12n
0270
(K12n
0270
)
A knot diagram
1
Linearized knot diagam
3 5 9 6 2 11 10 12 4 7 9 11
Solving Sequence
6,11 2,7
5 3 1 4 10 8 9 12
c
6
c
5
c
2
c
1
c
4
c
10
c
7
c
9
c
11
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h159u
20
− 349u
19
+ ··· + 1024b + 97, 65u
20
− 195u
19
+ ··· + 2048a + 2111, u
21
− 2u
20
+ ··· + 5u
2
− 1i
I
u
2
= h2u
7
+ 5u
6
+ 11u
5
+ 22u
4
+ 25u
3
+ 24u
2
+ 7b + 15u + 1,
− 19u
7
− 36u
6
− 87u
5
− 172u
4
− 186u
3
− 164u
2
+ 14a − 133u − 3,
u
8
+ 2u
7
+ 5u
6
+ 10u
5
+ 12u
4
+ 12u
3
+ 11u
2
+ 3u + 2i
I
u
3
= h−a
2
+ 2au + b + 2a − 2u − 1, a
4
− 3a
3
u − 4a
3
+ 9a
2
u + 5a
2
− 11au − 2a + 5u + 1, u
2
+ 1i
I
u
4
= h3642u
11
+ 10715u
10
+ ··· + 16346b + 454, −9302u
11
+ 5482u
10
+ ··· + 277882a − 125487,
u
12
+ 3u
11
+ 11u
10
+ 23u
9
+ 46u
8
+ 68u
7
+ 94u
6
+ 99u
5
+ 97u
4
+ 76u
3
+ 52u
2
+ 26u + 17i
I
u
5
= hb + 2a + 2, 4a
2
+ 10a + 7, u + 1i
* 5 irreducible components of dim
C
= 0, with total 51 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
= h159u
20
− 349u
19
+ · · · + 1024b + 97, 65u
20
− 195u
19
+ · · · + 2048a +
2111, u
21
− 2u
20
+ · · · + 5u
2
− 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
−0.0317383u
20
+ 0.0952148u
19
+ ··· + 6.03076u − 1.03076
−0.155273u
20
+ 0.340820u
19
+ ··· + 0.219727u − 0.0947266
a
7
=
1
−u
2
a
5
=
−0.395996u
20
+ 0.992676u
19
+ ··· + 0.588379u + 0.591309
0.114258u
20
− 0.327148u
19
+ ··· + 0.583008u − 0.442383
a
3
=
−0.340332u
20
+ 0.810059u
19
+ ··· + 1.36279u + 0.301270
0.106445u
20
− 0.0537109u
19
+ ··· + 0.825195u − 0.684570
a
1
=
u
0.0312500u
20
− 0.0312500u
19
+ ··· + 0.968750u − 0.0312500
a
4
=
−0.281738u
20
+ 0.665527u
19
+ ··· + 1.17139u + 0.148926
0.114258u
20
− 0.327148u
19
+ ··· + 0.583008u − 0.442383
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
−1
0.0312500u
20
− 0.0937500u
19
+ ··· − 0.0312500u + 0.0312500
a
12
=
u
0.0312500u
20
− 0.0312500u
19
+ ··· + 0.968750u − 0.0312500
(ii) Obstruction class = −1
(iii) Cusp Shapes =
9279
4096
u
20
−
13821
4096
u
19
+ ··· +
13503
4096
u +
26625
4096
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
21
+ 8u
20
+ ··· + 145u − 16
c
2
, c
5
u
21
+ 2u
20
+ ··· + 9u − 4
c
3
, c
9
u
21
+ 3u
20
+ ··· − 8u − 32
c
6
, c
7
, c
8
c
10
, c
11
u
21
− 2u
20
+ ··· + 5u
2
− 1
c
12
u
21
+ 26u
20
+ ··· + 10u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
21
+ 12y
20
+ ··· + 51681y −256
c
2
, c
5
y
21
+ 8y
20
+ ··· + 145y −16
c
3
, c
9
y
21
− 5y
20
+ ··· − 4928y −1024
c
6
, c
7
, c
8
c
10
, c
11
y
21
+ 26y
20
+ ··· + 10y −1
c
12
y
21
− 66y
20
+ ··· + 126y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.458142 + 0.833548I
a = 0.677568 − 0.339414I
b = −0.773041 + 0.928850I
4.17175 − 1.61049I 8.87690 − 1.72492I
u = 0.458142 − 0.833548I
a = 0.677568 + 0.339414I
b = −0.773041 − 0.928850I
4.17175 + 1.61049I 8.87690 + 1.72492I
u = 0.334381 + 0.773560I
a = 1.290350 − 0.376146I
b = −0.805389 − 0.873526I
4.35172 + 4.34513I 10.07573 − 8.03255I
u = 0.334381 − 0.773560I
a = 1.290350 + 0.376146I
b = −0.805389 + 0.873526I
4.35172 − 4.34513I 10.07573 + 8.03255I
u = 1.216790 + 0.212353I
a = 1.211710 + 0.267891I
b = −0.377864 − 0.854536I
1.30694 + 1.63824I −1.29573 + 4.22399I
u = 1.216790 − 0.212353I
a = 1.211710 − 0.267891I
b = −0.377864 + 0.854536I
1.30694 − 1.63824I −1.29573 − 4.22399I
u = 0.097170 + 0.403788I
a = 0.57317 + 1.41705I
b = 0.207107 + 0.829659I
−1.22812 + 1.66803I 2.33962 − 5.96953I
u = 0.097170 − 0.403788I
a = 0.57317 − 1.41705I
b = 0.207107 − 0.829659I
−1.22812 − 1.66803I 2.33962 + 5.96953I
u = 0.381501
a = 0.337636
b = 0.264712
0.708376 14.4470
u = −0.02913 + 1.62816I
a = −0.961297 + 0.342959I
b = 1.038150 − 0.513588I
−10.09800 − 1.80625I 3.06957 + 1.66115I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.02913 − 1.62816I
a = −0.961297 − 0.342959I
b = 1.038150 + 0.513588I
−10.09800 + 1.80625I 3.06957 − 1.66115I
u = −0.38269 + 1.61174I
a = 0.856171 + 0.551613I
b = −1.005420 − 0.467651I
−10.43370 − 7.89166I 3.87913 + 3.10640I
u = −0.38269 − 1.61174I
a = 0.856171 − 0.551613I
b = −1.005420 + 0.467651I
−10.43370 + 7.89166I 3.87913 − 3.10640I
u = 0.10115 + 1.67309I
a = −1.266450 + 0.124462I
b = 0.725547 + 1.193790I
−12.24280 + 4.61265I 1.46303 − 2.55091I
u = 0.10115 − 1.67309I
a = −1.266450 − 0.124462I
b = 0.725547 − 1.193790I
−12.24280 − 4.61265I 1.46303 + 2.55091I
u = −0.49023 + 1.63779I
a = 1.53317 + 0.26133I
b = −0.694270 + 1.177460I
−12.6524 − 14.0619I 2.23345 + 6.95334I
u = −0.49023 − 1.63779I
a = 1.53317 − 0.26133I
b = −0.694270 − 1.177460I
−12.6524 + 14.0619I 2.23345 − 6.95334I
u = −0.265665 + 0.100260I
a = −3.22189 + 1.30088I
b = 0.565254 − 0.857227I
0.38970 − 2.24826I 1.51589 + 3.88242I
u = −0.265665 − 0.100260I
a = −3.22189 − 1.30088I
b = 0.565254 + 0.857227I
0.38970 + 2.24826I 1.51589 − 3.88242I
u = −0.23068 + 1.77893I
a = −0.111321 − 0.270180I
b = −0.012425 − 1.345530I
−17.3796 − 4.8017I −0.50589 + 2.16688I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.23068 − 1.77893I
a = −0.111321 + 0.270180I
b = −0.012425 + 1.345530I
−17.3796 + 4.8017I −0.50589 − 2.16688I
7
II.
I
u
2
= h2u
7
+5u
6
+· · ·+7b+1, −19u
7
−36u
6
+· · ·+14a−3, u
8
+2u
7
+· · ·+3u+2i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
19
14
u
7
+
18
7
u
6
+ ··· +
19
2
u +
3
14
−
2
7
u
7
−
5
7
u
6
+ ··· −
15
7
u −
1
7
a
7
=
1
−u
2
a
5
=
0.785714u
7
+ 1.71429u
6
+ ··· + 8.64286u + 3.64286
−
2
7
u
7
−
4
7
u
6
+ ··· −
16
7
u −
5
7
a
3
=
u
7
+
16
7
u
6
+ ··· +
75
7
u +
20
7
−
2
7
u
7
−
6
7
u
6
+ ··· − 3u −
4
7
a
1
=
0.785714u
7
+ 1.57143u
6
+ ··· + 7.78571u + 2.21429
−
2
7
u
7
−
4
7
u
6
+ ··· −
16
7
u −
5
7
a
4
=
1
2
u
7
+
8
7
u
6
+ ··· +
89
14
u +
41
14
−
2
7
u
7
−
4
7
u
6
+ ··· −
16
7
u −
5
7
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
−
5
14
u
7
−
3
7
u
6
+ ··· −
9
14
u +
31
14
−
1
7
u
6
−
2
7
u
4
+ ··· +
1
7
u −
3
7
a
12
=
0.785714u
7
+ 1.57143u
6
+ ··· + 7.78571u + 2.21429
−
3
7
u
7
−
4
7
u
6
+ ··· −
5
7
u −
5
7
(ii) Obstruction class = −1
(iii) Cusp Shapes =
8
7
u
7
+
20
7
u
6
+
44
7
u
5
+
88
7
u
4
+
128
7
u
3
+
96
7
u
2
+
88
7
u +
74
7
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
4
+ 2u
3
+ 3u
2
+ u + 1)
2
c
2
, c
5
(u
4
+ u
2
− u + 1)
2
c
3
, c
9
(u
4
+ u
2
+ u + 1)
2
c
6
, c
7
, c
8
c
10
, c
11
u
8
+ 2u
7
+ 5u
6
+ 10u
5
+ 12u
4
+ 12u
3
+ 11u
2
+ 3u + 2
c
12
u
8
+ 6u
7
+ 9u
6
− 6u
5
+ 6u
4
+ 80u
3
+ 97u
2
+ 35u + 4
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)
2
c
2
, c
3
, c
5
c
9
(y
4
+ 2y
3
+ 3y
2
+ y + 1)
2
c
6
, c
7
, c
8
c
10
, c
11
y
8
+ 6y
7
+ 9y
6
− 6y
5
+ 6y
4
+ 80y
3
+ 97y
2
+ 35y + 4
c
12
y
8
− 18y
7
+ ··· − 449y + 16
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.003353 + 1.153470I
a = 0.283780 − 0.486090I
b = 0.547424 + 0.585652I
−2.30977 + 1.39709I 7.77019 − 3.86736I
u = 0.003353 − 1.153470I
a = 0.283780 + 0.486090I
b = 0.547424 − 0.585652I
−2.30977 − 1.39709I 7.77019 + 3.86736I
u = −1.281480 + 0.482756I
a = 1.44914 − 0.47651I
b = −0.547424 + 1.120870I
−5.91490 − 7.64338I 2.22981 + 6.51087I
u = −1.281480 − 0.482756I
a = 1.44914 + 0.47651I
b = −0.547424 − 1.120870I
−5.91490 + 7.64338I 2.22981 − 6.51087I
u = −0.046668 + 0.512275I
a = −2.11815 + 3.03669I
b = 0.547424 − 0.585652I
−2.30977 − 1.39709I 7.77019 + 3.86736I
u = −0.046668 − 0.512275I
a = −2.11815 − 3.03669I
b = 0.547424 + 0.585652I
−2.30977 + 1.39709I 7.77019 − 3.86736I
u = 0.32480 + 1.70994I
a = 1.135230 − 0.382122I
b = −0.547424 − 1.120870I
−5.91490 + 7.64338I 2.22981 − 6.51087I
u = 0.32480 − 1.70994I
a = 1.135230 + 0.382122I
b = −0.547424 + 1.120870I
−5.91490 − 7.64338I 2.22981 + 6.51087I
11
III. I
u
3
= h−a
2
+ 2au + b + 2a − 2u − 1, −3a
3
u + 9a
2
u + · · · − 2a + 1, u
2
+ 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
a
a
2
− 2au − 2a + 2u + 1
a
7
=
1
1
a
5
=
a
3
− 2a
2
u − 2a
2
+ 2au + a + 1
−a
3
u + 3a
2
u − 3a
2
− au + 6a − u − 4
a
3
=
−a
3
u + 4a
2
u − 2a
2
− 5au + 6a + 3u − 4
a
3
u − 3a
2
u + 4a
2
− 8a + 2u + 6
a
1
=
u
au − u + 2
a
4
=
−a
3
u + a
3
+ a
2
u − 5a
2
+ au + 7a − u − 3
−a
3
u + 3a
2
u − 3a
2
− au + 6a − u − 4
a
10
=
−u
0
a
8
=
0
1
a
9
=
−1
−a + 2u + 1
a
12
=
u
au + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a
3
u − 12a
2
u + 8a
2
+ 12au − 16a − 4u + 12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
2
(u
4
− u
3
+ u
2
+ 1)
2
c
3
, c
9
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
c
5
(u
4
+ u
3
+ u
2
+ 1)
2
c
6
, c
7
, c
8
c
10
, c
11
(u
2
+ 1)
4
c
12
(u + 1)
8
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
2
, c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
3
, c
9
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
c
6
, c
7
, c
8
c
10
, c
11
(y + 1)
8
c
12
(y −1)
8
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 0.674360 − 0.399232I
b = −0.851808 + 0.911292I
3.50087 − 3.16396I 3.82674 + 2.56480I
u = 1.000000I
a = 1.325640 − 0.399232I
b = −0.851808 − 0.911292I
3.50087 + 3.16396I 3.82674 − 2.56480I
u = 1.000000I
a = 0.59947 + 1.89923I
b = 0.351808 − 0.720342I
−3.50087 − 1.41510I 0.17326 + 4.90874I
u = 1.000000I
a = 1.40053 + 1.89923I
b = 0.351808 + 0.720342I
−3.50087 + 1.41510I 0.17326 − 4.90874I
u = − 1.000000I
a = 0.674360 + 0.399232I
b = −0.851808 − 0.911292I
3.50087 + 3.16396I 3.82674 − 2.56480I
u = − 1.000000I
a = 1.325640 + 0.399232I
b = −0.851808 + 0.911292I
3.50087 − 3.16396I 3.82674 + 2.56480I
u = − 1.000000I
a = 0.59947 − 1.89923I
b = 0.351808 + 0.720342I
−3.50087 + 1.41510I 0.17326 − 4.90874I
u = − 1.000000I
a = 1.40053 − 1.89923I
b = 0.351808 − 0.720342I
−3.50087 − 1.41510I 0.17326 + 4.90874I
15
IV. I
u
4
= h3642u
11
+ 10715u
10
+ · · · + 16346b + 454, −9302u
11
+ 5482u
10
+
· · · + 277882a − 125487, u
12
+ 3u
11
+ · · · + 26u + 17i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
0.0334746u
11
− 0.0197278u
10
+ ··· + 2.00276u + 0.451584
−0.222807u
11
− 0.655512u
10
+ ··· − 1.99700u − 0.0277744
a
7
=
1
−u
2
a
5
=
0.128213u
11
+ 0.278557u
10
+ ··· − 4.21560u − 3.08761
−0.0231249u
11
− 0.186651u
10
+ ··· − 0.439557u − 0.366145
a
3
=
0.0932842u
11
+ 0.0428527u
10
+ ··· − 4.81556u − 3.27790
−0.0855255u
11
− 0.333170u
10
+ ··· − 1.36376u − 0.854154
a
1
=
0.125996u
11
+ 0.290076u
10
+ ··· + 3.62135u + 1.59297
−0.0671724u
11
− 0.113606u
10
+ ··· − 0.562523u − 0.0635630
a
4
=
0.105088u
11
+ 0.0919059u
10
+ ··· − 4.65516u − 3.45376
−0.0231249u
11
− 0.186651u
10
+ ··· − 0.439557u − 0.366145
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
−0.00373900u
11
+ 0.0559554u
10
+ ··· + 0.431964u + 1.46531
0.0879114u
11
+ 0.316714u
10
+ ··· + 1.68292u + 0.141931
a
12
=
0.125996u
11
+ 0.290076u
10
+ ··· + 3.62135u + 1.59297
−0.0141931u
11
− 0.0431298u
10
+ ··· − 0.706289u − 1.55806
(ii) Obstruction class = −1
(iii) Cusp Shapes =
2796
8173
u
11
+
10892
8173
u
10
+
36956
8173
u
9
+
76592
8173
u
8
+
143424
8173
u
7
+
188928
8173
u
6
+
226188
8173
u
5
+
193280
8173
u
4
+
144560
8173
u
3
+
67608
8173
u
2
+
44584
8173
u +
44270
8173
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
2
c
2
, c
5
(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
2
c
3
, c
9
(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
2
c
6
, c
7
, c
8
c
10
, c
11
u
12
+ 3u
11
+ ··· + 26u + 17
c
12
u
12
+ 13u
11
+ ··· + 1092u + 289
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
2
c
2
, c
3
, c
5
c
9
(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
2
c
6
, c
7
, c
8
c
10
, c
11
y
12
+ 13y
11
+ ··· + 1092y + 289
c
12
y
12
− 19y
11
+ ··· − 7564y + 83521
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.942355 + 0.499238I
a = 1.51895 + 0.47306I
b = −0.713912 − 0.305839I
−3.55561 − 2.82812I 5.50976 + 2.97945I
u = −0.942355 − 0.499238I
a = 1.51895 − 0.47306I
b = −0.713912 + 0.305839I
−3.55561 + 2.82812I 5.50976 − 2.97945I
u = 0.343993 + 0.784320I
a = −3.38338 − 0.25597I
b = 0.498832 + 1.001300I
−3.55561 + 2.82812I 5.50976 − 2.97945I
u = 0.343993 − 0.784320I
a = −3.38338 + 0.25597I
b = 0.498832 − 1.001300I
−3.55561 − 2.82812I 5.50976 + 2.97945I
u = 0.072139 + 1.221000I
a = −0.36108 − 1.66788I
b = 0.498832 − 1.001300I
−3.55561 − 2.82812I 5.50976 + 2.97945I
u = 0.072139 − 1.221000I
a = −0.36108 + 1.66788I
b = 0.498832 + 1.001300I
−3.55561 + 2.82812I 5.50976 − 2.97945I
u = −0.98583 + 1.05129I
a = 0.337035 + 0.395158I
b = −0.284920 − 1.115140I
−7.69319 −6 − 1.019511 + 0.10I
u = −0.98583 − 1.05129I
a = 0.337035 − 0.395158I
b = −0.284920 + 1.115140I
−7.69319 −6 − 1.019511 + 0.10I
u = 0.18858 + 1.49820I
a = 0.690257 − 0.163478I
b = −0.713912 + 0.305839I
−3.55561 + 2.82812I 5.50976 − 2.97945I
u = 0.18858 − 1.49820I
a = 0.690257 + 0.163478I
b = −0.713912 − 0.305839I
−3.55561 − 2.82812I 5.50976 + 2.97945I
19
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.17653 + 1.68674I
a = 0.786457 + 0.514816I
b = −0.284920 + 1.115140I
−7.69319 −6 − 1.019511 + 0.10I
u = −0.17653 − 1.68674I
a = 0.786457 − 0.514816I
b = −0.284920 − 1.115140I
−7.69319 −6 − 1.019511 + 0.10I
20
V. I
u
5
= hb + 2a + 2, 4a
2
+ 10a + 7, u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
−1
a
2
=
a
−2a − 2
a
7
=
1
−1
a
5
=
3a +
9
2
−2a − 3
a
3
=
a +
3
2
−2a − 3
a
1
=
−1
0
a
4
=
a +
3
2
−2a − 3
a
10
=
1
−2
a
8
=
2
−3
a
9
=
1
−2
a
12
=
−1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =
31
2
a +
59
2
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
u
2
− u + 1
c
2
u
2
+ u + 1
c
3
, c
9
u
2
c
6
, c
7
, c
8
(u + 1)
2
c
10
, c
11
, c
12
(u − 1)
2
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
y
2
+ y + 1
c
3
, c
9
y
2
c
6
, c
7
, c
8
c
10
, c
11
, c
12
(y −1)
2
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.250000 + 0.433013I
b = 0.500000 − 0.866025I
1.64493 − 2.02988I 10.12500 + 6.71170I
u = −1.00000
a = −1.250000 − 0.433013I
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 10.12500 − 6.71170I
24
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
2
− u + 1)(u
4
− u
3
+ 3u
2
− 2u + 1)
2
(u
4
+ 2u
3
+ 3u
2
+ u + 1)
2
· ((u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
2
)(u
21
+ 8u
20
+ ··· + 145u − 16)
c
2
(u
2
+ u + 1)(u
4
+ u
2
− u + 1)
2
(u
4
− u
3
+ u
2
+ 1)
2
· ((u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
2
)(u
21
+ 2u
20
+ ··· + 9u − 4)
c
3
, c
9
u
2
(u
4
+ u
2
+ u + 1)
2
(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
2
· (u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1)(u
21
+ 3u
20
+ ··· − 8u − 32)
c
5
(u
2
− u + 1)(u
4
+ u
2
− u + 1)
2
(u
4
+ u
3
+ u
2
+ 1)
2
· ((u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
2
)(u
21
+ 2u
20
+ ··· + 9u − 4)
c
6
, c
7
, c
8
(u + 1)
2
(u
2
+ 1)
4
· (u
8
+ 2u
7
+ 5u
6
+ 10u
5
+ 12u
4
+ 12u
3
+ 11u
2
+ 3u + 2)
· (u
12
+ 3u
11
+ ··· + 26u + 17)(u
21
− 2u
20
+ ··· + 5u
2
− 1)
c
10
, c
11
(u − 1)
2
(u
2
+ 1)
4
· (u
8
+ 2u
7
+ 5u
6
+ 10u
5
+ 12u
4
+ 12u
3
+ 11u
2
+ 3u + 2)
· (u
12
+ 3u
11
+ ··· + 26u + 17)(u
21
− 2u
20
+ ··· + 5u
2
− 1)
c
12
((u − 1)
2
)(u + 1)
8
(u
8
+ 6u
7
+ ··· + 35u + 4)
· (u
12
+ 13u
11
+ ··· + 1092u + 289)(u
21
+ 26u
20
+ ··· + 10u − 1)
25
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
2
+ y + 1)(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)
2
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· ((y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
2
)(y
21
+ 12y
20
+ ··· + 51681y −256)
c
2
, c
5
(y
2
+ y + 1)(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
(y
4
+ 2y
3
+ 3y
2
+ y + 1)
2
· ((y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
2
)(y
21
+ 8y
20
+ ··· + 145y −16)
c
3
, c
9
y
2
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
(y
4
+ 2y
3
+ 3y
2
+ y + 1)
2
· ((y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
2
)(y
21
− 5y
20
+ ··· − 4928y −1024)
c
6
, c
7
, c
8
c
10
, c
11
((y −1)
2
)(y + 1)
8
(y
8
+ 6y
7
+ ··· + 35y + 4)
· (y
12
+ 13y
11
+ ··· + 1092y + 289)(y
21
+ 26y
20
+ ··· + 10y −1)
c
12
((y −1)
10
)(y
8
− 18y
7
+ ··· − 449y + 16)
· (y
12
− 19y
11
+ ··· − 7564y + 83521)(y
21
− 66y
20
+ ··· + 126y −1)
26
