12a
0619
(K12a
0619
)
A knot diagram
1
Linearized knot diagam
3 7 9 1 12 11 2 10 4 8 6 5
Solving Sequence
1,4 5,9
10 3 2 8 7 12 6 11
c
4
c
9
c
3
c
1
c
8
c
7
c
12
c
5
c
11
c
2
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.85075 × 10
26
u
54
− 1.82177 × 10
26
u
53
+ ··· + 5.53593 × 10
26
b + 1.34110 × 10
27
,
2.89480 × 10
27
u
54
+ 2.68152 × 10
27
u
53
+ ··· + 5.53593 × 10
26
a − 2.16054 × 10
28
, u
55
+ u
54
+ ··· − 12u − 1i
I
u
2
= h−u
3
a + u
2
a + 5u
3
− 4au + 5b − a + 10u, −u
3
a − u
2
a + 2u
3
+ a
2
− 2au + u
2
− 2a + 6u + 2, u
4
+ 3u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 63 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−1.85×10
26
u
54
−1.82×10
26
u
53
+· · ·+5.54×10
26
b+1.34×10
27
, 2.89×
10
27
u
54
+2.68×10
27
u
53
+· · ·+5.54×10
26
a−2.16×10
28
, u
55
+u
54
+· · ·−12u−1i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
−5.22911u
54
− 4.84385u
53
+ ··· + 184.977u + 39.0276
0.334317u
54
+ 0.329082u
53
+ ··· − 18.1099u − 2.42254
a
10
=
−5.56342u
54
− 5.17293u
53
+ ··· + 203.087u + 41.4501
0.334317u
54
+ 0.329082u
53
+ ··· − 18.1099u − 2.42254
a
3
=
2.01049u
54
+ 1.47507u
53
+ ··· − 60.3680u − 18.1292
−1.17670u
54
− 0.737240u
53
+ ··· + 28.3449u + 5.00114
a
2
=
−3.21401u
54
− 2.21018u
53
+ ··· + 93.1627u + 24.4667
1.00340u
54
+ 0.579236u
53
+ ··· − 25.9928u − 4.57326
a
8
=
−6.41607u
54
− 5.63866u
53
+ ··· + 226.145u + 38.8714
−0.416211u
54
− 0.438802u
53
+ ··· + 21.9914u + 4.79282
a
7
=
u
4
+ 3u
2
+ 1
u
6
+ 4u
4
+ 3u
2
a
12
=
u
u
3
+ u
a
6
=
u
2
+ 1
u
4
+ 2u
2
a
11
=
u
3
+ 2u
u
5
+ 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1012232232711983767701675065
276796444336859447752049744
u
54
+
676329581383746073717624245
276796444336859447752049744
u
53
+
··· −
6753618946943674742901539099
69199111084214861938012436
u −
9459757630078072988532524263
276796444336859447752049744
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
55
+ 23u
54
+ ··· + 88u − 16
c
2
, c
7
u
55
+ u
54
+ ··· − 8u + 4
c
3
, c
9
u
55
+ u
54
+ ··· + 6u + 5
c
4
, c
5
, c
6
c
11
, c
12
u
55
− u
54
+ ··· − 12u + 1
c
8
, c
10
u
55
+ 17u
54
+ ··· + 56u + 25
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
55
+ 27y
54
+ ··· + 50976y −256
c
2
, c
7
y
55
+ 23y
54
+ ··· + 88y −16
c
3
, c
9
y
55
− 17y
54
+ ··· + 56y −25
c
4
, c
5
, c
6
c
11
, c
12
y
55
+ 75y
54
+ ··· + 84y −1
c
8
, c
10
y
55
+ 47y
54
+ ··· + 41236y −625
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.242174 + 0.965422I
a = 1.84242 + 0.74241I
b = 1.088040 − 0.231202I
−0.05193 + 5.45928I 0
u = −0.242174 − 0.965422I
a = 1.84242 − 0.74241I
b = 1.088040 + 0.231202I
−0.05193 − 5.45928I 0
u = −0.054318 + 0.978179I
a = 0.29476 − 1.79156I
b = −0.865648 + 0.695285I
1.45344 + 2.67085I 0
u = −0.054318 − 0.978179I
a = 0.29476 + 1.79156I
b = −0.865648 − 0.695285I
1.45344 − 2.67085I 0
u = 0.118045 + 1.055120I
a = −0.217543 + 0.039268I
b = −0.093474 − 0.707979I
3.78484 − 2.38987I 0
u = 0.118045 − 1.055120I
a = −0.217543 − 0.039268I
b = −0.093474 + 0.707979I
3.78484 + 2.38987I 0
u = −0.025625 + 0.906620I
a = −1.46648 + 0.30670I
b = −1.021430 − 0.412711I
0.99023 − 1.37586I −3.05121 + 0.I
u = −0.025625 − 0.906620I
a = −1.46648 − 0.30670I
b = −1.021430 + 0.412711I
0.99023 + 1.37586I −3.05121 + 0.I
u = −0.391057 + 1.070450I
a = −1.53923 − 1.26504I
b = −1.015910 + 0.768566I
6.48872 + 11.16090I 0
u = −0.391057 − 1.070450I
a = −1.53923 + 1.26504I
b = −1.015910 − 0.768566I
6.48872 − 11.16090I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.352898 + 1.104100I
a = 0.266006 + 0.266601I
b = −0.735018 + 0.866010I
7.35517 − 5.07871I 0
u = 0.352898 − 1.104100I
a = 0.266006 − 0.266601I
b = −0.735018 − 0.866010I
7.35517 + 5.07871I 0
u = 0.286912 + 1.149410I
a = 1.08185 − 1.14553I
b = 0.950341 + 0.788022I
8.18462 − 5.12776I 0
u = 0.286912 − 1.149410I
a = 1.08185 + 1.14553I
b = 0.950341 − 0.788022I
8.18462 + 5.12776I 0
u = −0.225018 + 1.198970I
a = −0.0360231 − 0.1138210I
b = 0.817032 + 0.826405I
8.59101 − 0.90551I 0
u = −0.225018 − 1.198970I
a = −0.0360231 + 0.1138210I
b = 0.817032 − 0.826405I
8.59101 + 0.90551I 0
u = −0.566330 + 0.506423I
a = −0.847540 − 0.749555I
b = −0.903233 − 0.759567I
3.06251 − 3.58876I −4.22276 + 2.27079I
u = −0.566330 − 0.506423I
a = −0.847540 + 0.749555I
b = −0.903233 + 0.759567I
3.06251 + 3.58876I −4.22276 − 2.27079I
u = −0.260846 + 0.684203I
a = 1.25638 + 1.88980I
b = 0.786041 + 0.088863I
−1.81365 − 0.48899I −10.00862 − 1.42707I
u = −0.260846 − 0.684203I
a = 1.25638 − 1.88980I
b = 0.786041 − 0.088863I
−1.81365 + 0.48899I −10.00862 + 1.42707I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.578616 + 0.424067I
a = −1.86336 + 0.36259I
b = −0.859374 − 0.768588I
3.19815 − 2.18751I −4.02245 + 3.31324I
u = 0.578616 − 0.424067I
a = −1.86336 − 0.36259I
b = −0.859374 + 0.768588I
3.19815 + 2.18751I −4.02245 − 3.31324I
u = −0.654791 + 0.269416I
a = 2.28996 + 0.24061I
b = 0.965160 − 0.755306I
2.33083 + 7.60802I −6.45547 − 7.83198I
u = −0.654791 − 0.269416I
a = 2.28996 − 0.24061I
b = 0.965160 + 0.755306I
2.33083 − 7.60802I −6.45547 + 7.83198I
u = 0.622881 + 0.326038I
a = 0.734126 − 0.979583I
b = 0.784366 − 0.800341I
2.88441 − 1.75138I −4.95873 + 3.04143I
u = 0.622881 − 0.326038I
a = 0.734126 + 0.979583I
b = 0.784366 + 0.800341I
2.88441 + 1.75138I −4.95873 − 3.04143I
u = 0.160604 + 0.679165I
a = −0.997813 + 0.742503I
b = −0.743529 − 0.424149I
1.13258 − 1.82586I −0.68429 + 5.26457I
u = 0.160604 − 0.679165I
a = −0.997813 − 0.742503I
b = −0.743529 + 0.424149I
1.13258 + 1.82586I −0.68429 − 5.26457I
u = −0.466981 + 0.130221I
a = −3.35094 + 0.41526I
b = −0.962907 + 0.180222I
−3.42043 + 3.05241I −14.8881 − 6.0447I
u = −0.466981 − 0.130221I
a = −3.35094 − 0.41526I
b = −0.962907 − 0.180222I
−3.42043 − 3.05241I −14.8881 + 6.0447I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.298953 + 0.304108I
a = 0.473629 + 0.940791I
b = 0.076869 + 0.393888I
−0.455535 − 1.051460I −6.61895 + 6.28679I
u = 0.298953 − 0.304108I
a = 0.473629 − 0.940791I
b = 0.076869 − 0.393888I
−0.455535 + 1.051460I −6.61895 − 6.28679I
u = 0.01077 + 1.58006I
a = 0.756237 − 0.449907I
b = 0.812235 + 0.598969I
8.74683 − 2.33878I 0
u = 0.01077 − 1.58006I
a = 0.756237 + 0.449907I
b = 0.812235 − 0.598969I
8.74683 + 2.33878I 0
u = −0.04280 + 1.62916I
a = −0.857697 − 1.116840I
b = −0.725530 + 0.107591I
6.26450 + 0.46865I 0
u = −0.04280 − 1.62916I
a = −0.857697 + 1.116840I
b = −0.725530 − 0.107591I
6.26450 − 0.46865I 0
u = 0.366996
a = 2.18866
b = 0.695683
−0.945108 −11.2280
u = 0.00008 + 1.71308I
a = 1.247670 − 0.423656I
b = 1.129780 + 0.387208I
10.44000 − 1.33130I 0
u = 0.00008 − 1.71308I
a = 1.247670 + 0.423656I
b = 1.129780 − 0.387208I
10.44000 + 1.33130I 0
u = −0.05786 + 1.71749I
a = −1.41471 − 0.54631I
b = −1.169820 + 0.244407I
9.52562 + 6.62877I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.05786 − 1.71749I
a = −1.41471 + 0.54631I
b = −1.169820 − 0.244407I
9.52562 − 6.62877I 0
u = −0.01271 + 1.72544I
a = −0.088994 + 1.301550I
b = 0.885426 − 0.778612I
11.19810 + 2.93315I 0
u = −0.01271 − 1.72544I
a = −0.088994 − 1.301550I
b = 0.885426 + 0.778612I
11.19810 − 2.93315I 0
u = 0.02762 + 1.74047I
a = 0.102112 − 0.218682I
b = 0.093171 + 0.855980I
13.8627 − 2.9756I 0
u = 0.02762 − 1.74047I
a = 0.102112 + 0.218682I
b = 0.093171 − 0.855980I
13.8627 + 2.9756I 0
u = −0.10658 + 1.74213I
a = 1.06647 + 1.20290I
b = 1.054570 − 0.779432I
16.4678 + 13.2408I 0
u = −0.10658 − 1.74213I
a = 1.06647 − 1.20290I
b = 1.054570 + 0.779432I
16.4678 − 13.2408I 0
u = 0.09329 + 1.75197I
a = −0.270695 + 0.136458I
b = 0.708239 − 0.923393I
17.5515 − 6.9610I 0
u = 0.09329 − 1.75197I
a = −0.270695 − 0.136458I
b = 0.708239 + 0.923393I
17.5515 + 6.9610I 0
u = 0.07231 + 1.76162I
a = −0.736108 + 1.089440I
b = −1.006010 − 0.819655I
18.6505 − 6.6546I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.07231 − 1.76162I
a = −0.736108 − 1.089440I
b = −1.006010 + 0.819655I
18.6505 + 6.6546I 0
u = −0.05292 + 1.76973I
a = 0.111166 + 0.401269I
b = −0.795250 − 0.910655I
19.3144 + 0.2705I 0
u = −0.05292 − 1.76973I
a = 0.111166 − 0.401269I
b = −0.795250 + 0.910655I
19.3144 − 0.2705I 0
u = −0.146463 + 0.018325I
a = 5.57002 + 5.35103I
b = 0.898022 − 0.525426I
−1.72390 + 2.04138I −15.5710 − 3.1366I
u = −0.146463 − 0.018325I
a = 5.57002 − 5.35103I
b = 0.898022 + 0.525426I
−1.72390 − 2.04138I −15.5710 + 3.1366I
10
II. I
u
2
=
h−u
3
a+u
2
a+5u
3
−4au+5b−a +10u, −u
3
a+2u
3
+· · ·−2a+2, u
4
+3 u
2
+1 i
(i) Arc colorings
a
1
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
a
1
5
u
3
a − u
3
+ ··· +
1
5
a − 2u
a
10
=
−
1
5
u
3
a + u
3
+ ··· +
4
5
a + 2u
1
5
u
3
a − u
3
+ ··· +
1
5
a − 2u
a
3
=
u
3
+ 3u
2
5
u
3
a −
2
5
u
2
a +
3
5
au −
3
5
a
a
2
=
u
3
+ 3u
2
5
u
3
a −
2
5
u
2
a + ··· −
3
5
a + u
a
8
=
u
2
+ 2
1
5
u
3
a −
1
5
u
2
a +
4
5
au +
1
5
a
a
7
=
0
−u
2
− 1
a
12
=
u
u
3
+ u
a
6
=
u
2
+ 1
−u
2
− 1
a
11
=
u
3
+ 2u
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
8
5
u
3
a +
8
5
u
2
a −
12
5
au +
12
5
a − 8
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
8
c
2
, c
7
(u
2
+ 1)
4
c
3
, c
9
(u
4
− u
2
+ 1)
2
c
4
, c
5
, c
6
c
11
, c
12
(u
4
+ 3u
2
+ 1)
2
c
8
(u
2
− u + 1)
4
c
10
(u
2
+ u + 1)
4
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
8
c
2
, c
7
(y + 1)
8
c
3
, c
9
(y
2
− y + 1)
4
c
4
, c
5
, c
6
c
11
, c
12
(y
2
+ 3y + 1)
4
c
8
, c
10
(y
2
+ y + 1)
4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034I
a = 1.67504 − 0.90126I
b = 0.866025 − 0.500000I
−0.65797 + 2.02988I −6.00000 − 3.46410I
u = 0.618034I
a = −0.05701 + 1.90126I
b = −0.866025 − 0.500000I
−0.65797 − 2.02988I −6.00000 + 3.46410I
u = − 0.618034I
a = 1.67504 + 0.90126I
b = 0.866025 + 0.500000I
−0.65797 − 2.02988I −6.00000 + 3.46410I
u = − 0.618034I
a = −0.05701 − 1.90126I
b = −0.866025 + 0.500000I
−0.65797 + 2.02988I −6.00000 − 3.46410I
u = 1.61803I
a = −1.175040 + 0.035233I
b = −0.866025 + 0.500000I
7.23771 − 2.02988I −6.00000 + 3.46410I
u = 1.61803I
a = 0.557008 − 1.035230I
b = 0.866025 + 0.500000I
7.23771 + 2.02988I −6.00000 − 3.46410I
u = − 1.61803I
a = −1.175040 − 0.035233I
b = −0.866025 − 0.500000I
7.23771 + 2.02988I −6.00000 − 3.46410I
u = − 1.61803I
a = 0.557008 + 1.035230I
b = 0.866025 − 0.500000I
7.23771 − 2.02988I −6.00000 + 3.46410I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
55
+ 23u
54
+ ··· + 88u − 16)
c
2
, c
7
((u
2
+ 1)
4
)(u
55
+ u
54
+ ··· − 8u + 4)
c
3
, c
9
((u
4
− u
2
+ 1)
2
)(u
55
+ u
54
+ ··· + 6u + 5)
c
4
, c
5
, c
6
c
11
, c
12
((u
4
+ 3u
2
+ 1)
2
)(u
55
− u
54
+ ··· − 12u + 1)
c
8
((u
2
− u + 1)
4
)(u
55
+ 17u
54
+ ··· + 56u + 25)
c
10
((u
2
+ u + 1)
4
)(u
55
+ 17u
54
+ ··· + 56u + 25)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
55
+ 27y
54
+ ··· + 50976y −256)
c
2
, c
7
((y + 1)
8
)(y
55
+ 23y
54
+ ··· + 88y −16)
c
3
, c
9
((y
2
− y + 1)
4
)(y
55
− 17y
54
+ ··· + 56y −25)
c
4
, c
5
, c
6
c
11
, c
12
((y
2
+ 3y + 1)
4
)(y
55
+ 75y
54
+ ··· + 84y −1)
c
8
, c
10
((y
2
+ y + 1)
4
)(y
55
+ 47y
54
+ ··· + 41236y −625)
16
