12n
0824
(K12n
0824
)
A knot diagram
1
Linearized knot diagam
4 5 11 9 3 12 1 4 6 3 7 6
Solving Sequence
7,12
6 1
3,8
5 2 11 4 10 9
c
6
c
12
c
7
c
5
c
2
c
11
c
3
c
10
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
19
− 4u
18
+ ··· + 2b − 16, u
19
− 3u
18
+ ··· + 2a + 7, u
20
− 4u
19
+ ··· − 10u + 4i
I
u
2
= hu
11
+ 2u
10
+ 5u
9
+ 7u
8
+ 7u
7
+ 7u
6
+ u
5
− 2u
4
− 3u
3
− 4u
2
+ b + 1,
− 2u
11
− 3u
10
− 11u
9
− 13u
8
− 21u
7
− 20u
6
− 14u
5
− 8u
4
+ 2u
3
+ 6u
2
+ a + 4u + 3,
u
12
+ u
11
+ 6u
10
+ 5u
9
+ 13u
8
+ 9u
7
+ 10u
6
+ 5u
5
− 2u
4
− 3u
3
− 4u
2
− 3u + 1i
I
u
3
= h−309u
5
a
3
+ 1269u
5
a
2
+ ··· + 3300a − 2645, −u
5
a
2
+ 5u
5
a + ··· + 20a − 22,
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 56 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
= hu
19
−4u
18
+· · ·+2b −16, u
19
−3u
18
+· · ·+2a +7, u
20
−4u
19
+· · ·−10u +4i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
6
=
1
u
2
a
1
=
u
u
3
+ u
a
3
=
−
1
2
u
19
+
3
2
u
18
+ ··· +
9
2
u −
7
2
−
1
2
u
19
+ 2u
18
+ ··· −
17
2
u + 8
a
8
=
−u
4
− u
2
+ 1
−u
6
− 2u
4
− u
2
a
5
=
1
4
u
19
−
1
2
u
18
+ ··· −
9
4
u + 3
1
2
u
19
− 2u
18
+ ··· +
11
2
u − 5
a
2
=
−
1
4
u
19
+
1
2
u
18
+ ··· −
7
4
u + 3
−
1
2
u
19
+ 2u
18
+ ··· −
3
2
u − 1
a
11
=
−u
u
a
4
=
−
1
2
u
19
+
3
2
u
18
+ ··· +
11
2
u −
11
2
−
1
2
u
19
+ 2u
18
+ ··· −
19
2
u + 10
a
10
=
−
3
4
u
19
+
5
2
u
18
+ ··· −
21
4
u + 3
1
2
u
19
− 2u
18
+ ··· +
7
2
u − 1
a
9
=
−
1
4
u
19
+
1
2
u
18
+ ··· − 2u
2
+
1
4
u
1
2
u
19
− u
18
+ ··· +
3
2
u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −7u
19
+ 18u
18
− 80u
17
+ 154u
16
− 358u
15
+ 536u
14
− 813u
13
+ 973u
12
− 1035u
11
+
1062u
10
− 910u
9
+ 941u
8
− 875u
7
+ 831u
6
− 754u
5
+ 461u
4
− 286u
3
+ 94u
2
− 10u + 26
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
− 3u
19
+ ··· + u + 1
c
2
, c
5
u
20
+ 10u
19
+ ··· + 96u + 64
c
3
, c
4
, c
8
c
10
u
20
− u
19
+ ··· + u + 1
c
6
, c
11
, c
12
u
20
+ 4u
19
+ ··· + 10u + 4
c
7
u
20
− 4u
19
+ ··· − 702u + 180
c
9
u
20
+ u
19
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
+ 37y
19
+ ··· + 39y + 1
c
2
, c
5
y
20
− 18y
19
+ ··· + 15360y + 4096
c
3
, c
4
, c
8
c
10
y
20
+ 5y
19
+ ··· + 7y + 1
c
6
, c
11
, c
12
y
20
+ 16y
19
+ ··· + 84y + 16
c
7
y
20
− 16y
19
+ ··· + 203796y + 32400
c
9
y
20
+ 45y
19
+ ··· + 47y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.803867 + 0.553658I
a = −0.349654 − 0.706335I
b = 0.551090 − 0.058137I
4.62986 − 2.72937I 8.70348 + 10.27722I
u = −0.803867 − 0.553658I
a = −0.349654 + 0.706335I
b = 0.551090 + 0.058137I
4.62986 + 2.72937I 8.70348 − 10.27722I
u = 0.918506 + 0.116805I
a = −1.36529 + 0.79203I
b = 0.368555 + 0.693111I
12.0452 + 8.8841I 3.76642 − 4.75992I
u = 0.918506 − 0.116805I
a = −1.36529 − 0.79203I
b = 0.368555 − 0.693111I
12.0452 − 8.8841I 3.76642 + 4.75992I
u = 0.324780 + 1.157920I
a = −0.864339 + 0.054005I
b = 0.1230180 + 0.0513018I
0.778843 + 0.268408I 0.81841 + 2.57430I
u = 0.324780 − 1.157920I
a = −0.864339 − 0.054005I
b = 0.1230180 − 0.0513018I
0.778843 − 0.268408I 0.81841 − 2.57430I
u = 0.790212 + 0.106147I
a = 0.786114 − 0.120277I
b = −0.311017 − 0.987779I
3.96682 + 3.79390I 4.34323 − 7.18597I
u = 0.790212 − 0.106147I
a = 0.786114 + 0.120277I
b = −0.311017 + 0.987779I
3.96682 − 3.79390I 4.34323 + 7.18597I
u = 0.497474 + 1.173310I
a = 0.040811 + 0.371019I
b = 1.152440 − 0.664354I
8.80911 − 3.86307I 1.37632 + 1.51742I
u = 0.497474 − 1.173310I
a = 0.040811 − 0.371019I
b = 1.152440 + 0.664354I
8.80911 + 3.86307I 1.37632 − 1.51742I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.034804 + 1.372190I
a = −0.01189 + 1.58136I
b = 0.28579 − 2.05494I
−5.57112 − 1.57630I −4.91455 + 4.16839I
u = −0.034804 − 1.372190I
a = −0.01189 − 1.58136I
b = 0.28579 + 2.05494I
−5.57112 + 1.57630I −4.91455 − 4.16839I
u = 0.341712 + 1.335550I
a = 0.64325 − 1.79998I
b = −0.16720 + 2.35614I
−0.56408 + 7.88066I −0.02470 − 9.49068I
u = 0.341712 − 1.335550I
a = 0.64325 + 1.79998I
b = −0.16720 − 2.35614I
−0.56408 − 7.88066I −0.02470 + 9.49068I
u = 0.41375 + 1.36143I
a = −0.02752 + 2.02004I
b = −0.53548 − 3.06612I
7.4013 + 13.6532I −0.15677 − 6.88804I
u = 0.41375 − 1.36143I
a = −0.02752 − 2.02004I
b = −0.53548 + 3.06612I
7.4013 − 13.6532I −0.15677 + 6.88804I
u = −0.23877 + 1.45754I
a = 0.61208 − 1.29474I
b = −0.89507 + 1.82927I
−1.86609 − 6.35619I −4.51758 + 8.10478I
u = −0.23877 − 1.45754I
a = 0.61208 + 1.29474I
b = −0.89507 − 1.82927I
−1.86609 + 6.35619I −4.51758 − 8.10478I
u = −0.208998 + 0.404596I
a = −0.463566 + 0.573101I
b = −0.072115 + 0.396543I
−0.021058 − 0.934804I −0.39426 + 7.39454I
u = −0.208998 − 0.404596I
a = −0.463566 − 0.573101I
b = −0.072115 − 0.396543I
−0.021058 + 0.934804I −0.39426 − 7.39454I
6
II.
I
u
2
= hu
11
+2u
10
+· · ·+b+1, −2u
11
−3u
10
+· · ·+a+3, u
12
+u
11
+· · ·−3u+1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
6
=
1
u
2
a
1
=
u
u
3
+ u
a
3
=
2u
11
+ 3u
10
+ ··· − 4u − 3
−u
11
− 2u
10
− 5u
9
− 7u
8
− 7u
7
− 7u
6
− u
5
+ 2u
4
+ 3u
3
+ 4u
2
− 1
a
8
=
−u
4
− u
2
+ 1
−u
6
− 2u
4
− u
2
a
5
=
−u
9
− 4u
7
− u
6
− 6u
5
− 4u
4
− 2u
3
− 3u
2
+ u + 3
−u
8
− u
7
− 3u
6
− 2u
5
− u
4
− u
3
+ 3u
2
+ u
a
2
=
−u
11
− u
10
− 4u
9
− 4u
8
− 4u
7
− 4u
6
+ 3u
5
+ 3u
4
+ 4u
3
+ 5u
2
− u − 3
u
11
+ u
10
+ 5u
9
+ 5u
8
+ 9u
7
+ 8u
6
+ 5u
5
+ 2u
4
− u
3
− 5u
2
− u
a
11
=
−u
u
a
4
=
2u
11
+ 2u
10
+ ··· − 3u − 3
−u
11
− u
10
− 4u
9
− 3u
8
− 4u
7
− 2u
6
+ 2u
5
+ 3u
4
+ 3u
3
+ 3u
2
− u − 1
a
10
=
−u
9
− u
8
− 5u
7
− 4u
6
− 8u
5
− 5u
4
− 2u
3
+ u
2
+ 4u + 4
−u
11
− 2u
10
− 6u
9
− 9u
8
− 12u
7
− 13u
6
− 7u
5
− 2u
4
+ 4u
3
+ 6u
2
+ 3u
a
9
=
−u
10
− 2u
9
− 6u
8
− 9u
7
− 12u
6
− 13u
5
− 8u
4
− 2u
3
+ 3u
2
+ 7u + 4
−u
11
− u
10
− 5u
9
− 4u
8
− 8u
7
− 6u
6
− 2u
5
− 2u
4
+ 4u
3
+ 2u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 8u
11
+ 7u
10
+ 39u
9
+ 28u
8
+ 67u
7
+ 41u
6
+ 35u
5
+ 17u
4
− 12u
3
− 11u
2
− 9u − 9
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 3u
11
+ ··· − 3u − 1
c
2
u
12
+ 3u
11
+ ··· − 7u + 3
c
3
, c
8
u
12
+ u
11
− 4u
10
− 4u
9
+ 3u
8
+ 4u
7
+ 4u
6
+ u
5
− 3u
4
− 2u
3
− 2u
2
− u − 1
c
4
, c
10
u
12
− u
11
− 4u
10
+ 4u
9
+ 3u
8
− 4u
7
+ 4u
6
− u
5
− 3u
4
+ 2u
3
− 2u
2
+ u − 1
c
5
u
12
− 3u
11
+ ··· + 7u + 3
c
6
u
12
+ u
11
+ ··· − 3u + 1
c
7
u
12
− u
11
+ ··· − 5u + 1
c
9
u
12
+ u
11
+ 2u
10
+ 2u
9
+ 3u
8
− u
7
− 4u
6
− 4u
5
− 3u
4
+ 4u
3
+ 4u
2
− u − 1
c
11
, c
12
u
12
− u
11
+ ··· + 3u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 5y
11
+ ··· − 5y + 1
c
2
, c
5
y
12
− 15y
11
+ ··· − 79y + 9
c
3
, c
4
, c
8
c
10
y
12
− 9y
11
+ ··· + 3y + 1
c
6
, c
11
, c
12
y
12
+ 11y
11
+ ··· − 17y + 1
c
7
y
12
− 9y
11
+ ··· − 19y + 1
c
9
y
12
+ 3y
11
+ ··· − 9y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.779914 + 0.263433I
a = −0.739356 − 0.514285I
b = 0.685921 − 0.270227I
4.49844 − 1.95126I 6.47342 + 1.58269I
u = −0.779914 − 0.263433I
a = −0.739356 + 0.514285I
b = 0.685921 + 0.270227I
4.49844 + 1.95126I 6.47342 − 1.58269I
u = −0.207510 + 1.165490I
a = 1.28081 − 1.18072I
b = −0.553497 + 1.171330I
2.05176 − 1.39702I 4.75145 + 0.05437I
u = −0.207510 − 1.165490I
a = 1.28081 + 1.18072I
b = −0.553497 − 1.171330I
2.05176 + 1.39702I 4.75145 − 0.05437I
u = 0.725402
a = 2.12667
b = −0.0816291
−1.30199 3.72770
u = 0.074423 + 1.296140I
a = −1.09122 + 1.86397I
b = 0.98699 − 2.82215I
−7.89835 + 1.11402I −9.81262 + 0.65462I
u = 0.074423 − 1.296140I
a = −1.09122 − 1.86397I
b = 0.98699 + 2.82215I
−7.89835 − 1.11402I −9.81262 − 0.65462I
u = 0.298860 + 1.278450I
a = 0.58549 − 1.60488I
b = −0.90103 + 2.69883I
−5.28054 + 3.69650I −1.82268 − 3.88848I
u = 0.298860 − 1.278450I
a = 0.58549 + 1.60488I
b = −0.90103 − 2.69883I
−5.28054 − 3.69650I −1.82268 + 3.88848I
u = −0.36930 + 1.39020I
a = 0.050962 − 1.233380I
b = −0.31739 + 1.65322I
−0.69950 − 6.22445I 1.48925 + 7.33691I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.36930 − 1.39020I
a = 0.050962 + 1.233380I
b = −0.31739 − 1.65322I
−0.69950 + 6.22445I 1.48925 − 7.33691I
u = 0.241471
a = −4.30005
b = −0.720367
−3.78082 −11.8850
11
III. I
u
3
= h−309u
5
a
3
+ 1269u
5
a
2
+ · · · + 3300a − 2645, −u
5
a
2
+ 5u
5
a + · · · +
20a − 22, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
6
=
1
u
2
a
1
=
u
u
3
+ u
a
3
=
a
0.121797a
3
u
5
− 0.500197a
2
u
5
+ ··· − 1.30075a + 1.04257
a
8
=
−u
4
− u
2
+ 1
u
5
+ u
4
+ 2u
3
+ u
2
+ u − 1
a
5
=
0.164762a
3
u
5
+ 0.0555775a
2
u
5
+ ··· − 1.46788a + 3.80922
−0.338195a
3
u
5
+ 0.146630a
2
u
5
+ ··· + 1.05952a − 0.137170
a
2
=
0.0429641a
3
u
5
+ 0.555775a
2
u
5
+ ··· − 0.167127a + 1.76665
0.272369a
3
u
5
− 0.547103a
2
u
5
+ ··· + 0.214032a − 0.430430
a
11
=
−u
u
a
4
=
0.610564a
3
u
5
− 0.693733a
2
u
5
+ ··· + 0.154513a − 0.293260
−0.488766a
3
u
5
+ 0.193536a
2
u
5
+ ··· − 0.455262a + 1.33583
a
10
=
−0.00827749a
3
u
5
+ 0.296807a
2
u
5
+ ··· − 0.118644a + 1.19196
0.144265a
3
u
5
− 0.368940a
2
u
5
+ ··· − 1.35081a + 1.45842
a
9
=
−0.140323a
3
u
5
− 0.357115a
2
u
5
+ ··· − 0.864013a + 2.53212
−0.00827749a
3
u
5
+ 0.296807a
2
u
5
+ ··· − 0.118644a + 1.19196
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
+ 4u
3
+ 8u
2
+ 4u + 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
− 5u
23
+ ··· + 3370u − 89
c
2
, c
5
(u
2
− u − 1)
12
c
3
, c
4
, c
8
c
10
u
24
− u
23
+ ··· − 64u − 31
c
6
, c
11
, c
12
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
4
c
7
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
4
c
9
u
24
+ u
23
+ ··· + 244u − 509
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
+ 7y
23
+ ··· − 10179964y + 7921
c
2
, c
5
(y
2
− 3y + 1)
12
c
3
, c
4
, c
8
c
10
y
24
− 5y
23
+ ··· + 120y + 961
c
6
, c
11
, c
12
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
4
c
7
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
4
c
9
y
24
+ 19y
23
+ ··· − 1812532y + 259081
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873214
a = −0.659674 + 0.470538I
b = 0.115030 + 0.358787I
3.71224 4.26950
u = −0.873214
a = −0.659674 − 0.470538I
b = 0.115030 − 0.358787I
3.71224 4.26950
u = −0.873214
a = 1.72705 + 0.77873I
b = −0.301154 + 0.593781I
11.6079 4.26950
u = −0.873214
a = 1.72705 − 0.77873I
b = −0.301154 − 0.593781I
11.6079 4.26950
u = 0.138835 + 1.234450I
a = −0.715076 − 0.696779I
b = −0.303312 + 0.803256I
0.98760 + 1.97241I −3.42428 − 3.68478I
u = 0.138835 + 1.234450I
a = 1.46538 − 0.91785I
b = −1.27213 + 1.88327I
−6.90809 + 1.97241I −3.42428 − 3.68478I
u = 0.138835 + 1.234450I
a = −0.40722 + 2.05651I
b = 0.52582 − 3.23377I
−6.90809 + 1.97241I −3.42428 − 3.68478I
u = 0.138835 + 1.234450I
a = −2.05522 − 2.28427I
b = 2.25718 + 2.73240I
0.98760 + 1.97241I −3.42428 − 3.68478I
u = 0.138835 − 1.234450I
a = −0.715076 + 0.696779I
b = −0.303312 − 0.803256I
0.98760 − 1.97241I −3.42428 + 3.68478I
u = 0.138835 − 1.234450I
a = 1.46538 + 0.91785I
b = −1.27213 − 1.88327I
−6.90809 − 1.97241I −3.42428 + 3.68478I
15
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.138835 − 1.234450I
a = −0.40722 − 2.05651I
b = 0.52582 + 3.23377I
−6.90809 − 1.97241I −3.42428 + 3.68478I
u = 0.138835 − 1.234450I
a = −2.05522 + 2.28427I
b = 2.25718 − 2.73240I
0.98760 − 1.97241I −3.42428 + 3.68478I
u = −0.408802 + 1.276380I
a = −0.236388 − 0.995320I
b = −0.07505 + 1.70186I
−0.25226 − 4.59213I 0.58114 + 3.20482I
u = −0.408802 + 1.276380I
a = 0.271082 + 0.518597I
b = −1.47317 − 1.04109I
7.64342 − 4.59213I 0.58114 + 3.20482I
u = −0.408802 + 1.276380I
a = 0.0568064 + 0.0049770I
b = 0.540176 − 0.066558I
−0.25226 − 4.59213I 0.58114 + 3.20482I
u = −0.408802 + 1.276380I
a = 0.19907 + 2.07416I
b = 0.25546 − 3.24019I
7.64342 − 4.59213I 0.58114 + 3.20482I
u = −0.408802 − 1.276380I
a = −0.236388 + 0.995320I
b = −0.07505 − 1.70186I
−0.25226 + 4.59213I 0.58114 − 3.20482I
u = −0.408802 − 1.276380I
a = 0.271082 − 0.518597I
b = −1.47317 + 1.04109I
7.64342 + 4.59213I 0.58114 − 3.20482I
u = −0.408802 − 1.276380I
a = 0.0568064 − 0.0049770I
b = 0.540176 + 0.066558I
−0.25226 + 4.59213I 0.58114 − 3.20482I
u = −0.408802 − 1.276380I
a = 0.19907 − 2.07416I
b = 0.25546 + 3.24019I
7.64342 + 4.59213I 0.58114 − 3.20482I
16
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.413150
a = 1.61499
b = 0.893703
−3.20899 5.41680
u = 0.413150
a = 2.19113 + 1.72840I
b = −1.244020 + 0.295025I
4.68669 5.41680
u = 0.413150
a = 2.19113 − 1.72840I
b = −1.244020 − 0.295025I
4.68669 5.41680
u = 0.413150
a = −3.28887
b = 0.0566461
−3.20899 5.41680
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
+ 3u
11
+ ··· − 3u − 1)(u
20
− 3u
19
+ ··· + u + 1)
· (u
24
− 5u
23
+ ··· + 3370u − 89)
c
2
((u
2
− u − 1)
12
)(u
12
+ 3u
11
+ ··· − 7u + 3)(u
20
+ 10u
19
+ ··· + 96u + 64)
c
3
, c
8
(u
12
+ u
11
− 4u
10
− 4u
9
+ 3u
8
+ 4u
7
+ 4u
6
+ u
5
− 3u
4
− 2u
3
− 2u
2
− u − 1)
· (u
20
− u
19
+ ··· + u + 1)(u
24
− u
23
+ ··· − 64u − 31)
c
4
, c
10
(u
12
− u
11
− 4u
10
+ 4u
9
+ 3u
8
− 4u
7
+ 4u
6
− u
5
− 3u
4
+ 2u
3
− 2u
2
+ u − 1)
· (u
20
− u
19
+ ··· + u + 1)(u
24
− u
23
+ ··· − 64u − 31)
c
5
((u
2
− u − 1)
12
)(u
12
− 3u
11
+ ··· + 7u + 3)(u
20
+ 10u
19
+ ··· + 96u + 64)
c
6
((u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
4
)(u
12
+ u
11
+ ··· − 3u + 1)
· (u
20
+ 4u
19
+ ··· + 10u + 4)
c
7
((u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
4
)(u
12
− u
11
+ ··· − 5u + 1)
· (u
20
− 4u
19
+ ··· − 702u + 180)
c
9
(u
12
+ u
11
+ 2u
10
+ 2u
9
+ 3u
8
− u
7
− 4u
6
− 4u
5
− 3u
4
+ 4u
3
+ 4u
2
− u − 1)
· (u
20
+ u
19
+ ··· + u + 1)(u
24
+ u
23
+ ··· + 244u − 509)
c
11
, c
12
((u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
4
)(u
12
− u
11
+ ··· + 3u + 1)
· (u
20
+ 4u
19
+ ··· + 10u + 4)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
12
− 5y
11
+ ··· − 5y + 1)(y
20
+ 37y
19
+ ··· + 39y + 1)
· (y
24
+ 7y
23
+ ··· − 10179964y + 7921)
c
2
, c
5
((y
2
− 3y + 1)
12
)(y
12
− 15y
11
+ ··· − 79y + 9)
· (y
20
− 18y
19
+ ··· + 15360y + 4096)
c
3
, c
4
, c
8
c
10
(y
12
− 9y
11
+ ··· + 3y + 1)(y
20
+ 5y
19
+ ··· + 7y + 1)
· (y
24
− 5y
23
+ ··· + 120y + 961)
c
6
, c
11
, c
12
((y
6
+ 5y
5
+ ··· − 5y + 1)
4
)(y
12
+ 11y
11
+ ··· − 17y + 1)
· (y
20
+ 16y
19
+ ··· + 84y + 16)
c
7
((y
6
− 7y
5
+ ··· − 5y + 1)
4
)(y
12
− 9y
11
+ ··· − 19y + 1)
· (y
20
− 16y
19
+ ··· + 203796y + 32400)
c
9
(y
12
+ 3y
11
+ ··· − 9y + 1)(y
20
+ 45y
19
+ ··· + 47y + 1)
· (y
24
+ 19y
23
+ ··· − 1812532y + 259081)
19
