071720 + 0.909787I
b = −1.071720 + 0.909787I
−5.85852 − 2.82812I −16.5384 + 2.7162I
u = −0.877439 − 0.744862I
a = 0.409360 − 0.347508I
b = 0.409360 − 0.347508I
2.03717 − 2.82812I −19.0485 + 4.3818I
u = 0.754878
a = −0.818721
b = −0.818721
−2.10041 −18.9930
u = 0.754878
a = 2.14344
b = 2.14344
−9.99610 −12.8330
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
((u
3
− u
2
+ 2u − 1)
2
)(u
61
+ 15u
60
+ ··· + 32u + 1)
c
2
((u
3
+ u
2
− 1)
2
)(u
61
+ 3u
60
+ ··· + 6u + 1)
c
3
, c
9
u
6
(u
61
+ u
60
+ ··· + 288u + 64)
c
5
((u
3
− u
2
+ 1)
2
)(u
61
+ 3u
60
+ ··· + 6u + 1)
c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
61
+ 15u
60
+ ··· + 32u + 1)
c
7
, c
8
((u
2
+ u − 1)
3
)(u
61
+ 4u
60
+ ··· + u + 1)
c
10
((u
2
+ u − 1)
3
)(u
61
− 14u
60
+ ··· + 1883u + 233)
c
11
, c
12
((u
2
− u − 1)
3
)(u
61
+ 4u
60
+ ··· + u + 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
((y
3
+ 3y
2
+ 2y −1)
2
)(y
61
+ 65y
60
+ ··· + 360y −1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
2
)(y
61
− 15y
60
+ ··· + 32y −1)
c
3
, c
9
y
6
(y
61
+ 35y
60
+ ··· − 48128y −4096)
c
7
, c
8
, c
11
c
12
((y
2
− 3y + 1)
3
)(y
61
− 70y
60
+ ··· + 33y −1)
c
10
((y
2
− 3y + 1)
3
)(y
61
+ 14y
60
+ ··· − 820731y −54289)
17
