12n
0435
(K12n
0435
)
A knot diagram
1
Linearized knot diagam
3 5 8 10 2 11 3 12 5 9 7 4
Solving Sequence
4,10 5,8
3 2 1 7 9 11 6 12
c
4
c
3
c
2
c
1
c
7
c
9
c
10
c
6
c
12
c
5
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.59109 × 10
70
u
47
+ 3.54965 × 10
69
u
46
+ ··· + 6.67625 × 10
70
b − 1.64708 × 10
71
,
1.45455 × 10
70
u
47
+ 1.03237 × 10
71
u
46
+ ··· + 1.93611 × 10
72
a − 2.56576 × 10
72
, u
48
+ 18u
46
+ ··· + 13u + 1i
I
u
2
= h−2u
18
+ 2u
17
+ ··· + b − 6, −3u
19
+ 4u
18
+ ··· + a + 6, u
20
− u
19
+ ··· − u + 1i
* 2 irreducible components of dim
C
= 0, with total 68 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−1.59 × 10
70
u
47
+ 3.55 × 10
69
u
46
+ · · · + 6.68 × 10
70
b − 1.65 ×
10
71
, 1.45 × 10
70
u
47
+ 1.03 × 10
71
u
46
+ · · · + 1.94 × 10
72
a − 2.57 ×
10
72
, u
48
+ 18u
46
+ · · · + 13u + 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
−u
2
a
8
=
−0.00751276u
47
− 0.0533219u
46
+ ··· − 12.6223u + 1.32521
0.238322u
47
− 0.0531683u
46
+ ··· + 12.7021u + 2.46708
a
3
=
0.553394u
47
− 0.134182u
46
+ ··· + 21.9136u + 6.68234
0.355711u
47
− 0.0573944u
46
+ ··· + 21.1330u + 2.83673
a
2
=
0.171681u
47
− 0.0635417u
46
+ ··· − 0.410390u + 3.71143
0.377230u
47
− 0.0534903u
46
+ ··· + 21.6696u + 2.90737
a
1
=
−1.01739u
47
− 0.00588462u
46
+ ··· − 41.8000u − 8.31915
−0.575290u
47
+ 0.0697542u
46
+ ··· − 22.4788u − 3.42421
a
7
=
−1.22875u
47
+ 0.234716u
46
+ ··· − 73.9234u − 8.65813
−0.0478120u
47
+ 0.0300471u
46
+ ··· − 4.36142u + 1.13699
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
6
=
−1.20137u
47
+ 0.213398u
46
+ ··· − 74.0842u − 8.67754
0.0502293u
47
+ 0.000898209u
46
+ ··· − 2.60784u + 1.37666
a
12
=
−0.442100u
47
− 0.0756388u
46
+ ··· − 19.3212u − 4.89494
−0.575290u
47
+ 0.0697542u
46
+ ··· − 22.4788u − 3.42421
(ii) Obstruction class = −1
(iii) Cusp Shapes = −0.724487u
47
+ 0.0435195u
46
+ ··· − 74.1390u − 1.70909
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
48
+ 67u
47
+ ··· + 604670586u + 32137561
c
2
, c
5
u
48
+ 3u
47
+ ··· + 13144u + 5669
c
3
, c
7
u
48
+ u
47
+ ··· + 12u + 1
c
4
, c
9
u
48
+ 18u
46
+ ··· + 13u + 1
c
6
, c
11
u
48
+ u
47
+ ··· − 2534u + 1167
c
8
u
48
− 3u
47
+ ··· + 3u + 1
c
10
u
48
+ 36u
47
+ ··· − 39u + 1
c
12
u
48
+ 11u
47
+ ··· + 7155479u + 5008881
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
48
− 157y
47
+ ··· + 26058532373273318y + 1032822827028721
c
2
, c
5
y
48
+ 67y
47
+ ··· + 604670586y + 32137561
c
3
, c
7
y
48
− 9y
47
+ ··· − 22y + 1
c
4
, c
9
y
48
+ 36y
47
+ ··· − 39y + 1
c
6
, c
11
y
48
− 53y
47
+ ··· + 12171488y + 1361889
c
8
y
48
− 3y
47
+ ··· + 33y + 1
c
10
y
48
− 36y
47
+ ··· − 575y + 1
c
12
y
48
+ 45y
47
+ ··· + 12614337897293y + 25088888872161
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.624375 + 0.790739I
a = 0.652451 + 0.017302I
b = −0.987977 − 0.377013I
3.11335 + 0.59534I 8.72046 − 1.05598I
u = −0.624375 − 0.790739I
a = 0.652451 − 0.017302I
b = −0.987977 + 0.377013I
3.11335 − 0.59534I 8.72046 + 1.05598I
u = −0.441932 + 0.974715I
a = −1.49605 − 0.17508I
b = 0.527184 − 0.376923I
−3.50502 − 4.85959I −3.75126 + 8.82096I
u = −0.441932 − 0.974715I
a = −1.49605 + 0.17508I
b = 0.527184 + 0.376923I
−3.50502 + 4.85959I −3.75126 − 8.82096I
u = −0.673784 + 0.878510I
a = −1.13342 − 1.57768I
b = 1.000680 − 0.446035I
2.90064 − 5.69851I 8.81108 + 5.67017I
u = −0.673784 − 0.878510I
a = −1.13342 + 1.57768I
b = 1.000680 + 0.446035I
2.90064 + 5.69851I 8.81108 − 5.67017I
u = −0.032468 + 0.886041I
a = 0.75785 + 1.61125I
b = 0.435941 + 0.357265I
−1.78883 + 1.42161I −2.69512 − 4.63338I
u = −0.032468 − 0.886041I
a = 0.75785 − 1.61125I
b = 0.435941 − 0.357265I
−1.78883 − 1.42161I −2.69512 + 4.63338I
u = 0.846760 + 0.741793I
a = 0.152370 − 0.285782I
b = 0.381842 + 0.020417I
0.97742 + 3.01246I 6.86160 − 7.48318I
u = 0.846760 − 0.741793I
a = 0.152370 + 0.285782I
b = 0.381842 − 0.020417I
0.97742 − 3.01246I 6.86160 + 7.48318I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.655494 + 0.992944I
a = 0.422584 − 0.642594I
b = −0.135634 − 0.580747I
0.46539 + 2.67049I 2.00000 − 1.39704I
u = 0.655494 − 0.992944I
a = 0.422584 + 0.642594I
b = −0.135634 + 0.580747I
0.46539 − 2.67049I 2.00000 + 1.39704I
u = 0.514823 + 1.094190I
a = −0.087680 + 0.763881I
b = 0.882038 + 0.252747I
−1.61783 + 4.43732I 2.00000 − 5.85728I
u = 0.514823 − 1.094190I
a = −0.087680 − 0.763881I
b = 0.882038 − 0.252747I
−1.61783 − 4.43732I 2.00000 + 5.85728I
u = 1.227510 + 0.143214I
a = 0.507883 + 0.381592I
b = −0.99480 + 1.00630I
−10.62720 + 0.57436I 0
u = 1.227510 − 0.143214I
a = 0.507883 − 0.381592I
b = −0.99480 − 1.00630I
−10.62720 − 0.57436I 0
u = 0.058897 + 1.244960I
a = 0.47320 − 1.49785I
b = −1.064760 − 0.441088I
−4.30551 − 1.03154I 0
u = 0.058897 − 1.244960I
a = 0.47320 + 1.49785I
b = −1.064760 + 0.441088I
−4.30551 + 1.03154I 0
u = −1.257530 + 0.042178I
a = 0.451983 − 0.394677I
b = −1.01210 − 0.99277I
−10.56360 + 7.91020I 0. − 4.35250I
u = −1.257530 − 0.042178I
a = 0.451983 + 0.394677I
b = −1.01210 + 0.99277I
−10.56360 − 7.91020I 0. + 4.35250I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.081966 + 1.278240I
a = −0.165976 + 1.161460I
b = −1.39247 + 0.57052I
−3.15567 − 4.84062I 0
u = −0.081966 − 1.278240I
a = −0.165976 − 1.161460I
b = −1.39247 − 0.57052I
−3.15567 + 4.84062I 0
u = −0.135469 + 1.297940I
a = −0.45559 + 1.67340I
b = −1.05029 + 1.24464I
−8.08347 − 4.38691I 0
u = −0.135469 − 1.297940I
a = −0.45559 − 1.67340I
b = −1.05029 − 1.24464I
−8.08347 + 4.38691I 0
u = −0.037303 + 0.688643I
a = 1.18323 + 2.50638I
b = 0.513502 + 0.031469I
−1.84510 + 1.37647I −3.10892 − 4.85405I
u = −0.037303 − 0.688643I
a = 1.18323 − 2.50638I
b = 0.513502 − 0.031469I
−1.84510 − 1.37647I −3.10892 + 4.85405I
u = −0.238967 + 1.325580I
a = −0.10468 + 1.41597I
b = −0.389767 + 1.147250I
−6.65584 − 1.72851I 0
u = −0.238967 − 1.325580I
a = −0.10468 − 1.41597I
b = −0.389767 − 1.147250I
−6.65584 + 1.72851I 0
u = 0.076869 + 1.361260I
a = −0.56627 − 1.73054I
b = −0.936074 − 0.890702I
−8.79264 + 3.29679I 0
u = 0.076869 − 1.361260I
a = −0.56627 + 1.73054I
b = −0.936074 + 0.890702I
−8.79264 − 3.29679I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.589410 + 0.100511I
a = 0.567538 − 0.049914I
b = −0.746651 − 0.034665I
1.122130 + 0.019409I 9.50095 − 0.01156I
u = 0.589410 − 0.100511I
a = 0.567538 + 0.049914I
b = −0.746651 + 0.034665I
1.122130 − 0.019409I 9.50095 + 0.01156I
u = −0.501349 + 0.282660I
a = 1.161770 + 0.357456I
b = 0.124161 − 0.520948I
−1.93771 + 1.02545I −1.89001 − 1.40862I
u = −0.501349 − 0.282660I
a = 1.161770 − 0.357456I
b = 0.124161 + 0.520948I
−1.93771 − 1.02545I −1.89001 + 1.40862I
u = 0.21012 + 1.44602I
a = −0.495221 − 1.182570I
b = −0.621138 − 0.800411I
−6.03656 + 5.90639I 0
u = 0.21012 − 1.44602I
a = −0.495221 + 1.182570I
b = −0.621138 + 0.800411I
−6.03656 − 5.90639I 0
u = 0.67234 + 1.40412I
a = −0.52296 + 1.53340I
b = 1.14683 + 0.92820I
−14.5255 + 6.1758I 0
u = 0.67234 − 1.40412I
a = −0.52296 − 1.53340I
b = 1.14683 − 0.92820I
−14.5255 − 6.1758I 0
u = −0.61912 + 1.43345I
a = −0.37937 − 1.57791I
b = 1.16220 − 0.96300I
−14.9365 − 14.5337I 0
u = −0.61912 − 1.43345I
a = −0.37937 + 1.57791I
b = 1.16220 + 0.96300I
−14.9365 + 14.5337I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.50632 + 1.50428I
a = 0.688530 − 0.755252I
b = 0.88540 − 1.18990I
−15.9240 + 6.7617I 0
u = 0.50632 − 1.50428I
a = 0.688530 + 0.755252I
b = 0.88540 + 1.18990I
−15.9240 − 6.7617I 0
u = −0.57393 + 1.51903I
a = 0.650441 + 0.646597I
b = 0.860711 + 1.121730I
−15.5031 + 1.2996I 0
u = −0.57393 − 1.51903I
a = 0.650441 − 0.646597I
b = 0.860711 − 1.121730I
−15.5031 − 1.2996I 0
u = −0.003644 + 0.307346I
a = 3.59056 − 1.52243I
b = 1.030320 + 0.343288I
0.26991 + 4.16666I 3.28056 − 8.29206I
u = −0.003644 − 0.307346I
a = 3.59056 + 1.52243I
b = 1.030320 − 0.343288I
0.26991 − 4.16666I 3.28056 + 8.29206I
u = −0.136716 + 0.053529I
a = 2.64684 − 0.41040I
b = 0.880852 + 0.819546I
−4.05966 + 3.04369I 7.93184 − 4.57606I
u = −0.136716 − 0.053529I
a = 2.64684 + 0.41040I
b = 0.880852 − 0.819546I
−4.05966 − 3.04369I 7.93184 + 4.57606I
9
II.
I
u
2
= h−2u
18
+2u
17
+· · ·+b−6, −3u
19
+4u
18
+· · ·+a+6, u
20
−u
19
+· · ·−u+1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
−u
2
a
8
=
3u
19
− 4u
18
+ ··· − 30u
2
− 6
2u
18
− 2u
17
+ ··· − 6u + 6
a
3
=
−4u
19
+ 4u
18
+ ··· + 2u
2
− u
−4u
19
+ 4u
18
+ ··· − 11u + 1
a
2
=
−3u
19
+ 3u
18
+ ··· + 6u − 1
−4u
19
+ 4u
18
+ ··· − 10u + 1
a
1
=
−3u
19
+ 3u
18
+ ··· + u
2
+ 3u
u
19
− 2u
18
+ ··· + 7u − 5
a
7
=
−u
19
+ u
18
+ ··· − 2u − 5
u
19
+ 2u
18
+ ··· + 34u
2
+ 7
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
6
=
−u
19
+ 2u
18
+ ··· − 3u − 4
u
19
+ 2u
18
+ ··· + 31u
2
+ 6
a
12
=
−4u
19
+ 5u
18
+ ··· − 4u + 5
u
19
− 2u
18
+ ··· + 7u − 5
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 5u
19
− 19u
18
+ 41u
17
− 102u
16
+ 143u
15
− 297u
14
+ 330u
13
− 608u
12
+ 541u
11
−
904u
10
+ 653u
9
− 1017u
8
+ 581u
7
− 856u
6
+ 364u
5
− 527u
4
+ 139u
3
− 204u
2
+ 27u − 40
10
(iv) u-Polynomials at the component
11
Crossings u-Polynomials at each crossing
c
1
u
20
− 18u
19
+ ··· − 12u + 1
c
2
u
20
+ 9u
18
+ ··· + 6u
2
+ 1
c
3
u
20
− 5u
18
+ ··· − 6u
2
+ 1
c
4
u
20
− u
19
+ ··· − u + 1
c
5
u
20
+ 9u
18
+ ··· + 6u
2
+ 1
c
6
u
20
− 5u
18
+ ··· − 2u + 1
c
7
u
20
− 5u
18
+ ··· − 6u
2
+ 1
c
8
u
20
− 4u
19
+ ··· + 3u + 1
c
9
u
20
+ u
19
+ ··· + u + 1
c
10
u
20
+ 11u
19
+ ··· + 15u + 1
c
11
u
20
− 5u
18
+ ··· + 2u + 1
c
12
u
20
− 2u
18
+ ··· + 243u + 67
12
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 18y
19
+ ··· + 8y + 1
c
2
, c
5
y
20
+ 18y
19
+ ··· + 12y + 1
c
3
, c
7
y
20
− 10y
19
+ ··· − 12y + 1
c
4
, c
9
y
20
+ 11y
19
+ ··· + 15y + 1
c
6
, c
11
y
20
− 10y
19
+ ··· − 6y + 1
c
8
y
20
+ 4y
19
+ ··· − 17y + 1
c
10
y
20
+ 7y
19
+ ··· − 13y + 1
c
12
y
20
− 4y
19
+ ··· − 11345y + 4489
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.331553 + 1.017880I
a = −0.263137 − 0.594468I
b = 1.161910 − 0.350994I
−1.07687 − 5.96268I 2.09077 + 8.83123I
u = −0.331553 − 1.017880I
a = −0.263137 + 0.594468I
b = 1.161910 + 0.350994I
−1.07687 + 5.96268I 2.09077 − 8.83123I
u = −0.709835 + 0.819251I
a = 1.48006 + 1.76057I
b = −1.044880 + 0.458438I
2.17630 − 6.02563I −0.56028 + 9.12343I
u = −0.709835 − 0.819251I
a = 1.48006 − 1.76057I
b = −1.044880 − 0.458438I
2.17630 + 6.02563I −0.56028 − 9.12343I
u = 0.796602 + 0.775891I
a = −0.547851 − 0.034126I
b = −0.616511 + 0.410301I
0.67603 + 2.30760I 0.839526 + 0.812523I
u = 0.796602 − 0.775891I
a = −0.547851 + 0.034126I
b = −0.616511 − 0.410301I
0.67603 − 2.30760I 0.839526 − 0.812523I
u = 0.419524 + 1.077190I
a = −0.842248 + 0.809097I
b = 0.707292 − 0.140260I
−3.12109 + 3.82990I −1.85547 − 2.52429I
u = 0.419524 − 1.077190I
a = −0.842248 − 0.809097I
b = 0.707292 + 0.140260I
−3.12109 − 3.82990I −1.85547 + 2.52429I
u = −0.291962 + 0.766600I
a = −1.56953 + 0.25172I
b = −1.075650 − 0.352178I
−0.11610 + 3.27114I −0.199143 − 1.320670I
u = −0.291962 − 0.766600I
a = −1.56953 − 0.25172I
b = −1.075650 + 0.352178I
−0.11610 − 3.27114I −0.199143 + 1.320670I
15
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.713645 + 0.939894I
a = −0.698514 − 0.004869I
b = 1.066520 + 0.507122I
1.80246 + 0.56778I 0.483242 − 0.636305I
u = −0.713645 − 0.939894I
a = −0.698514 + 0.004869I
b = 1.066520 − 0.507122I
1.80246 − 0.56778I 0.483242 + 0.636305I
u = 0.825016 + 0.948148I
a = −0.304719 + 0.539253I
b = 0.574355 + 0.561311I
0.15761 + 3.77342I 1.85491 − 8.11212I
u = 0.825016 − 0.948148I
a = −0.304719 − 0.539253I
b = 0.574355 − 0.561311I
0.15761 − 3.77342I 1.85491 + 8.11212I
u = 0.308877 + 0.668487I
a = 2.53526 − 2.52532I
b = −0.730517 − 0.250987I
−1.51336 − 0.66802I 2.43579 − 3.63747I
u = 0.308877 − 0.668487I
a = 2.53526 + 2.52532I
b = −0.730517 + 0.250987I
−1.51336 + 0.66802I 2.43579 + 3.63747I
u = 0.126858 + 1.307690I
a = −0.50102 − 1.66176I
b = −0.904806 − 1.078820I
−7.62458 + 3.83881I −1.21681 − 1.31905I
u = 0.126858 − 1.307690I
a = −0.50102 + 1.66176I
b = −0.904806 + 1.078820I
−7.62458 − 3.83881I −1.21681 + 1.31905I
u = 0.070116 + 0.565143I
a = −0.788301 − 0.715423I
b = 0.862292 − 0.773239I
−4.51987 − 2.92216I −8.87254 + 0.70432I
u = 0.070116 − 0.565143I
a = −0.788301 + 0.715423I
b = 0.862292 + 0.773239I
−4.51987 + 2.92216I −8.87254 − 0.70432I
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
20
− 18u
19
+ ··· − 12u + 1)
· (u
48
+ 67u
47
+ ··· + 604670586u + 32137561)
c
2
(u
20
+ 9u
18
+ ··· + 6u
2
+ 1)(u
48
+ 3u
47
+ ··· + 13144u + 5669)
c
3
(u
20
− 5u
18
+ ··· − 6u
2
+ 1)(u
48
+ u
47
+ ··· + 12u + 1)
c
4
(u
20
− u
19
+ ··· − u + 1)(u
48
+ 18u
46
+ ··· + 13u + 1)
c
5
(u
20
+ 9u
18
+ ··· + 6u
2
+ 1)(u
48
+ 3u
47
+ ··· + 13144u + 5669)
c
6
(u
20
− 5u
18
+ ··· − 2u + 1)(u
48
+ u
47
+ ··· − 2534u + 1167)
c
7
(u
20
− 5u
18
+ ··· − 6u
2
+ 1)(u
48
+ u
47
+ ··· + 12u + 1)
c
8
(u
20
− 4u
19
+ ··· + 3u + 1)(u
48
− 3u
47
+ ··· + 3u + 1)
c
9
(u
20
+ u
19
+ ··· + u + 1)(u
48
+ 18u
46
+ ··· + 13u + 1)
c
10
(u
20
+ 11u
19
+ ··· + 15u + 1)(u
48
+ 36u
47
+ ··· − 39u + 1)
c
11
(u
20
− 5u
18
+ ··· + 2u + 1)(u
48
+ u
47
+ ··· − 2534u + 1167)
c
12
(u
20
− 2u
18
+ ··· + 243u + 67)
· (u
48
+ 11u
47
+ ··· + 7155479u + 5008881)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
20
− 18y
19
+ ··· + 8y + 1)
· (y
48
− 157y
47
+ ··· + 26058532373273318y + 1032822827028721)
c
2
, c
5
(y
20
+ 18y
19
+ ··· + 12y + 1)
· (y
48
+ 67y
47
+ ··· + 604670586y + 32137561)
c
3
, c
7
(y
20
− 10y
19
+ ··· − 12y + 1)(y
48
− 9y
47
+ ··· − 22y + 1)
c
4
, c
9
(y
20
+ 11y
19
+ ··· + 15y + 1)(y
48
+ 36y
47
+ ··· − 39y + 1)
c
6
, c
11
(y
20
− 10y
19
+ ··· − 6y + 1)
· (y
48
− 53y
47
+ ··· + 12171488y + 1361889)
c
8
(y
20
+ 4y
19
+ ··· − 17y + 1)(y
48
− 3y
47
+ ··· + 33y + 1)
c
10
(y
20
+ 7y
19
+ ··· − 13y + 1)(y
48
− 36y
47
+ ··· − 575y + 1)
c
12
(y
20
− 4y
19
+ ··· − 11345y + 4489)
· (y
48
+ 45y
47
+ ··· + 12614337897293y + 25088888872161)
18
