12n
0519
(K12n
0519
)
A knot diagram
1
Linearized knot diagam
3 5 12 8 2 11 4 5 3 12 6 9
Solving Sequence
3,9 5,10
2 6 1 8 4 12 11 7
c
9
c
2
c
5
c
1
c
8
c
4
c
12
c
11
c
6
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h58u
9
− 249u
8
− 444u
7
+ 2113u
6
+ 1475u
5
− 7574u
4
+ 2612u
3
+ 2889u
2
+ 349b − 1335u − 353,
− 2528u
9
+ 8422u
8
+ ··· + 1745a − 16349,
u
10
− 4u
9
− 3u
8
+ 29u
7
− 14u
6
− 89u
5
+ 163u
4
− 109u
3
+ 17u
2
+ 13u − 5i
I
u
2
= h−88u
6
+ 345u
5
− 33u
4
− 837u
3
− 116u
2
+ 719b + 189u + 50,
267u
6
− 1202u
5
+ 909u
4
+ 1747u
3
+ 1169u
2
+ 5033a − 5010u + 649,
u
7
− 5u
6
+ 6u
5
+ 4u
4
− 7u
3
+ 2u
2
+ 5u − 7i
I
u
3
= hu
3
+ b − 3u − 2, u
3
a − 3u
3
+ a
2
− 3au − a + 10u, u
4
+ u
3
− 3u
2
− 3u − 1i
I
u
4
= hu
3
+ 2u
2
+ b − u − 2, u
3
a + 2u
2
a − 3u
3
+ a
2
− au − 8u
2
− 3a + 10, u
4
+ 3u
3
+ u
2
− 3u − 1i
* 4 irreducible components of dim
C
= 0, with total 33 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
= h58u
9
− 249u
8
+ · · · + 349b − 353, −2528u
9
+ 8422u
8
+ · · · + 1745a −
16349, u
10
− 4u
9
+ · · · + 13u − 5i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
5
=
1.44871u
9
− 4.82636u
8
+ ··· − 13.0281u + 9.36905
−0.166189u
9
+ 0.713467u
8
+ ··· + 3.82521u + 1.01146
a
10
=
1
u
2
a
2
=
0.0710602u
9
− 0.436103u
8
+ ··· − 1.40802u + 1.50544
−0.260745u
9
+ 0.446991u
8
+ ··· + 1.43266u + 0.742120
a
6
=
1.07736u
9
− 3.41834u
8
+ ··· − 8.31519u + 5.85673
0.00286533u
9
+ 0.280802u
8
+ ··· + 1.95129u − 0.931232
a
1
=
0.0710602u
9
− 0.436103u
8
+ ··· − 1.40802u + 1.50544
0.229226u
9
− 0.535817u
8
+ ··· − 0.896848u + 1.50143
a
8
=
−0.148424u
9
+ 0.854441u
8
+ ··· + 1.72321u − 1.36218
1.39542u
9
− 4.24928u
8
+ ··· − 12.7221u + 5.48997
a
4
=
−0.0108883u
9
− 0.667049u
8
+ ··· − 2.61490u − 0.0613181
−0.710602u
9
+ 0.361032u
8
+ ··· + 1.08023u − 0.0544413
a
12
=
0.300287u
9
− 0.971920u
8
+ ··· − 2.30487u + 3.00688
0.229226u
9
− 0.535817u
8
+ ··· − 0.896848u + 1.50143
a
11
=
−0.699713u
9
+ 1.02808u
8
+ ··· + 2.69513u + 1.00688
−1.77077u
9
+ 3.46418u
8
+ ··· + 9.10315u − 3.49857
a
7
=
1.40802u
9
− 3.21375u
8
+ ··· − 6.53639u + 1.19255
1.23496u
9
− 2.97421u
8
+ ··· − 6.99427u + 1.63897
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
150
349
u
9
+
391
349
u
8
−
3146
349
u
7
−
2093
349
u
6
+
18942
349
u
5
−
417
349
u
4
−
51937
349
u
3
+
49159
349
u
2
−
6389
349
u −
6172
349
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
10
+ 6u
9
+ ··· + 4u + 1
c
2
, c
5
, c
6
c
11
u
10
+ 3u
8
− 6u
7
+ 4u
6
− 17u
5
+ 4u
4
− 11u
3
+ 4u
2
− 2u + 1
c
3
, c
12
u
10
+ 2u
9
+ 29u
7
+ 73u
6
+ 119u
5
+ 136u
4
+ 94u
3
+ 32u
2
+ 2u − 1
c
4
, c
7
, c
8
u
10
− 7u
9
+ ··· + 8u − 8
c
9
u
10
+ 4u
9
+ ··· − 13u − 5
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
10
− 2y
9
+ ··· − 56y + 1
c
2
, c
5
, c
6
c
11
y
10
+ 6y
9
+ ··· + 4y + 1
c
3
, c
12
y
10
− 4y
9
+ ··· − 68y + 1
c
4
, c
7
, c
8
y
10
− 19y
9
+ ··· − 352y + 64
c
9
y
10
− 22y
9
+ ··· − 339y + 25
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.710716 + 0.441346I
a = 0.304205 + 0.895172I
b = −1.53379 − 0.01592I
5.27109 − 1.67493I 4.82619 + 2.79664I
u = 0.710716 − 0.441346I
a = 0.304205 − 0.895172I
b = −1.53379 + 0.01592I
5.27109 + 1.67493I 4.82619 − 2.79664I
u = 0.610288 + 0.265211I
a = −0.344245 − 0.794267I
b = 0.382201 − 0.301034I
−1.12041 − 1.07831I −3.35767 + 4.98904I
u = 0.610288 − 0.265211I
a = −0.344245 + 0.794267I
b = 0.382201 + 0.301034I
−1.12041 + 1.07831I −3.35767 − 4.98904I
u = −0.348971
a = −1.18540
b = −0.681296
0.925697 11.8780
u = 1.85552
a = 0.910166
b = 2.25830
1.90296 4.31530
u = 2.12102 + 0.16943I
a = −0.392434 + 0.798856I
b = −2.19435 + 0.73311I
15.2945 − 10.5947I 2.32893 + 3.81440I
u = 2.12102 − 0.16943I
a = −0.392434 − 0.798856I
b = −2.19435 − 0.73311I
15.2945 + 10.5947I 2.32893 − 3.81440I
u = −2.19529 + 0.82711I
a = 0.170094 − 0.566049I
b = −0.942562 − 0.925156I
−8.52247 + 3.39717I 0.10586 − 5.05917I
u = −2.19529 − 0.82711I
a = 0.170094 + 0.566049I
b = −0.942562 + 0.925156I
−8.52247 − 3.39717I 0.10586 + 5.05917I
5
II. I
u
2
= h−88u
6
+ 345u
5
+ · · · + 719b + 50, 267u
6
− 1202u
5
+ · · · + 5033a +
649, u
7
− 5u
6
+ 6u
5
+ 4u
4
− 7u
3
+ 2u
2
+ 5u − 7i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
5
=
−0.0530499u
6
+ 0.238824u
5
+ ··· + 0.995430u − 0.128949
0.122392u
6
− 0.479833u
5
+ ··· − 0.262865u − 0.0695410
a
10
=
1
u
2
a
2
=
0.0202662u
6
− 0.271011u
5
+ ··· + 0.271409u + 0.397576
−0.0792768u
6
+ 0.413074u
5
+ ··· + 1.40890u − 1.11405
a
6
=
−0.0103318u
6
+ 0.0989470u
5
+ ··· − 0.412875u + 0.914961
−0.0152990u
6
+ 0.184979u
5
+ ··· + 0.657858u + 0.258693
a
1
=
0.0202662u
6
− 0.271011u
5
+ ··· + 0.271409u + 0.397576
−0.00973574u
6
+ 0.208623u
5
+ ··· + 0.418637u + 0.0737135
a
8
=
0.159150u
6
− 0.716471u
5
+ ··· − 0.986290u + 1.38685
−0.0695410u
6
+ 0.204451u
5
+ ··· + 0.990264u + 0.812239
a
4
=
0.00993443u
6
− 0.172064u
5
+ ··· − 0.141466u + 0.312537
−0.122392u
6
+ 0.479833u
5
+ ··· + 1.26287u + 0.0695410
a
12
=
0.0105305u
6
− 0.0623882u
5
+ ··· + 0.690046u + 0.471289
−0.00973574u
6
+ 0.208623u
5
+ ··· + 0.418637u + 0.0737135
a
11
=
−0.132327u
6
+ 0.651897u
5
+ ··· + 0.404331u + 0.757004
−0.00973574u
6
+ 0.208623u
5
+ ··· + 0.418637u − 0.926287
a
7
=
0.0431154u
6
− 0.0667594u
5
+ ··· − 0.853964u − 0.183588
0.132128u
6
− 0.688456u
5
+ ··· − 0.681502u + 0.856745
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
334
719
u
6
+
1816
719
u
5
−
2462
719
u
4
−
742
719
u
3
+
148
719
u
2
+
2139
719
u −
39
719
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
7
− 7u
6
+ 22u
5
− 41u
4
+ 48u
3
− 33u
2
+ 10u + 1
c
2
, c
6
u
7
+ u
6
+ 4u
5
+ 3u
4
+ 6u
3
+ 3u
2
+ 4u + 1
c
3
, c
12
u
7
− u
6
+ u
5
− u
4
− u
3
+ u
2
+ 1
c
4
u
7
− u
6
− 3u
5
+ u
4
+ 5u
3
− u
2
− 2u + 1
c
5
, c
11
u
7
− u
6
+ 4u
5
− 3u
4
+ 6u
3
− 3u
2
+ 4u − 1
c
7
, c
8
u
7
+ u
6
− 3u
5
− u
4
+ 5u
3
+ u
2
− 2u − 1
c
9
u
7
− 5u
6
+ 6u
5
+ 4u
4
− 7u
3
+ 2u
2
+ 5u − 7
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
7
− 5y
6
+ 6y
5
− 11y
4
+ 52y
3
− 47y
2
+ 166y − 1
c
2
, c
5
, c
6
c
11
y
7
+ 7y
6
+ 22y
5
+ 41y
4
+ 48y
3
+ 33y
2
+ 10y − 1
c
3
, c
12
y
7
+ y
6
− 3y
5
− y
4
+ 5y
3
+ y
2
− 2y − 1
c
4
, c
7
, c
8
y
7
− 7y
6
+ 21y
5
− 37y
4
+ 41y
3
− 23y
2
+ 6y − 1
c
9
y
7
− 13y
6
+ 62y
5
− 70y
4
+ 23y
3
− 18y
2
+ 53y − 49
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.942087 + 0.385621I
a = −0.961712 + 0.696018I
b = −0.470376 + 0.273309I
0.08815 − 5.09905I −0.97794 + 6.62021I
u = −0.942087 − 0.385621I
a = −0.961712 − 0.696018I
b = −0.470376 − 0.273309I
0.08815 + 5.09905I −0.97794 − 6.62021I
u = 0.401929 + 0.876655I
a = 0.865092 + 0.818149I
b = −1.53400 − 0.17432I
4.42380 − 3.02243I 3.79268 + 4.58771I
u = 0.401929 − 0.876655I
a = 0.865092 − 0.818149I
b = −1.53400 + 0.17432I
4.42380 + 3.02243I 3.79268 − 4.58771I
u = 1.08372
a = 0.257183
b = 0.861033
−0.272703 0.397920
u = 2.49830 + 0.67865I
a = −0.103399 − 0.517020I
b = 1.073860 − 0.702292I
−9.31040 − 2.64371I −5.01371 + 0.82640I
u = 2.49830 − 0.67865I
a = −0.103399 + 0.517020I
b = 1.073860 + 0.702292I
−9.31040 + 2.64371I −5.01371 − 0.82640I
9
III.
I
u
3
= hu
3
+ b − 3u − 2, u
3
a − 3u
3
+ a
2
− 3au − a + 10u, u
4
+ u
3
− 3u
2
− 3u − 1i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
5
=
a
−u
3
+ 3u + 2
a
10
=
1
u
2
a
2
=
−u
3
a + 3u
3
+ 2au + u
2
+ a − 9u − 3
−u
3
a + au + a + u
a
6
=
2u
2
a + u
3
− 3au − 2u
2
− a − u + 1
−3u
3
a + 2u
2
a + 2u
3
+ 5au − 2u
2
+ 2a − 4u
a
1
=
−u
3
a + 3u
3
+ 2au + u
2
+ a − 9u − 3
u
3
a − u
2
a − 2u
3
− au + 4u + 2
a
8
=
−u
3
a + 3au + 2a + 1
−2u
3
+ u
2
+ 5u + 2
a
4
=
2u
3
a − u
2
a + u
3
− 5au − a − 3u − 2
3u
3
− 4u
2
− 4u
a
12
=
−u
2
a + u
3
+ au + u
2
+ a − 5u − 1
u
3
a − u
2
a − 2u
3
− au + 4u + 2
a
11
=
−3u
3
a + u
2
a − u
3
+ 5au − 3u
2
+ 2a + 5u + 7
−3u
3
a + u
2
a − 2u
3
+ 5au + u
2
+ 2a + 4u + 1
a
7
=
−2u
3
a + 4u
2
a − 2u
3
+ au + u
2
− 2a + 5u + 1
−7u
3
+ 10u
2
+ 5u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
3
− u
2
+ 6u + 5
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
8
− u
7
+ 16u
6
+ 69u
5
+ 299u
4
+ 497u
3
+ 868u
2
+ 535u + 841
c
2
, c
5
, c
6
c
11
u
8
+ 3u
7
+ 4u
6
+ 7u
5
+ 21u
4
+ 15u
3
+ 20u
2
+ 25u + 29
c
3
, c
12
u
8
− 6u
7
+ 38u
6
− 114u
5
+ 133u
4
− 82u
3
+ 227u
2
− 449u + 431
c
4
, c
7
, c
8
(u
4
+ 6u
3
+ 12u
2
+ 11u + 5)
2
c
9
(u
4
− u
3
− 3u
2
+ 3u − 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
8
+ 31y
7
+ ··· + 1173751y + 707281
c
2
, c
5
, c
6
c
11
y
8
− y
7
+ 16y
6
+ 69y
5
+ 299y
4
+ 497y
3
+ 868y
2
+ 535y + 841
c
3
, c
12
y
8
+ 40y
7
+ ··· − 5927y + 185761
c
4
, c
7
, c
8
(y
4
− 12y
3
+ 22y
2
− y + 25)
2
c
9
(y
4
− 7y
3
+ 13y
2
− 3y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.447135 + 0.308371I
a = 2.01612 − 0.24150I
b = 0.620433 + 0.769480I
0.96275 − 3.58171I 2.13603 + 1.81473I
u = −0.447135 + 0.308371I
a = −2.39569 + 1.01098I
b = 0.620433 + 0.769480I
0.96275 − 3.58171I 2.13603 + 1.81473I
u = −0.447135 − 0.308371I
a = 2.01612 + 0.24150I
b = 0.620433 − 0.769480I
0.96275 + 3.58171I 2.13603 − 1.81473I
u = −0.447135 − 0.308371I
a = −2.39569 − 1.01098I
b = 0.620433 − 0.769480I
0.96275 + 3.58171I 2.13603 − 1.81473I
u = 1.78897
a = 0.320723 + 0.781310I
b = 1.64145
−7.09598 1.08250
u = 1.78897
a = 0.320723 − 0.781310I
b = 1.64145
−7.09598 1.08250
u = −1.89470
a = 1.058840 + 0.580699I
b = 3.11769
18.3299 3.64550
u = −1.89470
a = 1.058840 − 0.580699I
b = 3.11769
18.3299 3.64550
13
IV. I
u
4
= hu
3
+ 2u
2
+ b − u − 2, u
3
a + 2u
2
a − 3u
3
+ a
2
− au − 8u
2
− 3a +
10, u
4
+ 3u
3
+ u
2
− 3u − 1i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
5
=
a
−u
3
− 2u
2
+ u + 2
a
10
=
1
u
2
a
2
=
−u
3
a − 2u
2
a + u
3
+ 3u
2
+ a + u − 3
−u
3
a − 2u
2
a + au + a + u
a
6
=
u
3
+ au + 2u
2
+ a − u − 3
u
3
a − au
a
1
=
−u
3
a − 2u
2
a + u
3
+ 3u
2
+ a + u − 3
u
3
a + u
2
a − au
a
8
=
−u
3
a − 2u
2
a + au + 2a + 1
−u
2
− u + 2
a
4
=
u
2
a + u
3
+ au + 2u
2
− a − u − 2
u
3
+ 2u
2
− 2
a
12
=
−u
2
a + u
3
− au + 3u
2
+ a + u − 3
u
3
a + u
2
a − au
a
11
=
u
3
a + u
2
a + u
3
− au + 3u
2
+ u − 3
u
3
a + u
2
a − au + u
2
− 1
a
7
=
au + u
2
+ u − 1
u
3
+ 2u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
3
− 3u
2
+ 2u + 5
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
8
− 7u
7
+ 20u
6
− 33u
5
+ 39u
4
− 33u
3
+ 20u
2
− 7u + 1
c
2
, c
6
u
8
+ u
7
+ 4u
6
+ 3u
5
+ 5u
4
+ 3u
3
+ 4u
2
+ u + 1
c
3
, c
12
u
8
+ 4u
7
+ 6u
6
+ 4u
5
− u
4
− 4u
3
− u
2
+ u + 1
c
4
(u
4
− 2u
2
+ u + 1)
2
c
5
, c
11
u
8
− u
7
+ 4u
6
− 3u
5
+ 5u
4
− 3u
3
+ 4u
2
− u + 1
c
7
, c
8
(u
4
− 2u
2
− u + 1)
2
c
9
(u
4
+ 3u
3
+ u
2
− 3u − 1)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
8
− 9y
7
+ 16y
6
+ 49y
5
+ 47y
4
+ 49y
3
+ 16y
2
− 9y + 1
c
2
, c
5
, c
6
c
11
y
8
+ 7y
7
+ 20y
6
+ 33y
5
+ 39y
4
+ 33y
3
+ 20y
2
+ 7y + 1
c
3
, c
12
y
8
− 4y
7
+ 2y
6
+ 2y
5
+ 15y
4
− 10y
3
+ 7y
2
− 3y + 1
c
4
, c
7
, c
8
(y
4
− 4y
3
+ 6y
2
− 5y + 1)
2
c
9
(y
4
− 7y
3
+ 17y
2
− 11y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.905166
a = 0.762444 + 0.799496I
b = 0.524889
1.00996 2.86910
u = 0.905166
a = 0.762444 − 0.799496I
b = 0.524889
1.00996 2.86910
u = −0.328956
a = 1.24511 + 2.77323I
b = 1.49022
5.07273 4.08860
u = −0.328956
a = 1.24511 − 2.77323I
b = 1.49022
5.07273 4.08860
u = −1.78810 + 0.40136I
a = 0.102698 − 0.845065I
b = −1.007550 − 0.513116I
−6.33121 + 1.96274I 2.02113 − 2.46157I
u = −1.78810 + 0.40136I
a = −0.110250 + 0.331949I
b = −1.007550 − 0.513116I
−6.33121 + 1.96274I 2.02113 − 2.46157I
u = −1.78810 − 0.40136I
a = 0.102698 + 0.845065I
b = −1.007550 + 0.513116I
−6.33121 − 1.96274I 2.02113 + 2.46157I
u = −1.78810 − 0.40136I
a = −0.110250 − 0.331949I
b = −1.007550 + 0.513116I
−6.33121 − 1.96274I 2.02113 + 2.46157I
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
(u
7
− 7u
6
+ 22u
5
− 41u
4
+ 48u
3
− 33u
2
+ 10u + 1)
· (u
8
− 7u
7
+ 20u
6
− 33u
5
+ 39u
4
− 33u
3
+ 20u
2
− 7u + 1)
· (u
8
− u
7
+ 16u
6
+ 69u
5
+ 299u
4
+ 497u
3
+ 868u
2
+ 535u + 841)
· (u
10
+ 6u
9
+ ··· + 4u + 1)
c
2
, c
6
(u
7
+ u
6
+ 4u
5
+ 3u
4
+ 6u
3
+ 3u
2
+ 4u + 1)
· (u
8
+ u
7
+ 4u
6
+ 3u
5
+ 5u
4
+ 3u
3
+ 4u
2
+ u + 1)
· (u
8
+ 3u
7
+ 4u
6
+ 7u
5
+ 21u
4
+ 15u
3
+ 20u
2
+ 25u + 29)
· (u
10
+ 3u
8
− 6u
7
+ 4u
6
− 17u
5
+ 4u
4
− 11u
3
+ 4u
2
− 2u + 1)
c
3
, c
12
(u
7
− u
6
+ u
5
− u
4
− u
3
+ u
2
+ 1)
· (u
8
− 6u
7
+ 38u
6
− 114u
5
+ 133u
4
− 82u
3
+ 227u
2
− 449u + 431)
· (u
8
+ 4u
7
+ 6u
6
+ 4u
5
− u
4
− 4u
3
− u
2
+ u + 1)
· (u
10
+ 2u
9
+ 29u
7
+ 73u
6
+ 119u
5
+ 136u
4
+ 94u
3
+ 32u
2
+ 2u − 1)
c
4
(u
4
− 2u
2
+ u + 1)
2
(u
4
+ 6u
3
+ 12u
2
+ 11u + 5)
2
· (u
7
− u
6
+ ··· − 2u + 1)(u
10
− 7u
9
+ ··· + 8u − 8)
c
5
, c
11
(u
7
− u
6
+ 4u
5
− 3u
4
+ 6u
3
− 3u
2
+ 4u − 1)
· (u
8
− u
7
+ 4u
6
− 3u
5
+ 5u
4
− 3u
3
+ 4u
2
− u + 1)
· (u
8
+ 3u
7
+ 4u
6
+ 7u
5
+ 21u
4
+ 15u
3
+ 20u
2
+ 25u + 29)
· (u
10
+ 3u
8
− 6u
7
+ 4u
6
− 17u
5
+ 4u
4
− 11u
3
+ 4u
2
− 2u + 1)
c
7
, c
8
(u
4
− 2u
2
− u + 1)
2
(u
4
+ 6u
3
+ 12u
2
+ 11u + 5)
2
· (u
7
+ u
6
+ ··· − 2u − 1)(u
10
− 7u
9
+ ··· + 8u − 8)
c
9
(u
4
− u
3
− 3u
2
+ 3u − 1)
2
(u
4
+ 3u
3
+ u
2
− 3u − 1)
2
· (u
7
− 5u
6
+ ··· + 5u − 7)(u
10
+ 4u
9
+ ··· − 13u − 5)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
10
(y
7
− 5y
6
+ 6y
5
− 11y
4
+ 52y
3
− 47y
2
+ 166y − 1)
· (y
8
− 9y
7
+ 16y
6
+ 49y
5
+ 47y
4
+ 49y
3
+ 16y
2
− 9y + 1)
· (y
8
+ 31y
7
+ ··· + 1173751y + 707281)(y
10
− 2y
9
+ ··· − 56y + 1)
c
2
, c
5
, c
6
c
11
(y
7
+ 7y
6
+ 22y
5
+ 41y
4
+ 48y
3
+ 33y
2
+ 10y − 1)
· (y
8
− y
7
+ 16y
6
+ 69y
5
+ 299y
4
+ 497y
3
+ 868y
2
+ 535y + 841)
· (y
8
+ 7y
7
+ 20y
6
+ 33y
5
+ 39y
4
+ 33y
3
+ 20y
2
+ 7y + 1)
· (y
10
+ 6y
9
+ ··· + 4y + 1)
c
3
, c
12
(y
7
+ y
6
− 3y
5
− y
4
+ 5y
3
+ y
2
− 2y − 1)
· (y
8
− 4y
7
+ 2y
6
+ 2y
5
+ 15y
4
− 10y
3
+ 7y
2
− 3y + 1)
· (y
8
+ 40y
7
+ ··· − 5927y + 185761)(y
10
− 4y
9
+ ··· − 68y + 1)
c
4
, c
7
, c
8
(y
4
− 12y
3
+ 22y
2
− y + 25)
2
(y
4
− 4y
3
+ 6y
2
− 5y + 1)
2
· (y
7
− 7y
6
+ 21y
5
− 37y
4
+ 41y
3
− 23y
2
+ 6y − 1)
· (y
10
− 19y
9
+ ··· − 352y + 64)
c
9
(y
4
− 7y
3
+ 13y
2
− 3y + 1)
2
(y
4
− 7y
3
+ 17y
2
− 11y + 1)
2
· (y
7
− 13y
6
+ 62y
5
− 70y
4
+ 23y
3
− 18y
2
+ 53y − 49)
· (y
10
− 22y
9
+ ··· − 339y + 25)
19
