12n
0561
(K12n
0561
)
A knot diagram
1
Linearized knot diagam
3 7 12 8 11 2 10 12 3 6 5 10
Solving Sequence
2,6
7
3,10
8 11 1 5 4 9 12
c
6
c
2
c
7
c
10
c
1
c
5
c
4
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h19514643174787u
34
− 4572969237385u
33
+ ··· + 35208391096192b − 115134997157437,
− 44181034772241u
34
− 93208096612611u
33
+ ··· + 176041955480960a − 34686663928811,
u
35
+ u
34
+ ··· − 4u − 5i
I
u
2
= h−u
2
a − u
2
+ b + a, u
3
a − 2u
2
a + u
3
+ a
2
− au + 2a − 1, u
4
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 43 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
= h1.95×10
13
u
34
−4.57×10
12
u
33
+· · ·+3.52×10
13
b−1.15×10
14
, −4.42×
10
13
u
34
−9.32×10
13
u
33
+· · ·+1.76×10
14
a−3.47×10
13
, u
35
+u
34
+· · ·−4u−5i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
10
=
0.250969u
34
+ 0.529465u
33
+ ··· + 2.80218u + 0.197036
−0.554261u
34
+ 0.129883u
33
+ ··· + 0.868164u + 3.27010
a
8
=
0.791772u
34
+ 0.423142u
33
+ ··· + 3.34243u − 2.71039
−0.523385u
34
+ 0.0869282u
33
+ ··· + 0.937584u + 2.29735
a
11
=
0.805230u
34
+ 0.399582u
33
+ ··· + 1.93402u − 3.07306
−0.554261u
34
+ 0.129883u
33
+ ··· + 0.868164u + 3.27010
a
1
=
u
3
u
5
− u
3
+ u
a
5
=
−0.472936u
34
+ 0.0799123u
33
+ ··· + 0.438637u + 2.87741
−0.556115u
34
− 0.173472u
33
+ ··· − 2.34087u + 2.38011
a
4
=
−0.816166u
34
+ 0.145199u
33
+ ··· + 1.79451u + 3.42292
−1.06658u
34
− 0.102045u
33
+ ··· − 3.07330u + 6.87995
a
9
=
0.601350u
34
+ 0.306853u
33
+ ··· + 2.54938u − 1.70567
−0.455033u
34
+ 0.135445u
33
+ ··· + 0.580899u + 2.30784
a
12
=
−0.528222u
34
+ 0.0504600u
33
+ ··· − 1.61488u + 4.49023
0.662513u
34
+ 0.0170577u
33
+ ··· + 1.34187u − 2.86220
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
3308824036235
17604195548096
u
34
+
2338411391075
17604195548096
u
33
+ ··· +
68902332437973
8802097774048
u −
143545741438237
17604195548096
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
35
+ 21u
34
+ ··· − 74u + 25
c
2
, c
6
u
35
− u
34
+ ··· − 4u + 5
c
3
u
35
− 5u
34
+ ··· + 296u + 28
c
4
, c
9
u
35
− u
34
+ ··· − 16u + 4
c
5
, c
10
, c
11
u
35
+ u
34
+ ··· − 6u + 1
c
7
u
35
− 3u
34
+ ··· + 6720u − 1472
c
8
u
35
− 3u
34
+ ··· + 6u + 5
c
12
u
35
+ 3u
34
+ ··· + 12u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
35
− 9y
34
+ ··· + 39626y −625
c
2
, c
6
y
35
− 21y
34
+ ··· − 74y −25
c
3
y
35
− 57y
34
+ ··· − 32784y −784
c
4
, c
9
y
35
+ 43y
34
+ ··· + 184y −16
c
5
, c
10
, c
11
y
35
+ 39y
34
+ ··· + 34y −1
c
7
y
35
− 31y
34
+ ··· + 22678016y −2166784
c
8
y
35
+ 39y
34
+ ··· + 126y −25
c
12
y
35
+ 51y
34
+ ··· + 154y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.203911 + 0.973360I
a = −0.519106 + 0.963683I
b = −0.26045 − 1.53546I
−1.43478 − 6.20512I −1.86669 + 2.78198I
u = −0.203911 − 0.973360I
a = −0.519106 − 0.963683I
b = −0.26045 + 1.53546I
−1.43478 + 6.20512I −1.86669 − 2.78198I
u = 0.066320 + 0.984954I
a = −0.448541 − 0.317717I
b = −0.750046 + 0.533829I
−8.19601 + 2.49744I −5.06408 − 2.60195I
u = 0.066320 − 0.984954I
a = −0.448541 + 0.317717I
b = −0.750046 − 0.533829I
−8.19601 − 2.49744I −5.06408 + 2.60195I
u = 0.801249 + 0.559810I
a = 1.36028 − 1.03030I
b = 0.01451 + 1.59501I
8.83899 − 2.24678I 4.64462 + 3.81835I
u = 0.801249 − 0.559810I
a = 1.36028 + 1.03030I
b = 0.01451 − 1.59501I
8.83899 + 2.24678I 4.64462 − 3.81835I
u = 1.014440 + 0.137733I
a = −1.44237 + 0.41096I
b = −0.391237 + 0.679863I
−2.29834 − 0.76813I −7.21756 − 0.78239I
u = 1.014440 − 0.137733I
a = −1.44237 − 0.41096I
b = −0.391237 − 0.679863I
−2.29834 + 0.76813I −7.21756 + 0.78239I
u = 0.910408 + 0.541398I
a = −0.378721 + 0.991836I
b = −0.212665 − 0.039123I
−1.75949 − 2.05479I −11.09061 + 3.13209I
u = 0.910408 − 0.541398I
a = −0.378721 − 0.991836I
b = −0.212665 + 0.039123I
−1.75949 + 2.05479I −11.09061 − 3.13209I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.922478 + 0.127300I
a = −1.96495 + 0.18361I
b = −0.05491 + 1.64507I
5.90539 + 0.64255I −5.75392 + 0.64646I
u = −0.922478 − 0.127300I
a = −1.96495 − 0.18361I
b = −0.05491 − 1.64507I
5.90539 − 0.64255I −5.75392 − 0.64646I
u = −0.812370 + 0.424392I
a = 0.773162 + 0.362489I
b = 0.109550 − 0.685718I
1.05382 + 1.86722I 3.18128 − 4.59075I
u = −0.812370 − 0.424392I
a = 0.773162 − 0.362489I
b = 0.109550 + 0.685718I
1.05382 − 1.86722I 3.18128 + 4.59075I
u = 0.912261
a = 1.05610
b = 0.463900
−1.34236 −8.12570
u = −0.842169 + 0.777146I
a = −0.89892 − 1.28817I
b = −0.050273 + 1.397230I
3.07114 + 2.89052I −2.21100 − 2.95071I
u = −0.842169 − 0.777146I
a = −0.89892 + 1.28817I
b = −0.050273 − 1.397230I
3.07114 − 2.89052I −2.21100 + 2.95071I
u = −1.137850 + 0.335834I
a = −1.53286 + 0.05803I
b = −0.587577 + 0.359881I
−3.27817 + 4.26600I −8.05552 − 6.74990I
u = −1.137850 − 0.335834I
a = −1.53286 − 0.05803I
b = −0.587577 − 0.359881I
−3.27817 − 4.26600I −8.05552 + 6.74990I
u = −1.152990 + 0.388555I
a = 0.878906 − 1.034180I
b = 0.002872 − 1.337860I
2.24498 + 1.33286I −3.24945 − 0.69819I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.152990 − 0.388555I
a = 0.878906 + 1.034180I
b = 0.002872 + 1.337860I
2.24498 − 1.33286I −3.24945 + 0.69819I
u = 0.098226 + 0.753815I
a = 0.468379 + 0.420561I
b = 0.09057 − 1.46639I
5.88674 + 2.52249I 1.23506 − 2.87985I
u = 0.098226 − 0.753815I
a = 0.468379 − 0.420561I
b = 0.09057 + 1.46639I
5.88674 − 2.52249I 1.23506 + 2.87985I
u = 1.194500 + 0.470305I
a = −1.80401 − 0.60694I
b = −0.19201 − 1.46324I
2.66330 − 7.05772I −2.51905 + 5.88692I
u = 1.194500 − 0.470305I
a = −1.80401 + 0.60694I
b = −0.19201 + 1.46324I
2.66330 + 7.05772I −2.51905 − 5.88692I
u = 1.330550 + 0.340353I
a = 0.175592 − 0.683458I
b = 0.33732 − 1.47223I
−6.45071 + 1.73894I −5.86337 − 0.49500I
u = 1.330550 − 0.340353I
a = 0.175592 + 0.683458I
b = 0.33732 + 1.47223I
−6.45071 − 1.73894I −5.86337 + 0.49500I
u = −1.252450 + 0.585255I
a = 1.89880 + 0.08113I
b = 0.27080 − 1.59119I
−4.64525 + 11.83980I −4.29756 − 5.89282I
u = −1.252450 − 0.585255I
a = 1.89880 − 0.08113I
b = 0.27080 + 1.59119I
−4.64525 − 11.83980I −4.29756 + 5.89282I
u = −1.326130 + 0.448294I
a = 0.903307 + 0.583518I
b = 0.842224 + 0.446428I
−12.58030 + 2.55008I −8.43526 − 0.60252I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.326130 − 0.448294I
a = 0.903307 − 0.583518I
b = 0.842224 − 0.446428I
−12.58030 − 2.55008I −8.43526 + 0.60252I
u = 1.297550 + 0.525631I
a = 1.45281 − 0.36988I
b = 0.780540 + 0.640043I
−11.9977 − 7.9047I −7.36295 + 5.50788I
u = 1.297550 − 0.525631I
a = 1.45281 + 0.36988I
b = 0.780540 − 0.640043I
−11.9977 + 7.9047I −7.36295 − 5.50788I
u = −0.019019 + 0.438455I
a = 0.850181 + 0.183537I
b = 0.318829 + 0.391878I
−0.204022 − 1.109310I −3.01108 + 6.25704I
u = −0.019019 − 0.438455I
a = 0.850181 − 0.183537I
b = 0.318829 − 0.391878I
−0.204022 + 1.109310I −3.01108 − 6.25704I
8
II. I
u
2
= h−u
2
a − u
2
+ b + a, u
3
a − 2u
2
a + u
3
+ a
2
− au + 2a − 1, u
4
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
10
=
a
u
2
a + u
2
− a
a
8
=
u
3
a − u
2
a + u
3
− au + 2a
−u
3
a + u
2
a + u
2
− a − u
a
11
=
−u
2
a − u
2
+ 2a
u
2
a + u
2
− a
a
1
=
u
3
0
a
5
=
−u
3
a + u
2
a + u
3
− au − u
2
− 2u + 1
−u
3
+ au + u + 1
a
4
=
−2u
3
a + 2u
2
a + u
3
− 2au − u
2
− 2u + 2
−2u
3
+ 2au + u
2
− a + 2u
a
9
=
−u
2
+ a + 1
u
2
a + 2u
2
− a
a
12
=
−u
3
a + u
3
+ au + u
2
− a − u − 1
−u
2
a − u
3
− u
2
+ a
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 4
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
4
c
2
, c
6
, c
8
(u
4
− u
2
+ 1)
2
c
3
u
8
+ 4u
7
+ 16u
6
+ 34u
5
+ 57u
4
+ 62u
3
+ 46u
2
+ 20u + 4
c
4
, c
9
(u
2
+ 1)
4
c
5
, c
10
, c
11
(u
4
+ 3u
2
+ 1)
2
c
7
u
8
− 8u
7
+ 25u
6
− 38u
5
+ 33u
4
− 28u
3
+ 28u
2
− 16u + 4
c
12
(u
2
+ u − 1)
4
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)
4
c
2
, c
6
, c
8
(y
2
− y + 1)
4
c
3
y
8
+ 16y
7
+ 98y
6
+ 264y
5
+ 353y
4
+ 168y
3
+ 92y
2
− 32y + 16
c
4
, c
9
(y + 1)
8
c
5
, c
10
, c
11
(y
2
+ 3y + 1)
4
c
7
y
8
− 14y
7
+ 83y
6
− 186y
5
+ 113y
4
+ 48y
3
+ 152y
2
− 32y + 16
c
12
(y
2
− 3y + 1)
4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.866025 + 0.500000I
a = 0.901259 + 0.057008I
b = 1.61803I
7.23771 − 2.02988I −2.00000 + 3.46410I
u = 0.866025 + 0.500000I
a = −1.03523 + 1.17504I
b = − 0.618034I
−0.65797 − 2.02988I −2.00000 + 3.46410I
u = 0.866025 − 0.500000I
a = 0.901259 − 0.057008I
b = − 1.61803I
7.23771 + 2.02988I −2.00000 − 3.46410I
u = 0.866025 − 0.500000I
a = −1.03523 − 1.17504I
b = 0.618034I
−0.65797 + 2.02988I −2.00000 − 3.46410I
u = −0.866025 + 0.500000I
a = 0.035233 − 0.557008I
b = − 0.618034I
−0.65797 + 2.02988I −2.00000 − 3.46410I
u = −0.866025 + 0.500000I
a = −1.90126 − 1.67504I
b = 1.61803I
7.23771 + 2.02988I −2.00000 − 3.46410I
u = −0.866025 − 0.500000I
a = 0.035233 + 0.557008I
b = 0.618034I
−0.65797 − 2.02988I −2.00000 + 3.46410I
u = −0.866025 − 0.500000I
a = −1.90126 + 1.67504I
b = − 1.61803I
7.23771 − 2.02988I −2.00000 + 3.46410I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
35
+ 21u
34
+ ··· − 74u + 25)
c
2
, c
6
((u
4
− u
2
+ 1)
2
)(u
35
− u
34
+ ··· − 4u + 5)
c
3
(u
8
+ 4u
7
+ 16u
6
+ 34u
5
+ 57u
4
+ 62u
3
+ 46u
2
+ 20u + 4)
· (u
35
− 5u
34
+ ··· + 296u + 28)
c
4
, c
9
((u
2
+ 1)
4
)(u
35
− u
34
+ ··· − 16u + 4)
c
5
, c
10
, c
11
((u
4
+ 3u
2
+ 1)
2
)(u
35
+ u
34
+ ··· − 6u + 1)
c
7
(u
8
− 8u
7
+ 25u
6
− 38u
5
+ 33u
4
− 28u
3
+ 28u
2
− 16u + 4)
· (u
35
− 3u
34
+ ··· + 6720u − 1472)
c
8
((u
4
− u
2
+ 1)
2
)(u
35
− 3u
34
+ ··· + 6u + 5)
c
12
((u
2
+ u − 1)
4
)(u
35
+ 3u
34
+ ··· + 12u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
35
− 9y
34
+ ··· + 39626y −625)
c
2
, c
6
((y
2
− y + 1)
4
)(y
35
− 21y
34
+ ··· − 74y −25)
c
3
(y
8
+ 16y
7
+ 98y
6
+ 264y
5
+ 353y
4
+ 168y
3
+ 92y
2
− 32y + 16)
· (y
35
− 57y
34
+ ··· − 32784y −784)
c
4
, c
9
((y + 1)
8
)(y
35
+ 43y
34
+ ··· + 184y −16)
c
5
, c
10
, c
11
((y
2
+ 3y + 1)
4
)(y
35
+ 39y
34
+ ··· + 34y −1)
c
7
(y
8
− 14y
7
+ 83y
6
− 186y
5
+ 113y
4
+ 48y
3
+ 152y
2
− 32y + 16)
· (y
35
− 31y
34
+ ··· + 22678016y −2166784)
c
8
((y
2
− y + 1)
4
)(y
35
+ 39y
34
+ ··· + 126y −25)
c
12
((y
2
− 3y + 1)
4
)(y
35
+ 51y
34
+ ··· + 154y −1)
14
