12n
0253
(K12n
0253
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 12 10 11 3 7 5 7 10
Solving Sequence
7,11 3,8
4 9 10 12 1 6 5 2
c
7
c
3
c
8
c
9
c
11
c
12
c
6
c
5
c
2
c
1
, c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h7195073323u
17
− 14120105470u
16
+ ··· + 28350336356b + 9967888544,
10882383441u
17
− 7621235445u
16
+ ··· + 28350336356a − 6834662691, u
18
− u
17
+ ··· + 2u − 1i
I
u
2
= hu
3
+ b + 1, −u
2
+ a − u − 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
I
u
3
= h3.21010 × 10
22
u
19
+ 1.39979 × 10
23
u
18
+ ··· + 1.27572 × 10
24
b − 1.65660 × 10
24
,
4.70076 × 10
24
u
19
+ 1.58931 × 10
25
u
18
+ ··· + 1.28848 × 10
26
a − 8.29304 × 10
26
,
u
20
+ 3u
19
+ ··· − 376u − 101i
I
u
4
= h−u
4
− u
2
+ b − 2u − 2, 2u
4
− u
3
+ 3u
2
+ a + 4u + 2, u
5
− u
4
+ u
3
+ 2u
2
− u − 1i
I
u
5
= h2b + 1, 2a + u − 2, u
2
− u − 1i
* 5 irreducible components of dim
C
= 0, with total 49 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
= h7.20 × 10
9
u
17
− 1.41 × 10
10
u
16
+ · · · + 2.84 × 10
10
b + 9.97 × 10
9
, 1.09 ×
10
10
u
17
−7.62×10
9
u
16
+· · ·+2.84×10
10
a−6.83×10
9
, u
18
−u
17
+· · ·+2u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
−0.383854u
17
+ 0.268823u
16
+ ··· + 8.54749u + 0.241079
−0.253791u
17
+ 0.498058u
16
+ ··· + 0.630885u − 0.351597
a
8
=
1
−u
2
a
4
=
−0.777946u
17
+ 0.831176u
16
+ ··· + 9.02458u − 0.225548
−0.526419u
17
+ 0.757093u
16
+ ··· + 1.36150u − 0.519857
a
9
=
−0.397400u
17
+ 0.367268u
16
+ ··· + 0.173895u − 0.480447
0.397400u
17
− 0.367268u
16
+ ··· − 1.17390u + 0.480447
a
10
=
−u
0.397400u
17
− 0.367268u
16
+ ··· − 1.17390u + 0.480447
a
12
=
u
u
a
1
=
−0.0564932u
17
+ 0.000418696u
16
+ ··· + 1.33714u + 0.0301320
0.397400u
17
− 0.367268u
16
+ ··· − 1.17390u + 0.480447
a
6
=
−0.0301320u
17
+ 0.0866252u
16
+ ··· + 0.314353u + 0.602600
−0.510579u
17
+ 0.169672u
16
+ ··· + 1.32635u + 0.815602
a
5
=
1
−0.480447u
17
+ 0.0830471u
16
+ ··· + 1.01199u + 1.21300
a
2
=
−0.0850210u
17
− 0.406163u
16
+ ··· + 4.28732u + 2.74100
0.642918u
17
− 0.279464u
16
+ ··· − 1.90934u − 0.721817
(ii) Obstruction class = −1
(iii) Cusp Shapes =
48478562013
14175168178
u
17
−
74014314443
56700672712
u
16
+ ···−
808636576713
56700672712
u −
444360496187
56700672712
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
18
+ 17u
17
+ ··· + 9840u + 256
c
2
, c
4
u
18
− 3u
17
+ ··· − 108u + 16
c
3
, c
8
u
18
+ 5u
17
+ ··· − 144u + 64
c
5
, c
6
, c
9
u
18
− 10u
16
+ ··· + 3u + 1
c
7
, c
10
, c
11
u
18
+ u
17
+ ··· − 2u − 1
c
12
u
18
+ 19u
17
+ ··· − 352u − 32
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
18
− 29y
17
+ ··· − 79539968y + 65536
c
2
, c
4
y
18
− 17y
17
+ ··· − 9840y + 256
c
3
, c
8
y
18
+ 9y
17
+ ··· − 45824y + 4096
c
5
, c
6
, c
9
y
18
− 20y
17
+ ··· − 13y + 1
c
7
, c
10
, c
11
y
18
+ 17y
17
+ ··· + 22y + 1
c
12
y
18
− 7y
17
+ ··· + 2560y + 1024
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.118044 + 0.790449I
a = 0.241077 − 0.719892I
b = −1.304910 − 0.071727I
3.00375 + 0.37123I 4.92078 − 0.99425I
u = 0.118044 − 0.790449I
a = 0.241077 + 0.719892I
b = −1.304910 + 0.071727I
3.00375 − 0.37123I 4.92078 + 0.99425I
u = −0.065288 + 1.324970I
a = −0.241397 + 1.248350I
b = −0.138781 + 0.489222I
−2.87372 − 1.22079I 4.21569 + 1.79625I
u = −0.065288 − 1.324970I
a = −0.241397 − 1.248350I
b = −0.138781 − 0.489222I
−2.87372 + 1.22079I 4.21569 − 1.79625I
u = 0.077894 + 0.510339I
a = −0.365385 + 1.013810I
b = 1.153220 − 0.268641I
1.16211 − 5.14367I 0.72332 + 8.80281I
u = 0.077894 − 0.510339I
a = −0.365385 − 1.013810I
b = 1.153220 + 0.268641I
1.16211 + 5.14367I 0.72332 − 8.80281I
u = 0.29181 + 1.48433I
a = −0.145718 + 0.537802I
b = 1.77462 + 0.55736I
−4.07395 + 5.05034I 3.51840 − 3.93444I
u = 0.29181 − 1.48433I
a = −0.145718 − 0.537802I
b = 1.77462 − 0.55736I
−4.07395 − 5.05034I 3.51840 + 3.93444I
u = 0.82095 + 1.28564I
a = 0.739381 − 0.956915I
b = −0.835011 − 0.397109I
−13.28510 + 3.46125I 10.05857 − 2.89058I
u = 0.82095 − 1.28564I
a = 0.739381 + 0.956915I
b = −0.835011 + 0.397109I
−13.28510 − 3.46125I 10.05857 + 2.89058I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.53352
a = 0.113874
b = −0.624141
7.37422 37.5360
u = −0.465205
a = −0.483543
b = −0.301112
0.706459 14.1730
u = −0.54788 + 1.47697I
a = −0.066154 − 1.204480I
b = 1.025710 − 0.899512I
−0.56915 − 8.13906I 5.11000 + 5.47218I
u = −0.54788 − 1.47697I
a = −0.066154 + 1.204480I
b = 1.025710 + 0.899512I
−0.56915 + 8.13906I 5.11000 − 5.47218I
u = 0.126997 + 0.287207I
a = 0.04063 + 2.28012I
b = 0.432312 − 0.429086I
−1.64243 + 0.66705I −2.34520 − 2.39491I
u = 0.126997 − 0.287207I
a = 0.04063 − 2.28012I
b = 0.432312 + 0.429086I
−1.64243 − 0.66705I −2.34520 + 2.39491I
u = −0.85668 + 1.69020I
a = 0.232404 + 1.080370I
b = −1.64453 + 1.59807I
−6.3235 − 13.9812I 2.81858 + 6.62298I
u = −0.85668 − 1.69020I
a = 0.232404 − 1.080370I
b = −1.64453 − 1.59807I
−6.3235 + 13.9812I 2.81858 − 6.62298I
6
II. I
u
2
= hu
3
+ b + 1, −u
2
+ a − u − 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
u
2
+ u + 1
−u
3
− 1
a
8
=
1
−u
2
a
4
=
−u
3
− u − 1
−u
3
− u − 1
a
9
=
u
3
+ u
2
+ u + 2
−u
3
− u
2
− 2u − 2
a
10
=
−u
−u
3
− u
2
− 2u − 2
a
12
=
u
u
a
1
=
u
3
+ 2u
u
3
+ u
2
+ 2u + 2
a
6
=
0
2u
3
+ u
2
+ 3u + 2
a
5
=
−1
2u
3
+ u
2
+ 3u + 1
a
2
=
u
u
3
+ 3u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
2
+ u + 8
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− 3u
3
+ 5u
2
− 3u + 1
c
2
u
4
+ u
3
− u
2
− u + 1
c
3
, c
5
, c
9
u
4
− 2u
3
+ 2u
2
− u + 1
c
4
, c
12
u
4
− u
3
− u
2
+ u + 1
c
6
, c
8
u
4
+ 2u
3
+ 2u
2
+ u + 1
c
7
, c
10
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
11
u
4
− u
3
+ 2u
2
− 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
4
+ y
3
+ 9y
2
+ y + 1
c
2
, c
4
, c
12
y
4
− 3y
3
+ 5y
2
− 3y + 1
c
3
, c
5
, c
6
c
8
, c
9
y
4
+ 2y
2
+ 3y + 1
c
7
, c
10
, c
11
y
4
+ 3y
3
+ 2y
2
+ 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = 0.570696 − 0.107280I
b = −1.121740 − 0.425428I
1.74699 + 4.62527I 8.34046 − 2.29879I
u = −0.621744 − 0.440597I
a = 0.570696 + 0.107280I
b = −1.121740 + 0.425428I
1.74699 − 4.62527I 8.34046 + 2.29879I
u = 0.121744 + 1.306620I
a = −0.57070 + 1.62477I
b = −0.37826 + 2.17265I
−5.03685 + 0.56550I −0.34046 + 2.89736I
u = 0.121744 − 1.306620I
a = −0.57070 − 1.62477I
b = −0.37826 − 2.17265I
−5.03685 − 0.56550I −0.34046 − 2.89736I
10
III. I
u
3
=
h3.21×10
22
u
19
+1.40×10
23
u
18
+· · ·+1.28×10
24
b−1.66×10
24
, 4.70×10
24
u
19
+
1.59 × 10
25
u
18
+ · · · + 1.29 × 10
26
a − 8.29 × 10
26
, u
20
+ 3u
19
+ · · · − 376u − 101i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
−0.0364831u
19
− 0.123348u
18
+ ··· − 2.99222u + 6.43631
−0.0251630u
19
− 0.109726u
18
+ ··· + 10.5249u + 1.29856
a
8
=
1
−u
2
a
4
=
−0.0190479u
19
− 0.0791729u
18
+ ··· − 1.37806u + 6.33110
0.00211539u
19
− 0.0452946u
18
+ ··· + 11.8210u + 2.11973
a
9
=
−0.0344491u
19
− 0.0902792u
18
+ ··· − 0.756139u + 5.71229
−0.0246550u
19
− 0.0750446u
18
+ ··· + 4.35620u − 0.000767038
a
10
=
−0.0591041u
19
− 0.165324u
18
+ ··· + 3.60006u + 5.71152
−0.0246550u
19
− 0.0750446u
18
+ ··· + 4.35620u − 0.000767038
a
12
=
u
u
a
1
=
0.0699474u
19
+ 0.209299u
18
+ ··· − 4.79549u − 9.53973
0.0255973u
19
+ 0.0893164u
18
+ ··· − 5.84866u − 0.104669
a
6
=
−0.0266289u
19
− 0.0263605u
18
+ ··· − 1.12655u − 5.56497
−0.0255925u
19
− 0.0488488u
18
+ ··· − 2.27935u − 0.105970
a
5
=
−0.0141044u
19
− 0.0539607u
18
+ ··· + 8.39337u − 2.97964
−0.0130681u
19
− 0.0764490u
18
+ ··· + 7.24057u + 2.47936
a
2
=
0.00280493u
19
− 0.0244296u
18
+ ··· + 1.75093u + 2.31489
−0.0326545u
19
− 0.134709u
18
+ ··· + 10.6296u + 2.01431
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1916248316412301273576
20913445507764897101425
u
19
+
6077402466516369822264
20913445507764897101425
u
18
+ ··· −
339832460854605717393908
20913445507764897101425
u +
38632741605919100523346
20913445507764897101425
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
+ 5u
4
+ 8u
3
+ 3u
2
− u + 1)
4
c
2
, c
4
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
4
c
3
, c
8
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
4
c
5
, c
6
, c
9
u
20
+ 3u
19
+ ··· − 690u − 209
c
7
, c
10
, c
11
u
20
− 3u
19
+ ··· + 376u − 101
c
12
(u
2
− u − 1)
10
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y −1)
4
c
2
, c
4
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
4
c
3
, c
8
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
4
c
5
, c
6
, c
9
y
20
− 9y
19
+ ··· − 194368y + 43681
c
7
, c
10
, c
11
y
20
+ 11y
19
+ ··· − 147436y + 10201
c
12
(y
2
− 3y + 1)
10
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.18182
a = −3.08950
b = 3.54825
1.54676 4.51890
u = 0.392602 + 1.119800I
a = 0.865506 − 0.649555I
b = 0.719869 − 0.653025I
−4.27694 + 1.53058I 5.48489 − 4.43065I
u = 0.392602 − 1.119800I
a = 0.865506 + 0.649555I
b = 0.719869 + 0.653025I
−4.27694 − 1.53058I 5.48489 + 4.43065I
u = −1.251030 + 0.315505I
a = −0.411298 + 0.093081I
b = 0.868398 + 0.361281I
3.61874 + 1.53058I 5.48489 − 4.43065I
u = −1.251030 − 0.315505I
a = −0.411298 − 0.093081I
b = 0.868398 − 0.361281I
3.61874 − 1.53058I 5.48489 + 4.43065I
u = 0.342814 + 0.586956I
a = −1.19319 − 3.16782I
b = −1.206580 + 0.583471I
3.61874 + 1.53058I 5.48489 − 4.43065I
u = 0.342814 − 0.586956I
a = −1.19319 + 3.16782I
b = −1.206580 − 0.583471I
3.61874 − 1.53058I 5.48489 + 4.43065I
u = −0.181709 + 1.389530I
a = −1.37865 − 0.57028I
b = −3.09521 − 1.77844I
−6.34892 4.51886 + 0.I
u = −0.181709 − 1.389530I
a = −1.37865 + 0.57028I
b = −3.09521 + 1.77844I
−6.34892 4.51886 + 0.I
u = −0.11218 + 1.41123I
a = 0.578243 + 0.977323I
b = 0.379194 + 0.917038I
−1.92472 + 4.40083I 1.25569 − 3.49859I
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.11218 − 1.41123I
a = 0.578243 − 0.977323I
b = 0.379194 − 0.917038I
−1.92472 − 4.40083I 1.25569 + 3.49859I
u = −0.04570 + 1.46451I
a = −0.003969 + 1.233620I
b = 0.16549 + 1.82037I
−4.27694 − 1.53058I 5.48489 + 4.43065I
u = −0.04570 − 1.46451I
a = −0.003969 − 1.233620I
b = 0.16549 − 1.82037I
−4.27694 + 1.53058I 5.48489 − 4.43065I
u = 1.15161 + 1.24448I
a = −0.402882 + 0.473976I
b = 0.813208 + 0.060105I
−9.82040 + 4.40083I 1.25569 − 3.49859I
u = 1.15161 − 1.24448I
a = −0.402882 − 0.473976I
b = 0.813208 − 0.060105I
−9.82040 − 4.40083I 1.25569 + 3.49859I
u = −0.230378
a = 7.85497
b = −1.18372
1.54676 4.51890
u = −1.88238 + 0.35860I
a = 0.067918 + 0.167881I
b = −0.91426 − 1.68174I
−1.92472 + 4.40083I 1.25569 − 3.49859I
u = −1.88238 − 0.35860I
a = 0.067918 − 0.167881I
b = −0.91426 + 1.68174I
−1.92472 − 4.40083I 1.25569 + 3.49859I
u = −0.38975 + 1.92049I
a = −0.068772 − 0.859380I
b = 0.58762 − 1.94190I
−9.82040 − 4.40083I 1.25569 + 3.49859I
u = −0.38975 − 1.92049I
a = −0.068772 + 0.859380I
b = 0.58762 + 1.94190I
−9.82040 + 4.40083I 1.25569 − 3.49859I
15
IV.
I
u
4
= h−u
4
−u
2
+b−2u−2, 2u
4
−u
3
+3u
2
+a+4u+2, u
5
−u
4
+u
3
+2u
2
−u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
−2u
4
+ u
3
− 3u
2
− 4u − 2
u
4
+ u
2
+ 2u + 2
a
8
=
1
−u
2
a
4
=
−3u
4
+ 2u
3
− 4u
2
− 5u − 1
u
4
− u
3
+ u
2
+ 3u + 2
a
9
=
−u
4
+ u
3
− u
2
− 3u + 1
u
4
− u
3
+ u
2
+ 2u − 1
a
10
=
−u
u
4
− u
3
+ u
2
+ 2u − 1
a
12
=
u
u
a
1
=
u
3
+ 2u
−u
4
+ u
3
− u
2
− 2u + 1
a
6
=
0
−u
4
+ 2u
3
− 2u
2
− u + 3
a
5
=
−1
−u
4
+ 2u
3
− 2u
2
− u + 2
a
2
=
−u
4
− u
2
− 3u − 3
2u
3
− u
2
+ u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
4
+ 3u
3
+ 5u
2
+ 16u + 19
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− 7u
4
+ 15u
3
− 7u
2
− 2u − 1
c
2
u
5
+ 3u
4
+ u
3
− 3u
2
− 2u − 1
c
3
u
5
+ 2u
4
+ 5u
3
+ 4u
2
− 1
c
4
u
5
− 3u
4
+ u
3
+ 3u
2
− 2u + 1
c
5
, c
9
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
6
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
7
, c
10
u
5
− u
4
+ u
3
+ 2u
2
− u − 1
c
8
u
5
− 2u
4
+ 5u
3
− 4u
2
+ 1
c
11
u
5
+ u
4
+ u
3
− 2u
2
− u + 1
c
12
u
5
− 4u
4
+ 9u
3
− 21u
2
+ 31u − 17
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
− 19y
4
+ 123y
3
− 123y
2
− 10y −1
c
2
, c
4
y
5
− 7y
4
+ 15y
3
− 7y
2
− 2y −1
c
3
, c
8
y
5
+ 6y
4
+ 9y
3
− 12y
2
+ 8y −1
c
5
, c
6
, c
9
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
7
, c
10
, c
11
y
5
+ y
4
+ 3y
3
− 8y
2
+ 5y −1
c
12
y
5
+ 2y
4
− 25y
3
− 19y
2
+ 247y −289
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.821196
a = −7.66362
b = 4.77152
2.16633 39.9010
u = −0.688402 + 0.106340I
a = −1.322140 + 0.434760I
b = 1.278340 − 0.069185I
4.53993 + 0.30358I 10.54519 + 0.60661I
u = −0.688402 − 0.106340I
a = −1.322140 − 0.434760I
b = 1.278340 + 0.069185I
4.53993 − 0.30358I 10.54519 − 0.60661I
u = 0.77780 + 1.38013I
a = 0.653954 − 0.923165I
b = −0.664098 − 0.673862I
−13.8478 + 3.3875I −4.49564 − 1.04146I
u = 0.77780 − 1.38013I
a = 0.653954 + 0.923165I
b = −0.664098 + 0.673862I
−13.8478 − 3.3875I −4.49564 + 1.04146I
19
V. I
u
5
= h2b + 1, 2a + u − 2, u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
−
1
2
u + 1
−0.5
a
8
=
1
−u − 1
a
4
=
−
1
2
u + 1
−0.5
a
9
=
1
−u − 1
a
10
=
−u
−u − 1
a
12
=
u
u
a
1
=
−1
−u − 1
a
6
=
2u + 2
3u + 2
a
5
=
1
u + 1
a
2
=
−
1
2
u
−u −
3
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
45
4
u −
13
4
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
8
u
2
c
4
(u + 1)
2
c
5
, c
6
u
2
+ 3u + 1
c
7
u
2
− u − 1
c
9
u
2
− 3u + 1
c
10
, c
11
, c
12
u
2
+ u − 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
2
c
3
, c
8
y
2
c
5
, c
6
, c
9
y
2
− 7y + 1
c
7
, c
10
, c
11
c
12
y
2
− 3y + 1
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 1.30902
b = −0.500000
−0.657974 3.70290
u = 1.61803
a = 0.190983
b = −0.500000
7.23771 −21.4530
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
(u
4
− 3u
3
+ 5u
2
− 3u + 1)(u
5
− 7u
4
+ 15u
3
− 7u
2
− 2u − 1)
· ((u
5
+ 5u
4
+ 8u
3
+ 3u
2
− u + 1)
4
)(u
18
+ 17u
17
+ ··· + 9840u + 256)
c
2
(u − 1)
2
(u
4
+ u
3
− u
2
− u + 1)(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
4
· (u
5
+ 3u
4
+ u
3
− 3u
2
− 2u − 1)(u
18
− 3u
17
+ ··· − 108u + 16)
c
3
u
2
(u
4
− 2u
3
+ 2u
2
− u + 1)(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
4
· (u
5
+ 2u
4
+ 5u
3
+ 4u
2
− 1)(u
18
+ 5u
17
+ ··· − 144u + 64)
c
4
(u + 1)
2
(u
4
− u
3
− u
2
+ u + 1)(u
5
− 3u
4
+ u
3
+ 3u
2
− 2u + 1)
· ((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
4
)(u
18
− 3u
17
+ ··· − 108u + 16)
c
5
(u
2
+ 3u + 1)(u
4
− 2u
3
+ 2u
2
− u + 1)(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
· (u
18
− 10u
16
+ ··· + 3u + 1)(u
20
+ 3u
19
+ ··· − 690u − 209)
c
6
(u
2
+ 3u + 1)(u
4
+ 2u
3
+ 2u
2
+ u + 1)(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
· (u
18
− 10u
16
+ ··· + 3u + 1)(u
20
+ 3u
19
+ ··· − 690u − 209)
c
7
(u
2
− u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
5
− u
4
+ u
3
+ 2u
2
− u − 1)
· (u
18
+ u
17
+ ··· − 2u − 1)(u
20
− 3u
19
+ ··· + 376u − 101)
c
8
u
2
(u
4
+ 2u
3
+ 2u
2
+ u + 1)(u
5
− 2u
4
+ 5u
3
− 4u
2
+ 1)
· ((u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
4
)(u
18
+ 5u
17
+ ··· − 144u + 64)
c
9
(u
2
− 3u + 1)(u
4
− 2u
3
+ 2u
2
− u + 1)(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
· (u
18
− 10u
16
+ ··· + 3u + 1)(u
20
+ 3u
19
+ ··· − 690u − 209)
c
10
(u
2
+ u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
5
− u
4
+ u
3
+ 2u
2
− u − 1)
· (u
18
+ u
17
+ ··· − 2u − 1)(u
20
− 3u
19
+ ··· + 376u − 101)
c
11
(u
2
+ u − 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
5
+ u
4
+ u
3
− 2u
2
− u + 1)
· (u
18
+ u
17
+ ··· − 2u − 1)(u
20
− 3u
19
+ ··· + 376u − 101)
c
12
(u
2
− u − 1)
10
(u
2
+ u − 1)(u
4
− u
3
− u
2
+ u + 1)
· (u
5
− 4u
4
+ 9u
3
− 21u
2
+ 31u − 17)(u
18
+ 19u
17
+ ··· − 352u − 32)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
2
)(y
4
+ y
3
+ 9y
2
+ y + 1)(y
5
− 19y
4
+ ··· − 10y −1)
· (y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y −1)
4
· (y
18
− 29y
17
+ ··· − 79539968y + 65536)
c
2
, c
4
(y −1)
2
(y
4
− 3y
3
+ 5y
2
− 3y + 1)(y
5
− 7y
4
+ 15y
3
− 7y
2
− 2y −1)
· ((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
4
)(y
18
− 17y
17
+ ··· − 9840y + 256)
c
3
, c
8
y
2
(y
4
+ 2y
2
+ 3y + 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
4
· (y
5
+ 6y
4
+ 9y
3
− 12y
2
+ 8y −1)(y
18
+ 9y
17
+ ··· − 45824y + 4096)
c
5
, c
6
, c
9
(y
2
− 7y + 1)(y
4
+ 2y
2
+ 3y + 1)(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
· (y
18
− 20y
17
+ ··· − 13y + 1)(y
20
− 9y
19
+ ··· − 194368y + 43681)
c
7
, c
10
, c
11
(y
2
− 3y + 1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
5
+ y
4
+ 3y
3
− 8y
2
+ 5y −1)
· (y
18
+ 17y
17
+ ··· + 22y + 1)(y
20
+ 11y
19
+ ··· − 147436y + 10201)
c
12
(y
2
− 3y + 1)
11
(y
4
− 3y
3
+ 5y
2
− 3y + 1)
· (y
5
+ 2y
4
− 25y
3
− 19y
2
+ 247y −289)
· (y
18
− 7y
17
+ ··· + 2560y + 1024)
25
