12n
0808
(K12n
0808
)
A knot diagram
1
Linearized knot diagam
4 7 12 7 9 3 10 12 5 1 4 8
Solving Sequence
8,12 1,4
3 11 10 7 5 2 6 9
c
12
c
3
c
11
c
10
c
7
c
4
c
2
c
6
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−1117319072383u
19
+ 155977493456u
18
+ ··· + 3124548433565b − 6750707248079,
− 3478787820073u
19
− 1267593674108u
18
+ ··· + 1874729060139a − 4936538211880,
u
20
+ 9u
18
+ ··· + 5u + 1i
I
u
2
= h2u
4
− u
3
+ 3u
2
+ b − 4u + 3, a, u
5
− u
4
+ 2u
3
− 3u
2
+ 3u − 1i
I
u
3
= h−18996421u
15
− 61047734u
14
+ ··· + 119799436b − 83873864,
50129469u
15
+ 202053023u
14
+ ··· + 119799436a + 472472676, u
16
+ 2u
15
+ ··· − 8u + 4i
I
u
4
= h−34278227166280u
15
+ 103194516923463u
14
+ ··· + 205378365871400b − 3319079207393740,
280790731208697u
15
− 809250943773254u
14
+ ··· + 1437648561099800a + 32596276646035500,
u
16
− 2u
15
+ ··· + 300u + 100i
* 4 irreducible components of dim
C
= 0, with total 57 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−1.12×10
12
u
19
+1.56×10
11
u
18
+· · ·+3.12×10
12
b−6.75×10
12
, −3.48×
10
12
u
19
−1.27×10
12
u
18
+· · ·+1.87×10
12
a−4.94×10
12
, u
20
+9u
18
+· · ·+5u+1i
(i) Arc colorings
a
8
=
0
u
a
12
=
1
0
a
1
=
1
−u
2
a
4
=
1.85562u
19
+ 0.676148u
18
+ ··· + 2.89591u + 2.63320
0.357594u
19
− 0.0499200u
18
+ ··· + 4.28276u + 2.16054
a
3
=
2.21322u
19
+ 0.626228u
18
+ ··· + 7.17867u + 4.79374
0.357594u
19
− 0.0499200u
18
+ ··· + 4.28276u + 2.16054
a
11
=
−0.428529u
19
+ 0.0562364u
18
+ ··· + 3.76713u + 0.479192
0.584730u
19
+ 0.277642u
18
+ ··· + 6.15595u + 1.31484
a
10
=
0.371174u
19
+ 0.479192u
18
+ ··· + 10.0704u + 1.73780
0.371174u
19
+ 0.479192u
18
+ ··· + 9.07043u + 1.73780
a
7
=
1.58160u
19
− 0.705052u
18
+ ··· − 9.81943u − 1.17548
1.15307u
19
− 0.648816u
18
+ ··· − 6.05229u − 0.696286
a
5
=
−1.73780u
19
+ 0.371174u
18
+ ··· − 0.103654u + 0.381445
−1.73780u
19
+ 0.371174u
18
+ ··· − 0.103654u − 0.618555
a
2
=
0.818867u
19
+ 0.0944094u
18
+ ··· − 7.05001u + 0.973427
−0.799703u
19
− 0.422955u
18
+ ··· − 6.30330u − 1.25860
a
6
=
1.73780u
19
− 0.371174u
18
+ ··· + 0.103654u − 0.381445
1.73780u
19
− 0.371174u
18
+ ··· + 0.103654u + 0.618555
a
9
=
u
u
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
1120455221723
240349879505
u
19
−
471292989666
240349879505
u
18
+ ··· +
3691766656988
240349879505
u +
320606903159
240349879505
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 3u
19
+ ··· + 218u − 13
c
2
, c
3
, c
6
c
11
u
20
− u
19
+ ··· − 10u
2
− 1
c
4
, c
10
u
20
− 8u
18
+ ··· − 4u + 1
c
5
, c
8
, c
9
c
12
u
20
+ 9u
18
+ ··· + 5u + 1
c
7
u
20
+ u
19
+ ··· − 143u + 19
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 63y
19
+ ··· − 9564y + 169
c
2
, c
3
, c
6
c
11
y
20
+ 5y
19
+ ··· + 20y + 1
c
4
, c
10
y
20
− 16y
19
+ ··· − 18y + 1
c
5
, c
8
, c
9
c
12
y
20
+ 18y
19
+ ··· − 15y + 1
c
7
y
20
− 11y
19
+ ··· − 15167y + 361
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.400816 + 0.884966I
a = −2.72954 + 2.09837I
b = −0.949109 − 0.132376I
−8.66277 − 4.49166I −15.7192 + 8.9902I
u = −0.400816 − 0.884966I
a = −2.72954 − 2.09837I
b = −0.949109 + 0.132376I
−8.66277 + 4.49166I −15.7192 − 8.9902I
u = 0.445075 + 0.664586I
a = −0.55825 − 1.54756I
b = 0.759984 + 0.115147I
−6.98935 + 2.26121I −8.63634 − 2.08867I
u = 0.445075 − 0.664586I
a = −0.55825 + 1.54756I
b = 0.759984 − 0.115147I
−6.98935 − 2.26121I −8.63634 + 2.08867I
u = 0.255620 + 0.731714I
a = −1.08398 + 0.96572I
b = −0.429423 + 0.879049I
−0.07650 + 4.92471I −2.33557 − 12.60053I
u = 0.255620 − 0.731714I
a = −1.08398 − 0.96572I
b = −0.429423 − 0.879049I
−0.07650 − 4.92471I −2.33557 + 12.60053I
u = −0.012360 + 0.770424I
a = 1.81430 − 0.38135I
b = 0.562883 − 0.868623I
−0.82447 + 2.67386I −9.08540 − 1.69155I
u = −0.012360 − 0.770424I
a = 1.81430 + 0.38135I
b = 0.562883 + 0.868623I
−0.82447 − 2.67386I −9.08540 + 1.69155I
u = 0.480017 + 0.602509I
a = 0.643469 + 0.157584I
b = −0.082504 − 0.310448I
0.60874 + 1.46463I 3.26499 − 6.09999I
u = 0.480017 − 0.602509I
a = 0.643469 − 0.157584I
b = −0.082504 + 0.310448I
0.60874 − 1.46463I 3.26499 + 6.09999I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.531335 + 1.189620I
a = 0.242043 − 0.317433I
b = 0.060234 + 0.456251I
−3.53143 − 6.83177I −9.5565 + 11.0135I
u = −0.531335 − 1.189620I
a = 0.242043 + 0.317433I
b = 0.060234 − 0.456251I
−3.53143 + 6.83177I −9.5565 − 11.0135I
u = −1.12739 + 1.01287I
a = 0.825387 − 1.140150I
b = 0.68802 + 1.97233I
8.73704 − 6.60530I −3.07657 + 3.57054I
u = −1.12739 − 1.01287I
a = 0.825387 + 1.140150I
b = 0.68802 − 1.97233I
8.73704 + 6.60530I −3.07657 − 3.57054I
u = −0.361658
a = −0.615435
b = −1.63927
−10.4764 −36.4690
u = −0.237363
a = 1.77115
b = 0.677320
−1.17003 −10.0210
u = 1.34920 + 1.48354I
a = 0.679909 + 0.785418I
b = 1.06269 − 2.17951I
6.2442 + 13.4988I −4.98661 − 5.91424I
u = 1.34920 − 1.48354I
a = 0.679909 − 0.785418I
b = 1.06269 + 2.17951I
6.2442 − 13.4988I −4.98661 + 5.91424I
u = −0.15850 + 2.40587I
a = −0.911199 + 0.120376I
b = −1.69180 − 0.15828I
−15.1787 + 0.9594I −7.12357 − 7.70746I
u = −0.15850 − 2.40587I
a = −0.911199 − 0.120376I
b = −1.69180 + 0.15828I
−15.1787 − 0.9594I −7.12357 + 7.70746I
6
II. I
u
2
= h2u
4
− u
3
+ 3u
2
+ b − 4u + 3, a, u
5
− u
4
+ 2u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
8
=
0
u
a
12
=
1
0
a
1
=
1
−u
2
a
4
=
0
−2u
4
+ u
3
− 3u
2
+ 4u − 3
a
3
=
−2u
4
+ u
3
− 3u
2
+ 4u − 3
−2u
4
+ u
3
− 3u
2
+ 4u − 3
a
11
=
1
−2u
4
+ u
3
− 4u
2
+ 4u − 4
a
10
=
−2u
4
+ u
3
− 3u
2
+ 4u − 3
−2u
4
+ u
3
− 3u
2
+ 3u − 3
a
7
=
u
4
+ 2u
2
− 2u + 2
u
4
+ 2u
2
− 2u + 3
a
5
=
3u
4
− u
3
+ 5u
2
− 6u + 6
3u
4
− u
3
+ 5u
2
− 6u + 7
a
2
=
1
2u
4
− u
3
+ 3u
2
− 4u + 4
a
6
=
3u
4
− u
3
+ 6u
2
− 6u + 6
3u
4
− u
3
+ 6u
2
− 6u + 7
a
9
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 19u
4
− 9u
3
+ 32u
2
− 38u + 32
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
11
u
5
− u
3
+ 2u
2
− 2u + 1
c
3
, c
6
u
5
− u
3
− 2u
2
− 2u − 1
c
4
, c
10
u
5
− 5u
4
+ 9u
3
− 9u
2
+ 4u − 1
c
5
, c
8
u
5
+ u
4
+ 2u
3
+ 3u
2
+ 3u + 1
c
7
u
5
+ 2u
4
+ u
3
− u
2
− u − 1
c
9
, c
12
u
5
− u
4
+ 2u
3
− 3u
2
+ 3u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
, c
11
y
5
− 2y
4
− 3y
3
− 1
c
4
, c
10
y
5
− 7y
4
− y
3
− 19y
2
− 2y −1
c
5
, c
8
, c
9
c
12
y
5
+ 3y
4
+ 4y
3
+ y
2
+ 3y −1
c
7
y
5
− 2y
4
+ 3y
3
+ y
2
− y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.692449 + 0.655213I
a = 0
b = 0.701186 + 0.377712I
−0.075375 + 0.838336I −3.26553 − 0.08174I
u = 0.692449 − 0.655213I
a = 0
b = 0.701186 − 0.377712I
−0.075375 − 0.838336I −3.26553 + 0.08174I
u = −0.45440 + 1.37619I
a = 0
b = 0.166160 − 0.938713I
−2.98113 − 6.24267I −2.74051 + 3.66349I
u = −0.45440 − 1.37619I
a = 0
b = 0.166160 + 0.938713I
−2.98113 + 6.24267I −2.74051 − 3.66349I
u = 0.523892
a = 0
b = −1.73469
−10.3363 21.0120
10
III.
I
u
3
= h−1.90 × 10
7
u
15
− 6.10 × 10
7
u
14
+ · · · + 1.20 × 10
8
b − 8.39 × 10
7
, 5.01 ×
10
7
u
15
+2.02×10
8
u
14
+· · ·+1.20×10
8
a+4.72×10
8
, u
16
+2u
15
+· · · −8u +4i
(i) Arc colorings
a
8
=
0
u
a
12
=
1
0
a
1
=
1
−u
2
a
4
=
−0.418445u
15
− 1.68659u
14
+ ··· − 6.15108u − 3.94386
0.158569u
15
+ 0.509583u
14
+ ··· + 0.0442522u + 0.700119
a
3
=
−0.259876u
15
− 1.17701u
14
+ ··· − 6.10683u − 3.24374
0.158569u
15
+ 0.509583u
14
+ ··· + 0.0442522u + 0.700119
a
11
=
0.519177u
15
+ 1.43811u
14
+ ··· + 9.32741u + 0.285737
−0.200027u
15
− 0.613205u
14
+ ··· − 0.0872469u − 0.165792
a
10
=
0.640403u
15
+ 1.35117u
14
+ ··· + 10.3615u − 1.47910
−0.0359168u
15
− 0.479565u
14
+ ··· + 3.03280u − 1.48337
a
7
=
−0.708674u
15
− 1.89413u
14
+ ··· − 6.60773u − 3.73765
−0.145182u
15
− 0.190232u
14
+ ··· − 2.63374u + 1.34969
a
5
=
−0.186181u
15
− 0.319098u
14
+ ··· + 0.454556u + 1.41790
0.267538u
15
+ 0.426352u
14
+ ··· + 2.44587u − 1.74924
a
2
=
−0.685361u
15
− 3.03767u
14
+ ··· − 5.13508u − 11.9169
−0.121226u
15
+ 0.0869414u
14
+ ··· − 1.03413u + 1.76483
a
6
=
−0.594605u
15
− 1.13328u
14
+ ··· − 2.65622u + 2.06585
−0.140886u
15
− 0.387830u
14
+ ··· − 0.664908u − 1.10129
a
9
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
8429911
59899718
u
15
−
17842718
29949859
u
14
+ ··· −
218282262
29949859
u −
26259324
29949859
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
8
− 8u
7
+ 18u
6
+ 4u
5
− 59u
4
+ 60u
3
− 16u
2
+ 8u − 4)
2
c
2
, c
11
u
16
+ 2u
15
+ ··· − 16u + 4
c
3
, c
6
u
16
− 2u
15
+ ··· + 16u + 4
c
4
, c
10
u
16
− 2u
15
+ ··· − 8u + 1
c
5
, c
8
u
16
− 2u
15
+ ··· + 8u + 4
c
7
(u
4
− 2u
3
+ u
2
+ 2u − 1)
4
c
9
, c
12
u
16
+ 2u
15
+ ··· − 8u + 4
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
8
− 28y
7
+ ··· + 64y + 16)
2
c
2
, c
3
, c
6
c
11
y
16
+ 2y
15
+ ··· − 48y + 16
c
4
, c
10
y
16
− 10y
15
+ ··· + 8y + 1
c
5
, c
8
, c
9
c
12
y
16
+ 14y
15
+ ··· + 112y + 16
c
7
(y
4
− 2y
3
+ 7y
2
− 6y + 1)
4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.521539 + 0.812354I
a = −1.103100 + 0.293517I
b = −0.297232 + 0.643334I
−0.16449 + 4.11697I −3.58579 − 1.95664I
u = 0.521539 − 0.812354I
a = −1.103100 − 0.293517I
b = −0.297232 − 0.643334I
−0.16449 − 4.11697I −3.58579 + 1.95664I
u = −0.429596 + 0.954915I
a = −0.780096 + 0.350949I
b = 0.658899 − 0.384785I
−7.31723 −7.76641 + 0.I
u = −0.429596 − 0.954915I
a = −0.780096 − 0.350949I
b = 0.658899 + 0.384785I
−7.31723 −7.76641 + 0.I
u = −0.186433 + 0.770599I
a = 0.669548 − 1.217980I
b = 0.606249 − 0.893781I
−0.16449 + 4.11697I −3.58579 − 1.95664I
u = −0.186433 − 0.770599I
a = 0.669548 + 1.217980I
b = 0.606249 + 0.893781I
−0.16449 − 4.11697I −3.58579 + 1.95664I
u = −0.490161 + 1.117010I
a = −1.91627 + 1.34689I
b = −1.287370 − 0.336211I
−8.06018 − 4.11697I −3.58579 + 1.95664I
u = −0.490161 − 1.117010I
a = −1.91627 − 1.34689I
b = −1.287370 + 0.336211I
−8.06018 + 4.11697I −3.58579 − 1.95664I
u = 0.581942 + 0.493509I
a = 0.738930 + 0.871340I
b = 0.916003 − 0.507301I
−0.907436 −5.06202 + 0.I
u = 0.581942 − 0.493509I
a = 0.738930 − 0.871340I
b = 0.916003 + 0.507301I
−0.907436 −5.06202 + 0.I
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.362162 + 0.512371I
a = −3.12506 − 3.31201I
b = 0.478355 + 0.319469I
−8.06018 + 4.11697I −3.58579 − 1.95664I
u = 0.362162 − 0.512371I
a = −3.12506 + 3.31201I
b = 0.478355 − 0.319469I
−8.06018 − 4.11697I −3.58579 + 1.95664I
u = −1.52354 + 0.89039I
a = 0.449712 − 0.769502I
b = −0.34988 + 2.39621I
6.98825 −5.06202 + 0.I
u = −1.52354 − 0.89039I
a = 0.449712 + 0.769502I
b = −0.34988 − 2.39621I
6.98825 −5.06202 + 0.I
u = 0.16409 + 2.41606I
a = −0.933673 − 0.063412I
b = −1.72502 + 0.37187I
−15.2129 −7.76641 + 0.I
u = 0.16409 − 2.41606I
a = −0.933673 + 0.063412I
b = −1.72502 − 0.37187I
−15.2129 −7.76641 + 0.I
15
IV. I
u
4
= h−3.43 × 10
13
u
15
+ 1.03 × 10
14
u
14
+ · · · + 2.05 × 10
14
b − 3.32 ×
10
15
, 2.81 × 10
14
u
15
− 8.09 × 10
14
u
14
+ · · · + 1.44 × 10
15
a + 3.26 ×
10
16
, u
16
− 2u
15
+ · · · + 300u + 100i
(i) Arc colorings
a
8
=
0
u
a
12
=
1
0
a
1
=
1
−u
2
a
4
=
−0.195312u
15
+ 0.562899u
14
+ ··· − 40.9348u − 22.6733
0.166903u
15
− 0.502461u
14
+ ··· + 33.4081u + 16.1608
a
3
=
−0.0284097u
15
+ 0.0604385u
14
+ ··· − 7.52666u − 6.51252
0.166903u
15
− 0.502461u
14
+ ··· + 33.4081u + 16.1608
a
11
=
−0.228188u
15
+ 0.663154u
14
+ ··· − 49.1027u − 24.8300
0.358944u
15
− 1.06222u
14
+ ··· + 76.6335u + 39.3872
a
10
=
−0.0462385u
15
+ 0.116397u
14
+ ··· − 11.6840u − 6.12062
0.189267u
15
− 0.550278u
14
+ ··· + 39.9708u + 21.1014
a
7
=
0.137787u
15
− 0.390781u
14
+ ··· + 30.8679u + 17.1770
0.0668156u
15
− 0.188222u
14
+ ··· + 14.9926u + 7.60864
a
5
=
−0.0655664u
15
+ 0.154266u
14
+ ··· − 16.4167u − 13.4236
0.0766237u
15
− 0.229597u
14
+ ··· + 16.2498u + 7.54204
a
2
=
0.161372u
15
− 0.474932u
14
+ ··· + 34.1101u + 18.2214
−0.181949u
15
+ 0.546757u
14
+ ··· − 37.4187u − 18.7094
a
6
=
−0.00353928u
15
+ 0.0318774u
14
+ ··· + 0.790632u + 3.47522
−0.145729u
15
+ 0.415741u
14
+ ··· − 31.8758u − 17.4904
a
9
=
u
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
682071438
1480097765
u
15
+
9582518138
7400488825
u
14
+ ··· −
28606345540
296019553
u −
90455921558
1480097765
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
8
− 4u
7
+ 14u
5
− 7u
4
− 14u
3
+ 22u
2
− 12u + 4)
2
c
2
, c
3
, c
6
c
11
u
16
+ 4u
15
+ ··· − 1324u + 244
c
4
, c
10
u
16
− 4u
15
+ ··· − 136u + 61
c
5
, c
8
, c
9
c
12
u
16
− 2u
15
+ ··· + 300u + 100
c
7
(u
4
− u
2
+ 1)
4
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
8
− 16y
7
+ 98y
6
− 264y
5
+ 353y
4
− 168y
3
+ 92y
2
+ 32y + 16)
2
c
2
, c
3
, c
6
c
11
y
16
+ 38y
15
+ ··· − 463680y + 59536
c
4
, c
10
y
16
+ 14y
15
+ ··· + 7612y + 3721
c
5
, c
8
, c
9
c
12
y
16
− 14y
15
+ ··· − 72000y + 10000
c
7
(y
2
− y + 1)
8
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.963502 + 0.055502I
a = 0.09642 − 1.67377I
b = −0.66220 + 2.05898I
7.23771 + 2.02988I −4.00000 − 3.46410I
u = −0.963502 − 0.055502I
a = 0.09642 + 1.67377I
b = −0.66220 − 2.05898I
7.23771 − 2.02988I −4.00000 + 3.46410I
u = 0.303110 + 0.990536I
a = 0.570516 + 0.174581I
b = 0.364193 − 0.687332I
−0.65797 + 2.02988I −4.00000 − 3.46410I
u = 0.303110 − 0.990536I
a = 0.570516 − 0.174581I
b = 0.364193 + 0.687332I
−0.65797 − 2.02988I −4.00000 + 3.46410I
u = 1.131100 + 0.420575I
a = −0.178491 − 0.480034I
b = 1.22413 + 1.19906I
−0.65797 + 2.02988I −4.00000 − 3.46410I
u = 1.131100 − 0.420575I
a = −0.178491 + 0.480034I
b = 1.22413 − 1.19906I
−0.65797 − 2.02988I −4.00000 + 3.46410I
u = −0.612127 + 0.162731I
a = 0.250693 − 0.943004I
b = −0.147418 − 0.121685I
−0.65797 + 2.02988I −4.00000 − 3.46410I
u = −0.612127 − 0.162731I
a = 0.250693 + 0.943004I
b = −0.147418 + 0.121685I
−0.65797 − 2.02988I −4.00000 + 3.46410I
u = −1.44011 + 0.50338I
a = 0.133675 − 0.382432I
b = 1.79516 + 0.39004I
−0.65797 − 2.02988I −4.00000 + 3.46410I
u = −1.44011 − 0.50338I
a = 0.133675 + 0.382432I
b = 1.79516 − 0.39004I
−0.65797 + 2.02988I −4.00000 − 3.46410I
19
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.77252 + 0.16127I
a = 0.082374 + 0.905350I
b = −0.49106 − 2.36800I
7.23771 + 2.02988I −4.00000 − 3.46410I
u = 1.77252 − 0.16127I
a = 0.082374 − 0.905350I
b = −0.49106 + 2.36800I
7.23771 − 2.02988I −4.00000 + 3.46410I
u = −1.15277 + 1.65532I
a = 0.658243 − 0.458401I
b = 0.19431 + 2.36522I
7.23771 − 2.02988I −4.00000 + 3.46410I
u = −1.15277 − 1.65532I
a = 0.658243 + 0.458401I
b = 0.19431 − 2.36522I
7.23771 + 2.02988I −4.00000 − 3.46410I
u = 1.96178 + 1.36397I
a = 0.386573 + 0.556004I
b = −0.27711 − 2.67423I
7.23771 − 2.02988I −4.00000 + 3.46410I
u = 1.96178 − 1.36397I
a = 0.386573 − 0.556004I
b = −0.27711 + 2.67423I
7.23771 + 2.02988I −4.00000 − 3.46410I
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
5
− u
3
+ 2u
2
− 2u + 1)
· (u
8
− 8u
7
+ 18u
6
+ 4u
5
− 59u
4
+ 60u
3
− 16u
2
+ 8u − 4)
2
· (u
8
− 4u
7
+ 14u
5
− 7u
4
− 14u
3
+ 22u
2
− 12u + 4)
2
· (u
20
+ 3u
19
+ ··· + 218u − 13)
c
2
, c
11
(u
5
− u
3
+ 2u
2
− 2u + 1)(u
16
+ 2u
15
+ ··· − 16u + 4)
· (u
16
+ 4u
15
+ ··· − 1324u + 244)(u
20
− u
19
+ ··· − 10u
2
− 1)
c
3
, c
6
(u
5
− u
3
− 2u
2
− 2u − 1)(u
16
− 2u
15
+ ··· + 16u + 4)
· (u
16
+ 4u
15
+ ··· − 1324u + 244)(u
20
− u
19
+ ··· − 10u
2
− 1)
c
4
, c
10
(u
5
− 5u
4
+ 9u
3
− 9u
2
+ 4u − 1)(u
16
− 4u
15
+ ··· − 136u + 61)
· (u
16
− 2u
15
+ ··· − 8u + 1)(u
20
− 8u
18
+ ··· − 4u + 1)
c
5
, c
8
(u
5
+ u
4
+ 2u
3
+ 3u
2
+ 3u + 1)(u
16
− 2u
15
+ ··· + 300u + 100)
· (u
16
− 2u
15
+ ··· + 8u + 4)(u
20
+ 9u
18
+ ··· + 5u + 1)
c
7
(u
4
− u
2
+ 1)
4
(u
4
− 2u
3
+ u
2
+ 2u − 1)
4
(u
5
+ 2u
4
+ u
3
− u
2
− u − 1)
· (u
20
+ u
19
+ ··· − 143u + 19)
c
9
, c
12
(u
5
− u
4
+ 2u
3
− 3u
2
+ 3u − 1)(u
16
− 2u
15
+ ··· + 300u + 100)
· (u
16
+ 2u
15
+ ··· − 8u + 4)(u
20
+ 9u
18
+ ··· + 5u + 1)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
5
− 2y
4
− 3y
3
− 1)(y
8
− 28y
7
+ ··· + 64y + 16)
2
· (y
8
− 16y
7
+ 98y
6
− 264y
5
+ 353y
4
− 168y
3
+ 92y
2
+ 32y + 16)
2
· (y
20
− 63y
19
+ ··· − 9564y + 169)
c
2
, c
3
, c
6
c
11
(y
5
− 2y
4
− 3y
3
− 1)(y
16
+ 2y
15
+ ··· − 48y + 16)
· (y
16
+ 38y
15
+ ··· − 463680y + 59536)(y
20
+ 5y
19
+ ··· + 20y + 1)
c
4
, c
10
(y
5
− 7y
4
− y
3
− 19y
2
− 2y −1)(y
16
− 10y
15
+ ··· + 8y + 1)
· (y
16
+ 14y
15
+ ··· + 7612y + 3721)(y
20
− 16y
19
+ ··· − 18y + 1)
c
5
, c
8
, c
9
c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
+ 3y −1)(y
16
− 14y
15
+ ··· − 72000y + 10000)
· (y
16
+ 14y
15
+ ··· + 112y + 16)(y
20
+ 18y
19
+ ··· − 15y + 1)
c
7
((y
2
− y + 1)
8
)(y
4
− 2y
3
+ ··· − 6y + 1)
4
(y
5
− 2y
4
+ ··· − y −1)
· (y
20
− 11y
19
+ ··· − 15167y + 361)
22
