12a
1219
(K12a
1219
)
A knot diagram
1
Linearized knot diagam
5 6 7 11 2 10 4 12 1 3 8 9
Solving Sequence
8,12
9 1
4,10
7 3 6 11 5 2
c
8
c
12
c
9
c
7
c
3
c
6
c
11
c
4
c
1
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h8.75661 × 10
64
u
63
+ 2.76997 × 10
65
u
62
+ ··· + 1.65696 × 10
65
b + 2.96457 × 10
65
,
1.00296 × 10
66
u
63
+ 3.44551 × 10
66
u
62
+ ··· + 8.28478 × 10
64
a + 2.84103 × 10
65
, u
64
+ 4u
63
+ ··· − 10u + 1i
I
u
2
= hb − 1, a + 2, u + 1i
I
u
3
= hb + 1, a
2
− 4a + 2, u + 1i
* 3 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
=
h8.76×10
64
u
63
+2.77×10
65
u
62
+· · ·+1.66×10
65
b+2.96×10
65
, 1.00×10
66
u
63
+
3.45 × 10
66
u
62
+ · · · + 8.28 × 10
64
a + 2.84 × 10
65
, u
64
+ 4u
63
+ · · · − 10u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
−12.1060u
63
− 41.5885u
62
+ ··· + 205.872u − 3.42922
−0.528476u
63
− 1.67172u
62
+ ··· + 7.65402u − 1.78916
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
13.7729u
63
+ 46.8393u
62
+ ··· − 229.931u + 4.46101
0.181537u
63
+ 0.533529u
62
+ ··· − 1.36851u + 1.37956
a
3
=
4.35303u
63
+ 10.6700u
62
+ ··· − 32.5922u − 0.381251
−3.47471u
63
− 8.31368u
62
+ ··· + 28.6251u − 2.38908
a
6
=
8.57909u
63
+ 30.0107u
62
+ ··· − 140.536u − 4.30529
5.95601u
63
+ 19.1387u
62
+ ··· − 97.4281u + 9.72466
a
11
=
−u
u
a
5
=
−7.27418u
63
− 25.7859u
62
+ ··· + 120.459u + 3.84863
−5.36032u
63
− 17.4743u
62
+ ··· + 93.0670u − 9.06702
a
2
=
−8.95289u
63
− 28.0262u
62
+ ··· + 182.432u − 36.5824
−5.59890u
63
− 20.4560u
62
+ ··· + 122.657u − 8.92048
(ii) Obstruction class = −1
(iii) Cusp Shapes = −112.789u
63
− 383.132u
62
+ ··· + 2107.64u − 184.302
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
64
+ 5u
63
+ ··· − 6u − 2
c
3
, c
7
u
64
− 2u
63
+ ··· + 6u − 1
c
4
u
64
+ 16u
63
+ ··· − 397046u + 45841
c
6
u
64
− 18u
63
+ ··· − 8u + 1
c
8
, c
9
, c
11
c
12
u
64
− 4u
63
+ ··· + 10u + 1
c
10
u
64
− 2u
63
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
64
− 61y
63
+ ··· + 12y + 4
c
3
, c
7
y
64
− 52y
63
+ ··· + 6y + 1
c
4
y
64
− 440y
63
+ ··· + 63549675954y + 2101397281
c
6
y
64
− 504y
63
+ ··· − 638y + 1
c
8
, c
9
, c
11
c
12
y
64
− 76y
63
+ ··· − 106y + 1
c
10
y
64
− 8y
63
+ ··· − 114y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.626252 + 0.772749I
a = −0.442645 − 0.836072I
b = 1.282870 − 0.069070I
1.37703 − 2.62298I 0
u = −0.626252 − 0.772749I
a = −0.442645 + 0.836072I
b = 1.282870 + 0.069070I
1.37703 + 2.62298I 0
u = 0.892186 + 0.475186I
a = 1.45927 − 1.01010I
b = −1.37477 − 0.41692I
5.33163 + 8.09564I 0
u = 0.892186 − 0.475186I
a = 1.45927 + 1.01010I
b = −1.37477 + 0.41692I
5.33163 − 8.09564I 0
u = −0.885933 + 0.573263I
a = 0.924110 + 0.935341I
b = −1.224840 − 0.045601I
4.82419 − 0.31393I 0
u = −0.885933 − 0.573263I
a = 0.924110 − 0.935341I
b = −1.224840 + 0.045601I
4.82419 + 0.31393I 0
u = 0.866924 + 0.614790I
a = −1.33885 + 1.11410I
b = 1.36865 + 0.42563I
−0.85685 + 11.92020I 0
u = 0.866924 − 0.614790I
a = −1.33885 − 1.11410I
b = 1.36865 − 0.42563I
−0.85685 − 11.92020I 0
u = −1.088320 + 0.183319I
a = −1.033310 + 0.040437I
b = 0.007127 − 0.240948I
−3.32236 + 0.09828I 0
u = −1.088320 − 0.183319I
a = −1.033310 − 0.040437I
b = 0.007127 + 0.240948I
−3.32236 − 0.09828I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.727414 + 0.497209I
a = 0.517608 − 0.186664I
b = −0.136718 − 0.955922I
−5.57662 + 7.00157I 0
u = 0.727414 − 0.497209I
a = 0.517608 + 0.186664I
b = −0.136718 + 0.955922I
−5.57662 − 7.00157I 0
u = 0.086190 + 0.869107I
a = −0.161462 − 0.186514I
b = 1.276990 − 0.329837I
−3.24005 − 7.03352I 0
u = 0.086190 − 0.869107I
a = −0.161462 + 0.186514I
b = 1.276990 + 0.329837I
−3.24005 + 7.03352I 0
u = 0.826064 + 0.274343I
a = −1.68330 + 0.81928I
b = 1.39853 + 0.42614I
4.75434 + 3.17966I 0
u = 0.826064 − 0.274343I
a = −1.68330 − 0.81928I
b = 1.39853 − 0.42614I
4.75434 − 3.17966I 0
u = −1.20392
a = 1.35869
b = −0.745445
2.67121 0
u = −0.735824
a = −4.70570
b = 1.05067
2.92949 −31.0790
u = −0.061375 + 0.731517I
a = −0.056369 + 0.272785I
b = −1.230620 + 0.271880I
2.42658 − 4.10040I 5.18854 + 6.65155I
u = −0.061375 − 0.731517I
a = −0.056369 − 0.272785I
b = −1.230620 − 0.271880I
2.42658 + 4.10040I 5.18854 − 6.65155I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.156350 + 0.575363I
a = −1.056810 − 0.610432I
b = 1.251050 + 0.169381I
0.45977 + 2.04940I 0
u = −1.156350 − 0.575363I
a = −1.056810 + 0.610432I
b = 1.251050 − 0.169381I
0.45977 − 2.04940I 0
u = 0.645610 + 0.280609I
a = −0.622762 − 0.084173I
b = 0.227169 + 0.986896I
0.39947 + 3.16035I 2.81338 − 10.76378I
u = 0.645610 − 0.280609I
a = −0.622762 + 0.084173I
b = 0.227169 − 0.986896I
0.39947 − 3.16035I 2.81338 + 10.76378I
u = −0.537409 + 0.416579I
a = −0.908169 + 1.049400I
b = −0.301281 + 0.283446I
−3.30500 − 1.49850I −1.71841 + 4.52498I
u = −0.537409 − 0.416579I
a = −0.908169 − 1.049400I
b = −0.301281 − 0.283446I
−3.30500 + 1.49850I −1.71841 − 4.52498I
u = 0.171438 + 0.641396I
a = −0.810739 + 0.661600I
b = −0.011504 + 0.735189I
−7.24617 − 3.15234I −4.81813 + 1.66839I
u = 0.171438 − 0.641396I
a = −0.810739 − 0.661600I
b = −0.011504 − 0.735189I
−7.24617 + 3.15234I −4.81813 − 1.66839I
u = −0.622251 + 0.147718I
a = 0.623442 − 0.530311I
b = 0.140050 − 0.093500I
1.153010 − 0.371044I 8.52101 + 0.53269I
u = −0.622251 − 0.147718I
a = 0.623442 + 0.530311I
b = 0.140050 + 0.093500I
1.153010 + 0.371044I 8.52101 − 0.53269I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.625236
a = −32.6722
b = −0.982597
−2.29108 −335.850
u = 0.571354 + 0.216286I
a = 1.058770 − 0.384713I
b = −1.136650 + 0.669884I
−2.52737 + 1.49035I −1.46490 − 10.14674I
u = 0.571354 − 0.216286I
a = 1.058770 + 0.384713I
b = −1.136650 − 0.669884I
−2.52737 − 1.49035I −1.46490 + 10.14674I
u = −1.58313
a = 2.77393
b = −1.86821
3.83757 0
u = 1.58050 + 0.10884I
a = 0.006317 − 0.340356I
b = −0.242493 − 0.631350I
3.96868 + 3.34904I 0
u = 1.58050 − 0.10884I
a = 0.006317 + 0.340356I
b = −0.242493 + 0.631350I
3.96868 − 3.34904I 0
u = −1.58979 + 0.03598I
a = 1.41098 + 0.68557I
b = −1.17421 − 1.06495I
4.95069 − 2.27479I 0
u = −1.58979 − 0.03598I
a = 1.41098 − 0.68557I
b = −1.17421 + 1.06495I
4.95069 + 2.27479I 0
u = −1.60546 + 0.05832I
a = −0.671440 + 0.764979I
b = 0.437967 − 1.302070I
8.17400 − 4.30296I 0
u = −1.60546 − 0.05832I
a = −0.671440 − 0.764979I
b = 0.437967 + 1.302070I
8.17400 + 4.30296I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.60877 + 0.03569I
a = 0.077213 + 0.292537I
b = 0.219375 + 0.411958I
8.93615 + 1.02069I 0
u = 1.60877 − 0.03569I
a = 0.077213 − 0.292537I
b = 0.219375 − 0.411958I
8.93615 − 1.02069I 0
u = 0.385731
a = 4.06256
b = −1.41196
−3.33490 −8.92110
u = 0.129453 + 0.356601I
a = 1.25594 − 0.67953I
b = −0.129998 − 0.580705I
−1.019940 − 0.841014I −4.44164 + 2.44385I
u = 0.129453 − 0.356601I
a = 1.25594 + 0.67953I
b = −0.129998 + 0.580705I
−1.019940 + 0.841014I −4.44164 − 2.44385I
u = −0.151171 + 0.347893I
a = 1.51053 − 0.33569I
b = 1.087870 − 0.202838I
1.93096 − 1.01785I 2.25227 − 1.57609I
u = −0.151171 − 0.347893I
a = 1.51053 + 0.33569I
b = 1.087870 + 0.202838I
1.93096 + 1.01785I 2.25227 + 1.57609I
u = −1.61625 + 0.13512I
a = 0.579253 − 0.473421I
b = −0.254582 + 1.128060I
2.39807 − 9.34049I 0
u = −1.61625 − 0.13512I
a = 0.579253 + 0.473421I
b = −0.254582 − 1.128060I
2.39807 + 9.34049I 0
u = 1.63116 + 0.01107I
a = −1.17537 + 1.13475I
b = −0.795270 + 0.143585I
5.71807 + 0.01867I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.63116 − 0.01107I
a = −1.17537 − 1.13475I
b = −0.795270 − 0.143585I
5.71807 − 0.01867I 0
u = 1.62771 + 0.23629I
a = −1.53783 + 0.68857I
b = 1.350810 + 0.224077I
8.99118 + 6.39006I 0
u = 1.62771 − 0.23629I
a = −1.53783 − 0.68857I
b = 1.350810 − 0.224077I
8.99118 − 6.39006I 0
u = −1.65093 + 0.07673I
a = −2.22460 − 0.22306I
b = 1.61684 − 0.49505I
13.36180 − 4.53477I 0
u = −1.65093 − 0.07673I
a = −2.22460 + 0.22306I
b = 1.61684 + 0.49505I
13.36180 + 4.53477I 0
u = −1.67065 + 0.13316I
a = 2.09031 + 0.47257I
b = −1.51390 + 0.48815I
14.1665 − 10.4624I 0
u = −1.67065 − 0.13316I
a = 2.09031 − 0.47257I
b = −1.51390 − 0.48815I
14.1665 + 10.4624I 0
u = −1.66778 + 0.18149I
a = −2.03991 − 0.65692I
b = 1.46024 − 0.47787I
7.7777 − 15.0108I 0
u = −1.66778 − 0.18149I
a = −2.03991 + 0.65692I
b = 1.46024 + 0.47787I
7.7777 + 15.0108I 0
u = 1.68055
a = −2.49868
b = 1.25013
11.5859 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.68516 + 0.14611I
a = 1.76330 − 0.59949I
b = −1.328680 − 0.151173I
13.75650 + 3.08420I 0
u = 1.68516 − 0.14611I
a = 1.76330 + 0.59949I
b = −1.328680 + 0.151173I
13.75650 − 3.08420I 0
u = 1.73966
a = −2.03600
b = 1.32403
11.5124 0
u = 0.102209
a = 13.6904
b = −1.15668
−3.39768 −3.12270
11
II. I
u
2
= hb − 1, a + 2, u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
−1
a
9
=
1
−1
a
1
=
−1
0
a
4
=
−2
1
a
10
=
0
−1
a
7
=
−1
1
a
3
=
−1
0
a
6
=
−1
0
a
11
=
1
−1
a
5
=
−1
0
a
2
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
c
3
, c
8
, c
9
c
10
u + 1
c
4
, c
6
, c
7
c
11
, c
12
u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
c
3
, c
4
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −2.00000
b = 1.00000
3.28987 12.0000
15
III. I
u
3
= hb + 1, a
2
− 4a + 2, u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
−1
a
9
=
1
−1
a
1
=
−1
0
a
4
=
a
−1
a
10
=
0
−1
a
7
=
−a + 1
1
a
3
=
1
0
a
6
=
−a + 1
−a + 2
a
11
=
1
−1
a
5
=
1
a − 2
a
2
=
−a + 1
−2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
2
− 2
c
3
, c
10
, c
11
c
12
(u − 1)
2
c
4
u
2
+ 2u − 1
c
6
u
2
− 2u − 1
c
7
, c
8
, c
9
(u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −2)
2
c
3
, c
7
, c
8
c
9
, c
10
, c
11
c
12
(y −1)
2
c
4
, c
6
y
2
− 6y + 1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.585786
b = −1.00000
−1.64493 4.00000
u = −1.00000
a = 3.41421
b = −1.00000
−1.64493 4.00000
19
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u(u
2
− 2)(u
64
+ 5u
63
+ ··· − 6u − 2)
c
3
((u − 1)
2
)(u + 1)(u
64
− 2u
63
+ ··· + 6u − 1)
c
4
(u − 1)(u
2
+ 2u − 1)(u
64
+ 16u
63
+ ··· − 397046u + 45841)
c
6
(u − 1)(u
2
− 2u − 1)(u
64
− 18u
63
+ ··· − 8u + 1)
c
7
(u − 1)(u + 1)
2
(u
64
− 2u
63
+ ··· + 6u − 1)
c
8
, c
9
((u + 1)
3
)(u
64
− 4u
63
+ ··· + 10u + 1)
c
10
((u − 1)
2
)(u + 1)(u
64
− 2u
63
+ ··· − 2u − 1)
c
11
, c
12
((u − 1)
3
)(u
64
− 4u
63
+ ··· + 10u + 1)
20
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y(y −2)
2
(y
64
− 61y
63
+ ··· + 12y + 4)
c
3
, c
7
((y −1)
3
)(y
64
− 52y
63
+ ··· + 6y + 1)
c
4
(y −1)(y
2
− 6y + 1)
· (y
64
− 440y
63
+ ··· + 63549675954y + 2101397281)
c
6
(y −1)(y
2
− 6y + 1)(y
64
− 504y
63
+ ··· − 638y + 1)
c
8
, c
9
, c
11
c
12
((y −1)
3
)(y
64
− 76y
63
+ ··· − 106y + 1)
c
10
((y −1)
3
)(y
64
− 8y
63
+ ··· − 114y + 1)
21
