12a
0623
(K12a
0623
)
A knot diagram
1
Linearized knot diagam
3 7 10 8 11 2 6 12 4 1 5 9
Solving Sequence
8,12
9
1,5
4 10 3 11 6 7 2
c
8
c
12
c
4
c
9
c
3
c
11
c
5
c
7
c
2
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−97455286u
69
− 687147084u
68
+ ··· + 1146617856b − 4289778370546,
− 2616564386981u
69
− 14513594810490u
68
+ ··· + 83529964191744a − 11174734769596142,
u
70
+ 8u
69
+ ··· + 286521u + 48566i
I
u
2
= ha
2
+ b, a
3
+ a + 1, u − 1i
I
u
3
= ha
6
b
3
+ 3a
4
b
3
+ ··· − 2a + 1, u − 1i
I
v
1
= ha, b
9
+ 3b
7
− b
6
+ 3b
5
− 2b
4
+ 3b
3
− b
2
+ 2b − 1, v − 1i
* 3 irreducible components of dim
C
= 0, with total 82 representations.
* 1 irreducible components of dim
C
= 1
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−9.75 × 10
7
u
69
− 6.87 × 10
8
u
68
+ · · · + 1.15 × 10
9
b − 4.29 ×
10
12
, −2.62 × 10
12
u
69
− 1.45 × 10
13
u
68
+ · · · + 8.35 × 10
13
a − 1.12 ×
10
16
, u
70
+ 8u
69
+ · · · + 286521u + 48566i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
u
2
a
1
=
−u
−u
3
+ u
a
5
=
0.0313249u
69
+ 0.173753u
68
+ ··· + 1396.40u + 133.781
0.0849937u
69
+ 0.599282u
68
+ ··· + 18949.7u + 3741.25
a
4
=
0.116319u
69
+ 0.773035u
68
+ ··· + 20346.1u + 3875.03
0.0849937u
69
+ 0.599282u
68
+ ··· + 18949.7u + 3741.25
a
10
=
0.00502741u
69
+ 0.0354161u
68
+ ··· + 1224.91u + 250.562
0.00488338u
69
+ 0.0343100u
68
+ ··· + 1186.75u + 242.634
a
3
=
0.195712u
69
+ 1.34410u
68
+ ··· + 39810.9u + 7746.66
0.0226364u
69
+ 0.194253u
68
+ ··· + 9789.24u + 2087.61
a
11
=
−0.000357427u
69
− 0.00242242u
68
+ ··· − 69.0145u − 13.3644
0.00530723u
69
+ 0.0369747u
68
+ ··· + 1240.78u + 252.067
a
6
=
−0.178615u
69
− 1.22581u
68
+ ··· − 36017.6u − 6978.08
0.0152291u
69
+ 0.0703555u
68
+ ··· − 1191.09u − 359.181
a
7
=
0.0167391u
69
+ 0.103977u
68
+ ··· + 1900.39u + 320.006
0.0149456u
69
+ 0.106887u
68
+ ··· + 3365.62u + 648.931
a
2
=
0.0165738u
69
+ 0.102484u
68
+ ··· + 1975.39u + 350.122
0.0170226u
69
+ 0.116505u
68
+ ··· + 3345.21u + 649.216
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
836940515
5159780352
u
69
+
588646361
573308928
u
68
+ ··· +
107767045126535
5159780352
u +
9383241457921
2579890176
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
70
+ 20u
69
+ ··· − 1331u + 3844
c
2
, c
6
u
70
+ 4u
69
+ ··· + 163u + 62
c
3
, c
9
27(27u
70
+ 27u
69
+ ··· − 6u + 1)
c
4
64(64u
70
− 128u
69
+ ··· − 118314u + 15039)
c
5
, c
11
27(27u
70
+ 27u
69
+ ··· + 8u + 1)
c
8
, c
12
u
70
− 8u
69
+ ··· − 286521u + 48566
c
10
64(64u
70
+ 96u
68
+ ··· + 356292u + 635013)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
70
+ 60y
69
+ ··· + 419446271y + 14776336
c
2
, c
6
y
70
− 20y
69
+ ··· + 1331y + 3844
c
3
, c
9
729(729y
70
+ 34263y
69
+ ··· + 36y + 1)
c
4
4096(4096y
70
+ 28672y
69
+ ··· + 4.67293 × 10
9
y + 2.26172 × 10
8
)
c
5
, c
11
729(729y
70
+ 31347y
69
+ ··· + 36y + 1)
c
8
, c
12
y
70
− 48y
69
+ ··· − 3572288981y + 2358656356
c
10
4096
· (4096y
70
+ 12288y
69
+ ··· + 402241554234y + 403241510169)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.893258 + 0.388589I
a = −0.552997 − 0.780965I
b = 0.717124 − 0.518987I
−0.32512 − 1.94198I −8.00000 + 0.I
u = 0.893258 − 0.388589I
a = −0.552997 + 0.780965I
b = 0.717124 + 0.518987I
−0.32512 + 1.94198I −8.00000 + 0.I
u = 0.885513 + 0.566554I
a = 0.648570 + 0.785318I
b = −0.921742 + 0.293488I
−1.00008 + 3.25130I 0
u = 0.885513 − 0.566554I
a = 0.648570 − 0.785318I
b = −0.921742 − 0.293488I
−1.00008 − 3.25130I 0
u = −0.976800 + 0.399121I
a = 0.294251 + 0.362673I
b = 1.260270 + 0.039825I
−1.84431 + 1.78184I 0
u = −0.976800 − 0.399121I
a = 0.294251 − 0.362673I
b = 1.260270 − 0.039825I
−1.84431 − 1.78184I 0
u = −0.215479 + 0.892593I
a = −0.085570 + 1.030780I
b = −0.163375 − 1.188040I
9.48735 − 0.61216I − 60.10 − 1.258234I
u = −0.215479 − 0.892593I
a = −0.085570 − 1.030780I
b = −0.163375 + 1.188040I
9.48735 + 0.61216I − 60.10 + 1.258234I
u = 1.095420 + 0.120474I
a = −0.219567 − 0.414390I
b = 0.221171 − 1.058020I
−2.53231 + 0.75873I 0
u = 1.095420 − 0.120474I
a = −0.219567 + 0.414390I
b = 0.221171 + 1.058020I
−2.53231 − 0.75873I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.199757 + 0.871147I
a = −0.003629 − 1.084200I
b = 0.234615 + 1.216000I
8.94654 − 6.79753I −1.43725 + 3.99331I
u = −0.199757 − 0.871147I
a = −0.003629 + 1.084200I
b = 0.234615 − 1.216000I
8.94654 + 6.79753I −1.43725 − 3.99331I
u = −0.047582 + 1.166560I
a = 1.014120 − 0.495065I
b = −0.731641 + 0.839705I
5.86206 + 12.07860I 0
u = −0.047582 − 1.166560I
a = 1.014120 + 0.495065I
b = −0.731641 − 0.839705I
5.86206 − 12.07860I 0
u = 0.351855 + 0.750985I
a = 1.106910 + 0.809645I
b = −0.921640 − 0.132570I
−4.19660 − 2.54473I −15.7326 + 4.0183I
u = 0.351855 − 0.750985I
a = 1.106910 − 0.809645I
b = −0.921640 + 0.132570I
−4.19660 + 2.54473I −15.7326 − 4.0183I
u = −0.082928 + 1.177070I
a = −0.924676 + 0.521308I
b = 0.643758 − 0.842940I
6.91710 + 5.79367I 0
u = −0.082928 − 1.177070I
a = −0.924676 − 0.521308I
b = 0.643758 + 0.842940I
6.91710 − 5.79367I 0
u = −1.086210 + 0.503549I
a = −0.574505 − 0.269617I
b = −0.770399 + 0.355339I
1.74908 + 3.93961I 0
u = −1.086210 − 0.503549I
a = −0.574505 + 0.269617I
b = −0.770399 − 0.355339I
1.74908 − 3.93961I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.134340 + 0.450358I
a = 0.649347 + 0.472023I
b = 0.936842 − 0.641523I
−0.87100 + 7.24185I 0
u = −1.134340 − 0.450358I
a = 0.649347 − 0.472023I
b = 0.936842 + 0.641523I
−0.87100 − 7.24185I 0
u = 0.163320 + 0.751919I
a = 1.28394 + 0.96111I
b = −1.070540 − 0.431024I
1.13016 − 7.73289I −7.91932 + 7.41023I
u = 0.163320 − 0.751919I
a = 1.28394 − 0.96111I
b = −1.070540 + 0.431024I
1.13016 + 7.73289I −7.91932 − 7.41023I
u = −0.312077 + 0.680537I
a = −0.354528 − 0.535888I
b = 0.398075 + 0.893488I
1.65608 − 2.83282I −3.61063 + 4.56409I
u = −0.312077 − 0.680537I
a = −0.354528 + 0.535888I
b = 0.398075 − 0.893488I
1.65608 + 2.83282I −3.61063 − 4.56409I
u = −1.267440 + 0.034085I
a = 0.013948 + 0.738018I
b = 0.27952 − 2.11838I
0.08596 + 3.02639I 0
u = −1.267440 − 0.034085I
a = 0.013948 − 0.738018I
b = 0.27952 + 2.11838I
0.08596 − 3.02639I 0
u = 0.145804 + 0.711603I
a = −1.23500 − 0.98762I
b = 0.980444 + 0.490899I
1.85042 − 2.01318I −6.24038 + 2.74940I
u = 0.145804 − 0.711603I
a = −1.23500 + 0.98762I
b = 0.980444 − 0.490899I
1.85042 + 2.01318I −6.24038 − 2.74940I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.175260 + 0.493835I
a = −0.883929 − 0.354754I
b = −0.629453 + 0.782986I
6.52426 + 5.60100I 0
u = −1.175260 − 0.493835I
a = −0.883929 + 0.354754I
b = −0.629453 − 0.782986I
6.52426 − 5.60100I 0
u = −1.179980 + 0.484001I
a = 0.901317 + 0.412933I
b = 0.676364 − 0.838798I
5.92952 + 11.69500I 0
u = −1.179980 − 0.484001I
a = 0.901317 − 0.412933I
b = 0.676364 + 0.838798I
5.92952 − 11.69500I 0
u = −0.638279 + 1.121600I
a = −0.342951 + 0.347949I
b = 0.012985 − 0.537783I
3.62133 + 1.34127I 0
u = −0.638279 − 1.121600I
a = −0.342951 − 0.347949I
b = 0.012985 + 0.537783I
3.62133 − 1.34127I 0
u = −1.285690 + 0.354856I
a = 0.220687 + 1.116770I
b = 1.27101 − 0.92407I
−2.48686 + 5.88009I 0
u = −1.285690 − 0.354856I
a = 0.220687 − 1.116770I
b = 1.27101 + 0.92407I
−2.48686 − 5.88009I 0
u = −0.651619 + 0.094400I
a = −0.051734 + 0.517565I
b = 0.39702 + 1.63749I
2.36312 − 2.67944I 0.83747 + 1.99355I
u = −0.651619 − 0.094400I
a = −0.051734 − 0.517565I
b = 0.39702 − 1.63749I
2.36312 + 2.67944I 0.83747 − 1.99355I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.293020 + 0.364510I
a = −0.199843 − 1.172110I
b = −1.29020 + 0.89246I
−3.28892 + 11.74160I 0
u = −1.293020 − 0.364510I
a = −0.199843 + 1.172110I
b = −1.29020 − 0.89246I
−3.28892 − 11.74160I 0
u = −1.327690 + 0.292888I
a = 0.061634 + 0.954982I
b = 1.05038 − 0.97956I
−5.50505 + 3.76007I 0
u = −1.327690 − 0.292888I
a = 0.061634 − 0.954982I
b = 1.05038 + 0.97956I
−5.50505 − 3.76007I 0
u = −0.959797 + 0.964946I
a = −0.363592 + 0.229548I
b = −0.107376 − 0.290881I
3.60962 + 1.37846I 0
u = −0.959797 − 0.964946I
a = −0.363592 − 0.229548I
b = −0.107376 + 0.290881I
3.60962 − 1.37846I 0
u = −1.329050 + 0.345273I
a = −0.045275 − 1.092050I
b = −1.15508 + 0.82929I
−9.22838 + 6.41936I 0
u = −1.329050 − 0.345273I
a = −0.045275 + 1.092050I
b = −1.15508 − 0.82929I
−9.22838 − 6.41936I 0
u = 1.384970 + 0.062679I
a = −0.0888781 + 0.0984540I
b = 0.004223 − 1.355380I
3.40691 + 3.12614I 0
u = 1.384970 − 0.062679I
a = −0.0888781 − 0.0984540I
b = 0.004223 + 1.355380I
3.40691 − 3.12614I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.01228 + 1.43319I
a = 0.765224 − 0.164117I
b = −0.600894 + 0.466851I
−1.44075 + 5.42106I 0
u = 0.01228 − 1.43319I
a = 0.765224 + 0.164117I
b = −0.600894 − 0.466851I
−1.44075 − 5.42106I 0
u = −1.40114 + 0.30543I
a = 0.071097 − 0.933368I
b = −0.855621 + 0.811549I
−8.14596 + 0.05000I 0
u = −1.40114 − 0.30543I
a = 0.071097 + 0.933368I
b = −0.855621 − 0.811549I
−8.14596 − 0.05000I 0
u = 1.38518 + 0.55050I
a = −0.015502 + 1.021740I
b = −1.48665 − 1.26538I
1.3953 − 18.0795I 0
u = 1.38518 − 0.55050I
a = −0.015502 − 1.021740I
b = −1.48665 + 1.26538I
1.3953 + 18.0795I 0
u = 1.39507 + 0.55017I
a = 0.027764 − 0.989036I
b = 1.41776 + 1.26916I
2.33215 − 11.82430I 0
u = 1.39507 − 0.55017I
a = 0.027764 + 0.989036I
b = 1.41776 − 1.26916I
2.33215 + 11.82430I 0
u = 1.39216 + 0.59468I
a = 0.093686 + 0.940274I
b = −1.39504 − 1.00978I
−5.80113 − 12.08780I 0
u = 1.39216 − 0.59468I
a = 0.093686 − 0.940274I
b = −1.39504 + 1.00978I
−5.80113 + 12.08780I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.273311 + 0.365651I
a = −1.17605 − 1.31593I
b = 0.455887 + 0.275732I
−0.556921 − 1.030040I −7.85697 + 6.36655I
u = 0.273311 − 0.365651I
a = −1.17605 + 1.31593I
b = 0.455887 − 0.275732I
−0.556921 + 1.030040I −7.85697 − 6.36655I
u = 1.44491 + 0.60643I
a = −0.064046 − 0.828480I
b = 1.16426 + 1.00461I
−1.97882 − 8.19603I 0
u = 1.44491 − 0.60643I
a = −0.064046 + 0.828480I
b = 1.16426 − 1.00461I
−1.97882 + 8.19603I 0
u = 1.42868 + 0.71532I
a = 0.208526 + 0.758552I
b = −1.108370 − 0.691160I
−6.08462 − 4.27997I 0
u = 1.42868 − 0.71532I
a = 0.208526 − 0.758552I
b = −1.108370 + 0.691160I
−6.08462 + 4.27997I 0
u = −1.61721 + 0.94908I
a = 0.353318 − 0.398464I
b = −0.174161 + 0.196426I
1.64897 − 5.03310I 0
u = −1.61721 − 0.94908I
a = 0.353318 + 0.398464I
b = −0.174161 − 0.196426I
1.64897 + 5.03310I 0
u = 1.92964 + 0.33599I
a = −0.004191 − 0.434882I
b = 0.260469 + 0.905894I
3.22024 − 3.60088I 0
u = 1.92964 − 0.33599I
a = −0.004191 + 0.434882I
b = 0.260469 − 0.905894I
3.22024 + 3.60088I 0
11
II. I
u
2
= ha
2
+ b, a
3
+ a + 1, u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
1
a
9
=
1
1
a
1
=
−1
0
a
5
=
a
−a
2
a
4
=
−a
2
+ a
−a
2
a
10
=
−a
2
− a
−a
a
3
=
−1
0
a
11
=
−a
2
−a
a
6
=
−1
0
a
7
=
1
0
a
2
=
−1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
u
3
c
3
, c
5
, c
9
c
10
, c
11
u
3
+ u + 1
c
4
u
3
+ 2u
2
+ u − 1
c
8
, c
12
(u + 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
y
3
c
3
, c
5
, c
9
c
10
, c
11
y
3
+ 2y
2
+ y − 1
c
4
y
3
− 2y
2
+ 5y − 1
c
8
, c
12
(y − 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.341164 + 1.161540I
b = 1.23279 − 0.79255I
−1.64493 −6.00000
u = 1.00000
a = 0.341164 − 1.161540I
b = 1.23279 + 0.79255I
−1.64493 −6.00000
u = 1.00000
a = −0.682328
b = −0.465571
−1.64493 −6.00000
15
III. I
u
3
= ha
6
b
3
+ 3a
4
b
3
+ · · · − 2a + 1, u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
1
a
9
=
1
1
a
1
=
−1
0
a
5
=
a
b
a
4
=
b + a
b
a
10
=
−ba − a
2
+ 1
−ba + 1
a
3
=
a
2
b + a
3
+ b
a
2
b + b − a
a
11
=
−a
2
−ba + 1
a
6
=
a
3
+ a
a
2
b + b − a
a
7
=
−a
5
b − 2a
3
b + a
4
− ba + a
2
+ 1
−a
4
b
2
− 2b
2
a
2
+ 2a
3
b − b
2
+ 2ba − a
2
a
2
=
−a
4
b
2
− a
5
b − 2b
2
a
2
+ a
4
− b
2
+ ba − 1
−a
4
b
2
− 2b
2
a
2
+ 2a
3
b − b
2
+ 2ba − a
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a
4
b
2
− 8b
2
a
2
+ 8a
3
b − 4a
2
b − 4b
2
+ 8ba − 4a
2
− 4b + 4a − 16
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
16
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
1.37919 + 2.82812I −8.49025 − 2.97943I
17
IV. I
v
1
= ha, b
9
+ 3b
7
− b
6
+ 3b
5
− 2b
4
+ 3b
3
− b
2
+ 2b − 1, v − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
1
0
a
9
=
1
0
a
1
=
1
0
a
5
=
0
b
a
4
=
b
b
a
10
=
b
2
+ 1
b
2
a
3
=
b
3
+ 2b
b
3
+ b
a
11
=
1
b
2
a
6
=
b
b
3
+ b
a
7
=
−b
4
− b
2
+ 1
−b
6
− 2b
4
− b
2
a
2
=
b
6
+ 3b
4
+ 2b
2
+ 1
b
6
+ 2b
4
+ b
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4b
6
− 8b
4
+ 4b
3
− 4b
2
+ 4b − 10
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
3
+ u
2
+ 2u + 1)
3
c
2
, c
6
(u
3
− u
2
+ 1)
3
c
3
, c
4
, c
5
c
9
, c
11
u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1
c
8
, c
12
u
9
c
10
u
9
+ 6u
8
+ 15u
7
+ 23u
6
+ 27u
5
+ 24u
4
+ 15u
3
+ 7u
2
+ 2u − 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
3
+ 3y
2
+ 2y − 1)
3
c
2
, c
6
(y
3
− y
2
+ 2y − 1)
3
c
3
, c
4
, c
5
c
9
, c
11
y
9
+ 6y
8
+ 15y
7
+ 23y
6
+ 27y
5
+ 24y
4
+ 15y
3
+ 7y
2
+ 2y − 1
c
8
, c
12
y
9
c
10
y
9
− 6y
8
+ 3y
7
+ 23y
6
− 5y
5
− 16y
4
+ 43y
3
+ 59y
2
+ 18y − 1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −0.656619 + 0.765660I
3.02413 − 2.82812I −2.49024 + 2.97945I
v = 1.00000
a = 0
b = −0.656619 − 0.765660I
3.02413 + 2.82812I −2.49024 − 2.97945I
v = 1.00000
a = 0
b = 0.701160 + 0.628458I
3.02413 − 2.82812I −2.49024 + 2.97945I
v = 1.00000
a = 0
b = 0.701160 − 0.628458I
3.02413 + 2.82812I −2.49024 − 2.97945I
v = 1.00000
a = 0
b = −0.233800 + 1.078880I
−1.11345 −9.01951 + 0.I
v = 1.00000
a = 0
b = −0.233800 − 1.078880I
−1.11345 −9.01951 + 0.I
v = 1.00000
a = 0
b = −0.044542 + 1.394120I
3.02413 + 2.82812I −2.49024 − 2.97945I
v = 1.00000
a = 0
b = −0.044542 − 1.394120I
3.02413 − 2.82812I −2.49024 + 2.97945I
v = 1.00000
a = 0
b = 0.467600
−1.11345 −9.01950
21
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
u
3
(u
3
+ u
2
+ 2u + 1)
3
(u
70
+ 20u
69
+ ··· − 1331u + 3844)
c
2
, c
6
u
3
(u
3
− u
2
+ 1)
3
(u
70
+ 4u
69
+ ··· + 163u + 62)
c
3
, c
9
27(u
3
+ u + 1)(u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1)
· (27u
70
+ 27u
69
+ ··· − 6u + 1)
c
4
64(u
3
+ 2u
2
+ u − 1)(u
9
+ 3u
7
+ ··· + 2u + 1)
· (64u
70
− 128u
69
+ ··· − 118314u + 15039)
c
5
, c
11
27(u
3
+ u + 1)(u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1)
· (27u
70
+ 27u
69
+ ··· + 8u + 1)
c
8
, c
12
u
9
(u + 1)
3
(u
70
− 8u
69
+ ··· − 286521u + 48566)
c
10
64(u
3
+ u + 1)
· (u
9
+ 6u
8
+ 15u
7
+ 23u
6
+ 27u
5
+ 24u
4
+ 15u
3
+ 7u
2
+ 2u − 1)
· (64u
70
+ 96u
68
+ ··· + 356292u + 635013)
22
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
3
(y
3
+ 3y
2
+ 2y − 1)
3
· (y
70
+ 60y
69
+ ··· + 419446271y + 14776336)
c
2
, c
6
y
3
(y
3
− y
2
+ 2y − 1)
3
(y
70
− 20y
69
+ ··· + 1331y + 3844)
c
3
, c
9
729(y
3
+ 2y
2
+ y − 1)
· (y
9
+ 6y
8
+ 15y
7
+ 23y
6
+ 27y
5
+ 24y
4
+ 15y
3
+ 7y
2
+ 2y − 1)
· (729y
70
+ 34263y
69
+ ··· + 36y + 1)
c
4
4096(y
3
− 2y
2
+ 5y − 1)
· (y
9
+ 6y
8
+ 15y
7
+ 23y
6
+ 27y
5
+ 24y
4
+ 15y
3
+ 7y
2
+ 2y − 1)
· (4096y
70
+ 28672y
69
+ ··· + 4672926450y + 226171521)
c
5
, c
11
729(y
3
+ 2y
2
+ y − 1)
· (y
9
+ 6y
8
+ 15y
7
+ 23y
6
+ 27y
5
+ 24y
4
+ 15y
3
+ 7y
2
+ 2y − 1)
· (729y
70
+ 31347y
69
+ ··· + 36y + 1)
c
8
, c
12
y
9
(y − 1)
3
(y
70
− 48y
69
+ ··· − 3.57229 × 10
9
y + 2.35866 × 10
9
)
c
10
4096(y
3
+ 2y
2
+ y − 1)
· (y
9
− 6y
8
+ 3y
7
+ 23y
6
− 5y
5
− 16y
4
+ 43y
3
+ 59y
2
+ 18y − 1)
· (4096y
70
+ 12288y
69
+ ··· + 402241554234y + 403241510169)
23
