12a
0941
(K12a
0941
)
A knot diagram
1
Linearized knot diagam
4 6 9 1 10 3 11 12 2 7 8 5
Solving Sequence
7,11
8 12
3,9
4 6 2 1 10 5
c
7
c
11
c
8
c
3
c
6
c
2
c
1
c
10
c
5
c
4
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.42999 × 10
69
u
73
+ 6.42092 × 10
69
u
72
+ ··· + 1.65516 × 10
69
b + 1.02869 × 10
70
,
5.07268 × 10
70
u
73
+ 2.49297 × 10
71
u
72
+ ··· + 1.48964 × 10
70
a + 6.53143 × 10
71
, u
74
6u
73
+ ··· 5u + 9i
I
u
2
= h5b 3a 1, 3a
2
3a + 7, u
2
+ u 1i
I
u
3
= hb + 1, a
2
+ 2a + 3, u 1i
I
u
4
= hb 1, a 1, u 1i
* 4 irreducible components of dim
C
= 0, with total 81 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−1.43×10
69
u
73
+6.42×10
69
u
72
+· · ·+1.66×10
69
b+1.03×10
70
, 5.07×
10
70
u
73
+2.49×10
71
u
72
+· · ·+1.49×10
70
a+6.53×10
71
, u
74
6u
73
+· · ·5u+9i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
u
2
a
12
=
u
u
3
+ u
a
3
=
3.40530u
73
16.7354u
72
+ ··· 59.8211u 43.8456
0.863958u
73
3.87934u
72
+ ··· 2.16199u 6.21506
a
9
=
u
2
+ 1
u
4
+ 2u
2
a
4
=
0.875009u
73
4.53530u
72
+ ··· 39.7678u 20.7618
1.73096u
73
7.25329u
72
+ ··· + 3.74996u 8.00046
a
6
=
4.28594u
73
19.7999u
72
+ ··· 55.8776u 44.5013
6.03936u
73
26.3358u
72
+ ··· 2.69922u 34.6744
a
2
=
11.7060u
73
52.9235u
72
+ ··· 13.7202u 74.9192
8.41966u
73
37.8131u
72
+ ··· 2.40204u 50.4595
a
1
=
5.78714u
73
29.6988u
72
+ ··· 74.5556u 72.6750
0.856731u
73
2.06038u
72
+ ··· + 50.8868u + 18.9089
a
10
=
u
u
a
5
=
10.6776u
73
48.4916u
72
+ ··· 51.7357u 80.3621
12.4311u
73
55.0275u
72
+ ··· + 1.44267u 70.5352
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5.31177u
73
18.4801u
72
+ ··· + 92.2939u + 19.8328
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
12
u
74
3u
73
+ ··· 12u + 2
c
2
, c
6
u
74
4u
73
+ ··· 5u 3
c
3
9(9u
74
+ 30u
73
+ ··· 18775u + 11591)
c
5
9(9u
74
39u
73
+ ··· + 17840u + 853)
c
7
, c
8
, c
10
c
11
u
74
+ 6u
73
+ ··· + 5u + 9
c
9
u
74
+ 2u
73
+ ··· + 576u 432
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
12
y
74
+ 69y
73
+ ··· 48y + 4
c
2
, c
6
y
74
38y
73
+ ··· 775y + 9
c
3
81(81y
74
2970y
73
+ ··· 1.79556 × 10
9
y + 1.34351 × 10
8
)
c
5
81(81y
74
729y
73
+ ··· 2.92408 × 10
8
y + 727609)
c
7
, c
8
, c
10
c
11
y
74
86y
73
+ ··· 655y + 81
c
9
y
74
28y
73
+ ··· 2104704y + 186624
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.780489 + 0.653369I
a = 0.48251 1.51030I
b = 1.252830 0.542624I
7.3329 + 12.3570I 0
u = 0.780489 0.653369I
a = 0.48251 + 1.51030I
b = 1.252830 + 0.542624I
7.3329 12.3570I 0
u = 0.926260 + 0.427283I
a = 0.703366 + 0.640455I
b = 0.349481 + 0.290897I
2.66093 0.33343I 0
u = 0.926260 0.427283I
a = 0.703366 0.640455I
b = 0.349481 0.290897I
2.66093 + 0.33343I 0
u = 0.766830 + 0.541355I
a = 0.35452 + 1.57607I
b = 1.236780 + 0.551197I
1.12893 + 8.60971I 0
u = 0.766830 0.541355I
a = 0.35452 1.57607I
b = 1.236780 0.551197I
1.12893 8.60971I 0
u = 0.722917 + 0.572819I
a = 0.197853 + 0.324737I
b = 0.555314 0.341105I
2.44985 0.53730I 0
u = 0.722917 0.572819I
a = 0.197853 0.324737I
b = 0.555314 + 0.341105I
2.44985 + 0.53730I 0
u = 0.745115 + 0.507872I
a = 0.505661 + 0.814141I
b = 0.133191 + 0.959429I
3.89877 + 6.97858I 0
u = 0.745115 0.507872I
a = 0.505661 0.814141I
b = 0.133191 0.959429I
3.89877 6.97858I 0
5
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.058370 + 0.376551I
a = 0.390460 0.275743I
b = 0.938072 + 0.240382I
0.571733 + 0.687366I 0
u = 1.058370 0.376551I
a = 0.390460 + 0.275743I
b = 0.938072 0.240382I
0.571733 0.687366I 0
u = 0.209015 + 0.842113I
a = 0.613156 0.007223I
b = 1.188410 + 0.452211I
9.07024 7.42311I 0
u = 0.209015 0.842113I
a = 0.613156 + 0.007223I
b = 1.188410 0.452211I
9.07024 + 7.42311I 0
u = 0.456967 + 0.724258I
a = 1.005020 + 0.543237I
b = 0.876049 + 0.413421I
3.31198 4.04047I 0
u = 0.456967 0.724258I
a = 1.005020 0.543237I
b = 0.876049 0.413421I
3.31198 + 4.04047I 0
u = 0.610837 + 0.528728I
a = 0.906740 0.910567I
b = 0.766366 0.275897I
0.93403 1.81717I 10.28717 + 6.80364I
u = 0.610837 0.528728I
a = 0.906740 + 0.910567I
b = 0.766366 + 0.275897I
0.93403 + 1.81717I 10.28717 6.80364I
u = 0.654850 + 0.418140I
a = 0.18276 1.79134I
b = 1.202110 0.564951I
1.73943 + 3.84518I 2.69909 5.52948I
u = 0.654850 0.418140I
a = 0.18276 + 1.79134I
b = 1.202110 + 0.564951I
1.73943 3.84518I 2.69909 + 5.52948I
6
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.689051 + 0.317719I
a = 0.557656 1.039220I
b = 0.192843 0.984641I
2.13148 + 3.14846I 7.44124 8.73816I
u = 0.689051 0.317719I
a = 0.557656 + 1.039220I
b = 0.192843 + 0.984641I
2.13148 3.14846I 7.44124 + 8.73816I
u = 0.743515
a = 0.472726
b = 0.111724
1.29107 8.03060
u = 0.129068 + 0.704485I
a = 0.719921 + 0.160016I
b = 1.159780 0.397161I
3.03854 4.45334I 0.92150 + 5.94186I
u = 0.129068 0.704485I
a = 0.719921 0.160016I
b = 1.159780 + 0.397161I
3.03854 + 4.45334I 0.92150 5.94186I
u = 0.506904 + 0.477161I
a = 0.0436695 + 0.0957579I
b = 1.324910 0.354848I
8.65870 + 2.36010I 2.52056 4.62942I
u = 0.506904 0.477161I
a = 0.0436695 0.0957579I
b = 1.324910 + 0.354848I
8.65870 2.36010I 2.52056 + 4.62942I
u = 1.257500 + 0.465420I
a = 0.340473 + 0.018485I
b = 1.047160 0.375595I
4.58375 + 2.82748I 0
u = 1.257500 0.465420I
a = 0.340473 0.018485I
b = 1.047160 + 0.375595I
4.58375 2.82748I 0
u = 0.113465 + 0.635641I
a = 0.809632 0.222688I
b = 0.032593 0.706254I
5.75300 3.13005I 0.47310 + 1.96058I
7
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.113465 0.635641I
a = 0.809632 + 0.222688I
b = 0.032593 + 0.706254I
5.75300 + 3.13005I 0.47310 1.96058I
u = 0.623383 + 0.158653I
a = 0.32416 + 3.34499I
b = 0.921135 + 0.058648I
0.485464 0.382772I 9.7234 10.9570I
u = 0.623383 0.158653I
a = 0.32416 3.34499I
b = 0.921135 0.058648I
0.485464 + 0.382772I 9.7234 + 10.9570I
u = 0.416679 + 0.480285I
a = 0.52731 + 2.30548I
b = 1.149740 + 0.482784I
8.91952 + 1.00599I 3.16723 4.49529I
u = 0.416679 0.480285I
a = 0.52731 2.30548I
b = 1.149740 0.482784I
8.91952 1.00599I 3.16723 + 4.49529I
u = 0.519626 + 0.103037I
a = 0.91143 + 1.58062I
b = 0.521721 + 0.994680I
0.77301 1.82140I 7.86976 8.66716I
u = 0.519626 0.103037I
a = 0.91143 1.58062I
b = 0.521721 0.994680I
0.77301 + 1.82140I 7.86976 + 8.66716I
u = 0.505606 + 0.052276I
a = 6.60990 + 5.71947I
b = 1.027770 + 0.072612I
5.81053 0.10803I 4.98191 7.31698I
u = 0.505606 0.052276I
a = 6.60990 5.71947I
b = 1.027770 0.072612I
5.81053 + 0.10803I 4.98191 + 7.31698I
u = 1.49577
a = 0.878655
b = 1.51662
2.46559 0
8
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.49843 + 0.08478I
a = 0.84047 1.98778I
b = 0.974646 0.722897I
2.64300 2.84023I 0
u = 1.49843 0.08478I
a = 0.84047 + 1.98778I
b = 0.974646 + 0.722897I
2.64300 + 2.84023I 0
u = 0.222562 + 0.436276I
a = 0.363985 0.713608I
b = 1.218890 + 0.287656I
2.97702 0.79262I 0.91124 2.75524I
u = 0.222562 0.436276I
a = 0.363985 + 0.713608I
b = 1.218890 0.287656I
2.97702 + 0.79262I 0.91124 + 2.75524I
u = 1.52960 + 0.11352I
a = 0.789714 + 0.238208I
b = 1.48319 + 0.25429I
1.87547 4.40034I 0
u = 1.52960 0.11352I
a = 0.789714 0.238208I
b = 1.48319 0.25429I
1.87547 + 4.40034I 0
u = 1.54223 + 0.22663I
a = 0.121615 1.194980I
b = 1.068200 0.517925I
3.32651 + 7.47374I 0
u = 1.54223 0.22663I
a = 0.121615 + 1.194980I
b = 1.068200 + 0.517925I
3.32651 7.47374I 0
u = 1.56341 + 0.01756I
a = 0.80268 1.74739I
b = 1.121450 0.237964I
1.346100 + 0.385297I 0
u = 1.56341 0.01756I
a = 0.80268 + 1.74739I
b = 1.121450 + 0.237964I
1.346100 0.385297I 0
9
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.57344 + 0.03838I
a = 0.73507 1.79325I
b = 0.632143 1.209970I
8.05402 + 1.24749I 0
u = 1.57344 0.03838I
a = 0.73507 + 1.79325I
b = 0.632143 + 1.209970I
8.05402 1.24749I 0
u = 1.59590 + 0.11628I
a = 0.62969 + 1.75927I
b = 1.22206 + 0.75829I
5.93811 5.80018I 0
u = 1.59590 0.11628I
a = 0.62969 1.75927I
b = 1.22206 0.75829I
5.93811 + 5.80018I 0
u = 1.60516 + 0.04809I
a = 0.24131 1.69697I
b = 0.947104 0.308897I
7.27045 + 1.14976I 0
u = 1.60516 0.04809I
a = 0.24131 + 1.69697I
b = 0.947104 + 0.308897I
7.27045 1.14976I 0
u = 1.59879 + 0.15088I
a = 0.011727 + 1.260060I
b = 0.996972 + 0.473584I
8.49571 + 4.32035I 0
u = 1.59879 0.15088I
a = 0.011727 1.260060I
b = 0.996972 0.473584I
8.49571 4.32035I 0
u = 1.60460 + 0.09213I
a = 0.56982 + 1.62025I
b = 0.352351 + 1.192960I
9.99806 4.68716I 0
u = 1.60460 0.09213I
a = 0.56982 1.62025I
b = 0.352351 1.192960I
9.99806 + 4.68716I 0
10
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.61043 + 0.13230I
a = 0.166508 + 0.655866I
b = 0.302997 + 0.654160I
5.48438 + 2.96748I 0
u = 1.61043 0.13230I
a = 0.166508 0.655866I
b = 0.302997 0.654160I
5.48438 2.96748I 0
u = 1.61900 + 0.14939I
a = 0.54408 1.44965I
b = 0.230645 1.110670I
4.13109 9.45778I 0
u = 1.61900 0.14939I
a = 0.54408 + 1.44965I
b = 0.230645 + 1.110670I
4.13109 + 9.45778I 0
u = 1.62753 + 0.15988I
a = 0.45575 1.70456I
b = 1.27669 0.67771I
7.00005 11.26650I 0
u = 1.62753 0.15988I
a = 0.45575 + 1.70456I
b = 1.27669 + 0.67771I
7.00005 + 11.26650I 0
u = 1.63663 + 0.20074I
a = 0.32200 + 1.71785I
b = 1.28702 + 0.62542I
0.8135 15.6125I 0
u = 1.63663 0.20074I
a = 0.32200 1.71785I
b = 1.28702 0.62542I
0.8135 + 15.6125I 0
u = 1.65905 + 0.04861I
a = 0.313274 0.637145I
b = 0.433660 0.467149I
10.06930 + 0.35205I 0
u = 1.65905 0.04861I
a = 0.313274 + 0.637145I
b = 0.433660 + 0.467149I
10.06930 0.35205I 0
11
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.077697 + 0.312068I
a = 1.157550 0.437372I
b = 0.018367 + 0.438500I
0.170863 0.995364I 3.42059 + 6.18920I
u = 0.077697 0.312068I
a = 1.157550 + 0.437372I
b = 0.018367 0.438500I
0.170863 + 0.995364I 3.42059 6.18920I
u = 1.71160 + 0.03432I
a = 0.311387 0.960043I
b = 0.725354 0.496651I
6.99846 + 2.04216I 0
u = 1.71160 0.03432I
a = 0.311387 + 0.960043I
b = 0.725354 + 0.496651I
6.99846 2.04216I 0
12
II. I
u
2
= h5b 3a 1, 3a
2
3a + 7, u
2
+ u 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
u + 1
a
12
=
u
u + 1
a
3
=
a
3
5
a +
1
5
a
9
=
u
u
a
4
=
2
5
au +
3
5
a
1
5
u +
1
5
2
5
au +
1
5
a
1
5
u +
2
5
a
6
=
4
5
a
2
5
3
5
a
4
5
a
2
=
3
5
a +
6
5
3
5
a +
6
5
a
1
=
1
5
au +
3
5
a
2
5
u +
1
5
1
5
au +
4
5
a
2
5
u +
3
5
a
10
=
u
u
a
5
=
1
5
au +
3
5
a +
2
5
u
4
5
1
5
au +
2
5
a +
2
5
u
6
5
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2au
29
5
a
1
3
u
124
15
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
12
(u
2
u + 1)
2
c
2
, c
4
(u
2
+ u + 1)
2
c
3
9(9u
4
+ 9u
2
+ 1)
c
5
9(9u
4
+ 9u
3
3u + 1)
c
7
, c
8
(u
2
+ u 1)
2
c
9
u
4
c
10
, c
11
(u
2
u 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
12
(y
2
+ y + 1)
2
c
3
81(9y
2
+ 9y + 1)
2
c
5
81(81y
4
81y
3
+ 72y
2
9y + 1)
c
7
, c
8
, c
10
c
11
(y
2
3y + 1)
2
c
9
y
4
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
1(vol +
1CS) Cusp shape
u = 0.618034
a = 0.50000 + 1.44338I
b = 0.500000 + 0.866025I
0.98696 + 2.02988I 11.9907 10.1557I
u = 0.618034
a = 0.50000 1.44338I
b = 0.500000 0.866025I
0.98696 2.02988I 11.9907 + 10.1557I
u = 1.61803
a = 0.50000 + 1.44338I
b = 0.500000 + 0.866025I
8.88264 + 2.02988I 9.00929 3.70072I
u = 1.61803
a = 0.50000 1.44338I
b = 0.500000 0.866025I
8.88264 2.02988I 9.00929 + 3.70072I
16
III. I
u
3
= hb + 1, a
2
+ 2a + 3, u 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
1
a
8
=
1
1
a
12
=
1
0
a
3
=
a
1
a
9
=
0
1
a
4
=
a
a 1
a
6
=
a + 1
1
a
2
=
1
0
a
1
=
a 2
2
a
10
=
1
1
a
5
=
1
a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
12
u
2
+ 2
c
2
, c
7
, c
8
(u 1)
2
c
3
u
2
+ 2u + 3
c
5
u
2
2u + 3
c
6
, c
9
, c
10
c
11
(u + 1)
2
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
12
(y + 2)
2
c
2
, c
6
, c
7
c
8
, c
9
, c
10
c
11
(y 1)
2
c
3
, c
5
y
2
+ 2y + 9
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000 + 1.41421I
b = 1.00000
4.93480 0
u = 1.00000
a = 1.00000 1.41421I
b = 1.00000
4.93480 0
20
IV. I
u
4
= hb 1, a 1, u 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
1
a
8
=
1
1
a
12
=
1
0
a
3
=
1
1
a
9
=
0
1
a
4
=
1
0
a
6
=
2
1
a
2
=
1
0
a
1
=
1
0
a
10
=
1
1
a
5
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
12
u
c
2
, c
10
, c
11
u + 1
c
3
, c
5
, c
6
c
7
, c
8
, c
9
u 1
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
12
y
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
9
, c
10
, c
11
y 1
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
0 0
24
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
12
u(u
2
+ 2)(u
2
u + 1)
2
(u
74
3u
73
+ ··· 12u + 2)
c
2
((u 1)
2
)(u + 1)(u
2
+ u + 1)
2
(u
74
4u
73
+ ··· 5u 3)
c
3
81(u 1)(u
2
+ 2u + 3)(9u
4
+ 9u
2
+ 1)
· (9u
74
+ 30u
73
+ ··· 18775u + 11591)
c
4
u(u
2
+ 2)(u
2
+ u + 1)
2
(u
74
3u
73
+ ··· 12u + 2)
c
5
81(u 1)(u
2
2u + 3)(9u
4
+ 9u
3
3u + 1)
· (9u
74
39u
73
+ ··· + 17840u + 853)
c
6
(u 1)(u + 1)
2
(u
2
u + 1)
2
(u
74
4u
73
+ ··· 5u 3)
c
7
, c
8
((u 1)
3
)(u
2
+ u 1)
2
(u
74
+ 6u
73
+ ··· + 5u + 9)
c
9
u
4
(u 1)(u + 1)
2
(u
74
+ 2u
73
+ ··· + 576u 432)
c
10
, c
11
((u + 1)
3
)(u
2
u 1)
2
(u
74
+ 6u
73
+ ··· + 5u + 9)
25
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
12
y(y + 2)
2
(y
2
+ y + 1)
2
(y
74
+ 69y
73
+ ··· 48y + 4)
c
2
, c
6
((y 1)
3
)(y
2
+ y + 1)
2
(y
74
38y
73
+ ··· 775y + 9)
c
3
6561(y 1)(y
2
+ 2y + 9)(9y
2
+ 9y + 1)
2
· (81y
74
2970y
73
+ ··· 1795556943y + 134351281)
c
5
6561(y 1)(y
2
+ 2y + 9)(81y
4
81y
3
+ 72y
2
9y + 1)
· (81y
74
729y
73
+ ··· 292407758y + 727609)
c
7
, c
8
, c
10
c
11
((y 1)
3
)(y
2
3y + 1)
2
(y
74
86y
73
+ ··· 655y + 81)
c
9
y
4
(y 1)
3
(y
74
28y
73
+ ··· 2104704y + 186624)
26