11n
76
(K11n
76
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 11 9 3 6 7 1 6
Solving Sequence
6,8
9
4,7 1,3
2 11 5 10
c
8
c
6
c
3
c
2
c
11
c
5
c
10
c
1
, c
4
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
+ u
8
− 6u
7
− 4u
6
+ 12u
5
+ u
4
− 8u
3
+ 8u
2
+ 2d − u,
u
10
+ 2u
9
− 6u
8
− 12u
7
+ 14u
6
+ 21u
5
− 21u
4
− 5u
3
+ 22u
2
+ 2c − 10u,
u
10
+ 2u
9
− 5u
8
− 10u
7
+ 8u
6
+ 11u
5
− 9u
4
+ 4u
3
+ 13u
2
+ 4b + u, a − 1,
u
11
+ 2u
10
− 6u
9
− 12u
8
+ 13u
7
+ 21u
6
− 17u
5
− 7u
4
+ 18u
3
− 3u
2
− u − 1i
I
u
2
= h3u
7
+ 5u
6
− 3u
5
− u
4
+ 3u
3
− 10u
2
+ 4d − 9u − 2, 7u
7
+ 9u
6
− 15u
5
− u
4
+ 15u
3
− 34u
2
+ 8c − 9u + 14,
u
7
+ u
6
− u
5
+ u
4
+ u
3
− 4u
2
+ 2b − 3u, −u
7
− 3u
6
− 3u
5
+ 3u
4
+ 3u
3
− 6u
2
+ 8a + 7u + 18,
u
8
+ u
7
− 3u
6
− u
5
+ 3u
4
− 4u
3
− 3u
2
+ 4u + 4i
I
u
3
= hd + u, c, −au + b + a + 1, a
2
+ a − u − 1, u
2
+ u − 1i
I
u
4
= hu
3
+ d − u + 1, u
3
− u
2
+ c, u
3
− u
2
+ b − u + 2, −u
3
+ a − 1, u
4
− u
3
+ 2u − 1i
I
u
5
= hd + u, c, b + u + 1, a − 1, u
2
+ u − 1i
I
u
6
= hd, c + 1, b, a + 1, u − 1i
I
u
7
= hd, c − 1, b + 1, a, u − 1i
I
u
8
= hd, cb + 1, a + 1, u − 1i
I
v
1
= ha, d, c − 1, b + 1, v − 1i
* 8 irreducible components of dim
C
= 0, with total 32 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
1
I. I
u
1
= hu
9
+ u
8
+ · · · + 2d − u, u
10
+ 2u
9
+ · · · + 2c − 10u, u
10
+ 2u
9
+ · · · +
4b + u, a − 1, u
11
+ 2u
10
+ · · · − u − 1i
(i) Arc colorings
a
6
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
4
=
−
1
2
u
10
− u
9
+ ··· − 11u
2
+ 5u
−
1
2
u
9
−
1
2
u
8
+ ··· − 4u
2
+
1
2
u
a
7
=
u
−u
3
+ u
a
1
=
1
−
1
4
u
10
−
1
2
u
9
+ ··· −
13
4
u
2
−
1
4
u
a
3
=
−
1
2
u
10
−
1
2
u
9
+ ··· − 7u
2
+
9
2
u
−
1
2
u
9
−
1
2
u
8
+ ··· − 4u
2
+
1
2
u
a
2
=
−
1
2
u
10
−
3
4
u
9
+ ··· +
19
4
u +
3
4
−
1
2
u
10
−
3
4
u
9
+ ··· +
3
4
u +
1
4
a
11
=
1
−
1
4
u
10
−
1
2
u
9
+ ··· −
9
4
u
2
−
1
4
u
a
5
=
−u
1
4
u
9
+
1
2
u
8
+ ··· +
5
4
u +
1
4
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
10
+
7
2
u
9
−4u
8
−
45
2
u
7
+ u
6
+ 47u
5
+
3
2
u
4
−
71
2
u
3
+ 20u
2
+
23
2
u +
1
2
in decimal forms when there is not enough margin.
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
− 2u
10
+ u
9
+ 2u
8
− 5u
6
+ 7u
5
+ 6u
4
− 13u
3
+ 3u
2
+ 8u − 4
c
2
u
11
+ 2u
10
+ ··· + 88u + 16
c
3
, c
7
u
11
+ 2u
10
− u
9
− 8u
8
− 11u
7
+ 46u
5
+ 76u
4
+ 32u
3
− 12u
2
− 16u − 8
c
5
, c
6
, c
8
c
9
, c
11
u
11
+ 2u
10
+ ··· − u − 1
c
10
u
11
− 16u
10
+ ··· − 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
− 2y
10
+ ··· + 88y − 16
c
2
y
11
+ 14y
10
+ ··· + 2336y − 256
c
3
, c
7
y
11
− 6y
10
+ ··· + 64y − 64
c
5
, c
6
, c
8
c
9
, c
11
y
11
− 16y
10
+ ··· − 5y − 1
c
10
y
11
− 36y
10
+ ··· − 93y − 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.552760 + 0.641799I
a = 1.00000
b = 0.82545 − 1.53098I
c = 0.712390 + 0.815288I
d = 1.044080 + 0.152224I
0.79689 + 3.53286I 6.46290 − 7.08687I
u = 0.552760 − 0.641799I
a = 1.00000
b = 0.82545 + 1.53098I
c = 0.712390 − 0.815288I
d = 1.044080 − 0.152224I
0.79689 − 3.53286I 6.46290 + 7.08687I
u = 0.590824
a = 1.00000
b = −1.42161
c = 0.0396568
d = −0.620148
0.987118 9.97440
u = 1.64391 + 0.11631I
a = 1.00000
b = 0.437522 − 0.637453I
c = 0.234439 − 1.284060I
d = 0.24685 − 1.67120I
10.83450 + 3.51232I 10.06687 − 2.29315I
u = 1.64391 − 0.11631I
a = 1.00000
b = 0.437522 + 0.637453I
c = 0.234439 + 1.284060I
d = 0.24685 + 1.67120I
10.83450 − 3.51232I 10.06687 + 2.29315I
u = −1.60901 + 0.41639I
a = 1.00000
b = 0.32965 + 2.04536I
c = 0.194428 − 1.371430I
d = 1.57467 − 0.87175I
14.9243 − 12.3125I 9.62929 + 5.75829I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.60901 − 0.41639I
a = 1.00000
b = 0.32965 − 2.04536I
c = 0.194428 + 1.371430I
d = 1.57467 + 0.87175I
14.9243 + 12.3125I 9.62929 − 5.75829I
u = −0.162723 + 0.277330I
a = 1.00000
b = 0.160409 + 0.252652I
c = −0.22673 + 2.50982I
d = 0.267436 + 0.517187I
−1.66390 − 0.61823I −3.63835 + 1.22407I
u = −0.162723 − 0.277330I
a = 1.00000
b = 0.160409 − 0.252652I
c = −0.22673 − 2.50982I
d = 0.267436 − 0.517187I
−1.66390 + 0.61823I −3.63835 − 1.22407I
u = −1.72035 + 0.28600I
a = 1.00000
b = −0.04222 + 1.42193I
c = −0.434360 + 0.920646I
d = −1.82296 + 0.61164I
17.3830 − 4.9116I 11.49209 + 1.65700I
u = −1.72035 − 0.28600I
a = 1.00000
b = −0.04222 − 1.42193I
c = −0.434360 − 0.920646I
d = −1.82296 − 0.61164I
17.3830 + 4.9116I 11.49209 − 1.65700I
6
II. I
u
2
= h3u
7
+ 5u
6
+ · · · + 4d − 2, 7u
7
+ 9u
6
+ · · · + 8c + 14, u
7
+ u
6
+ · · · +
2b − 3u, −u
7
− 3u
6
+ · · · + 8a + 18, u
8
+ u
7
+ · · · + 4u + 4i
(i) Arc colorings
a
6
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
4
=
−
7
8
u
7
−
9
8
u
6
+ ··· +
9
8
u −
7
4
−
3
4
u
7
−
5
4
u
6
+ ··· +
9
4
u +
1
2
a
7
=
u
−u
3
+ u
a
1
=
1
8
u
7
+
3
8
u
6
+ ··· −
7
8
u −
9
4
−
1
2
u
7
−
1
2
u
6
+ ··· + 2u
2
+
3
2
u
a
3
=
−
1
8
u
7
+
1
8
u
6
+ ··· −
9
8
u −
9
4
−
3
4
u
7
−
5
4
u
6
+ ··· +
9
4
u +
1
2
a
2
=
−
1
2
u
7
+ 2u
5
+ ··· −
1
2
u −
5
2
−
3
4
u
7
−
5
4
u
6
+ ··· +
13
4
u +
3
2
a
11
=
1
8
u
7
+
3
8
u
6
+ ··· −
7
8
u −
9
4
−1
a
5
=
1
2
u
7
+ u
6
+ ··· −
5
2
u +
1
2
1
4
u
7
+
3
4
u
6
+ ··· −
7
4
u −
1
2
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
7
+ 6u
6
− 4u
5
+ 6u
3
− 14u
2
− 14u + 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
4
− u
3
+ u + 1)
2
c
2
(u
4
+ u
3
+ 4u
2
+ u + 1)
2
c
3
, c
7
(u
4
− 3u
3
+ 3u
2
− 2u + 2)
2
c
5
, c
6
, c
8
c
9
, c
11
u
8
+ u
7
− 3u
6
− u
5
+ 3u
4
− 4u
3
− 3u
2
+ 4u + 4
c
10
u
8
− 7u
7
+ 17u
6
− 17u
5
+ 19u
4
− 50u
3
+ 65u
2
− 40u + 16
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
4
− y
3
+ 4y
2
− y + 1)
2
c
2
(y
4
+ 7y
3
+ 16y
2
+ 7y + 1)
2
c
3
, c
7
(y
4
− 3y
3
+ y
2
+ 8y + 4)
2
c
5
, c
6
, c
8
c
9
, c
11
y
8
− 7y
7
+ 17y
6
− 17y
5
+ 19y
4
− 50y
3
+ 65y
2
− 40y + 16
c
10
y
8
− 15y
7
+ 89y
6
− 213y
5
+ 343y
4
− 846y
3
+ 833y
2
+ 480y + 256
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.695289 + 0.428533I
a = −1.29532 − 0.84192I
b = −0.109976 − 0.519497I
c = −0.57622 − 2.77692I
d = 0.066121 − 0.864054I
2.62917 − 1.45022I 7.43990 + 4.72374I
u = −0.695289 − 0.428533I
a = −1.29532 + 0.84192I
b = −0.109976 + 0.519497I
c = −0.57622 + 2.77692I
d = 0.066121 + 0.864054I
2.62917 + 1.45022I 7.43990 − 4.72374I
u = 0.529919 + 1.081980I
a = −0.745137 + 1.110160I
b = −0.39002 + 1.84237I
c = −1.47609 − 1.17606I
d = −1.56612 − 0.45882I
8.06290 + 6.78371I 8.56010 − 4.72374I
u = 0.529919 − 1.081980I
a = −0.745137 − 1.110160I
b = −0.39002 − 1.84237I
c = −1.47609 + 1.17606I
d = −1.56612 + 0.45882I
8.06290 − 6.78371I 8.56010 + 4.72374I
u = 1.261410 + 0.030288I
a = −0.542730 + 0.352757I
b = −0.109976 − 0.519497I
c = 0.054288 − 0.560610I
d = 0.066121 − 0.864054I
2.62917 − 1.45022I 7.43990 + 4.72374I
u = 1.261410 − 0.030288I
a = −0.542730 − 0.352757I
b = −0.109976 + 0.519497I
c = 0.054288 + 0.560610I
d = 0.066121 + 0.864054I
2.62917 + 1.45022I 7.43990 − 4.72374I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.59604 + 0.21793I
a = −0.416815 + 0.621004I
b = −0.39002 − 1.84237I
c = −0.001980 + 0.777911I
d = −1.56612 + 0.45882I
8.06290 − 6.78371I 8.56010 + 4.72374I
u = −1.59604 − 0.21793I
a = −0.416815 − 0.621004I
b = −0.39002 + 1.84237I
c = −0.001980 − 0.777911I
d = −1.56612 − 0.45882I
8.06290 + 6.78371I 8.56010 − 4.72374I
11
III. I
u
3
= hd + u, c, −au + b + a + 1, a
2
+ a − u − 1, u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
8
=
1
0
a
9
=
1
u − 1
a
4
=
0
−u
a
7
=
u
−u + 1
a
1
=
a
au − a − 1
a
3
=
u
−u
a
2
=
a
−1
a
11
=
a
−1
a
5
=
au − 1
au + u
a
10
=
u
−u
a
10
=
u
−u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
11
u
4
− u
3
+ 2u − 1
c
2
u
4
+ u
3
+ 2u
2
+ 4u + 1
c
3
, c
6
, c
7
c
8
, c
9
(u
2
+ u − 1)
2
c
10
u
4
− u
3
+ 2u
2
− 4u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
11
y
4
− y
3
+ 2y
2
− 4y + 1
c
2
, c
10
y
4
+ 3y
3
− 2y
2
− 12y + 1
c
3
, c
6
, c
7
c
8
, c
9
(y
2
− 3y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0.866760
b = −1.33107
c = 0
d = −0.618034
0.986960 10.0000
u = 0.618034
a = −1.86676
b = −0.286961
c = 0
d = −0.618034
0.986960 10.0000
u = −1.61803
a = −0.500000 + 0.606658I
b = 0.30902 − 1.58825I
c = 0
d = 1.61803
8.88264 10.0000
u = −1.61803
a = −0.500000 − 0.606658I
b = 0.30902 + 1.58825I
c = 0
d = 1.61803
8.88264 10.0000
15
IV.
I
u
4
= hu
3
+d−u+1, u
3
−u
2
+c, u
3
−u
2
+b−u+2, −u
3
+a−1, u
4
−u
3
+2u−1i
(i) Arc colorings
a
6
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
4
=
−u
3
+ u
2
−u
3
+ u − 1
a
7
=
u
−u
3
+ u
a
1
=
u
3
+ 1
−u
3
+ u
2
+ u − 2
a
3
=
u
2
− u + 1
−u
3
+ u − 1
a
2
=
u
3
− 2u + 2
−1
a
11
=
u
3
+ 1
−1
a
5
=
u
2
− 1
u
3
+ 1
a
10
=
−u
2
+ 1
u
3
− 2u
2
− 2u + 1
a
10
=
−u
2
+ 1
u
3
− 2u
2
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
, c
9
u
4
− u
3
+ 2u − 1
c
2
u
4
+ u
3
+ 2u
2
+ 4u + 1
c
3
, c
5
, c
7
c
11
(u
2
+ u − 1)
2
c
10
(u
2
− 3u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
, c
9
y
4
− y
3
+ 2y
2
− 4y + 1
c
2
y
4
+ 3y
3
− 2y
2
− 12y + 1
c
3
, c
5
, c
7
c
11
(y
2
− 3y + 1)
2
c
10
(y
2
− 7y + 1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.15372
a = −0.535687
b = −0.286961
c = 2.86676
d = −0.618034
0.986960 10.0000
u = 0.809017 + 0.981593I
a = −0.809017 + 0.981593I
b = 0.30902 + 1.58825I
c = 1.50000 + 0.60666I
d = 1.61803
8.88264 10.0000
u = 0.809017 − 0.981593I
a = −0.809017 − 0.981593I
b = 0.30902 − 1.58825I
c = 1.50000 − 0.60666I
d = 1.61803
8.88264 10.0000
u = 0.535687
a = 1.15372
b = −1.33107
c = 0.133240
d = −0.618034
0.986960 10.0000
19
V. I
u
5
= hd + u, c, b + u + 1, a − 1, u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
8
=
1
0
a
9
=
1
u − 1
a
4
=
0
−u
a
7
=
u
−u + 1
a
1
=
1
−u − 1
a
3
=
u
−u
a
2
=
1
−2u
a
11
=
1
−2u
a
5
=
−u
−u + 2
a
10
=
u
−u
a
10
=
u
−u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
9
, c
11
u
2
+ u − 1
c
2
u
2
+ 3u + 1
c
10
u
2
− 3u + 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
9
, c
11
y
2
− 3y + 1
c
2
, c
10
y
2
− 7y + 1
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.00000
b = −1.61803
c = 0
d = −0.618034
0.986960 10.0000
u = −1.61803
a = 1.00000
b = 0.618034
c = 0
d = 1.61803
8.88264 10.0000
23
VI. I
u
6
= hd, c + 1, b, a + 1, u − 1i
(i) Arc colorings
a
6
=
0
1
a
8
=
1
0
a
9
=
1
−1
a
4
=
−1
0
a
7
=
1
0
a
1
=
−1
0
a
3
=
−1
0
a
2
=
−1
0
a
11
=
−1
−1
a
5
=
−1
0
a
10
=
0
−1
a
10
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
24
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
u
c
5
, c
8
, c
9
u − 1
c
6
, c
10
, c
11
u + 1
25
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
y
c
5
, c
6
, c
8
c
9
, c
10
, c
11
y − 1
26
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 0
c = −1.00000
d = 0
3.28987 12.0000
27
VII. I
u
7
= hd, c − 1, b + 1, a, u − 1i
(i) Arc colorings
a
6
=
0
1
a
8
=
1
0
a
9
=
1
−1
a
4
=
1
0
a
7
=
1
0
a
1
=
0
−1
a
3
=
1
0
a
2
=
1
−1
a
11
=
0
−1
a
5
=
0
1
a
10
=
0
−1
a
10
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
28
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
, c
9
u − 1
c
2
, c
4
, c
6
u + 1
c
3
, c
5
, c
7
c
10
, c
11
u
29
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
8
, c
9
y − 1
c
3
, c
5
, c
7
c
10
, c
11
y
30
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −1.00000
c = 1.00000
d = 0
0 0
31
VIII. I
u
8
= hd, cb + 1, a + 1, u − 1i
(i) Arc colorings
a
6
=
0
1
a
8
=
1
0
a
9
=
1
−1
a
4
=
c
0
a
7
=
1
0
a
1
=
−1
b
a
3
=
c
0
a
2
=
c − 1
b
a
11
=
−1
b − 1
a
5
=
−1
b
a
10
=
0
−1
a
10
=
0
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −c
2
− b
2
+ 8
(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.
32
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
c = ···
d = ···
1.64493 8.06956 − 0.34732I
33
IX. I
v
1
= ha, d, c − 1, b + 1, v − 1i
(i) Arc colorings
a
6
=
1
0
a
8
=
1
0
a
9
=
1
0
a
4
=
1
0
a
7
=
1
0
a
1
=
0
−1
a
3
=
1
0
a
2
=
1
−1
a
11
=
1
−1
a
5
=
0
1
a
10
=
1
0
a
10
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
34
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u − 1
c
2
, c
4
, c
5
c
10
u + 1
c
3
, c
6
, c
7
c
8
, c
9
u
35
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
10
, c
11
y − 1
c
3
, c
6
, c
7
c
8
, c
9
y
36
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
c = 1.00000
d = 0
0 0
37
X. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u − 1)
2
(u
2
+ u − 1)(u
4
− u
3
+ u + 1)
2
(u
4
− u
3
+ 2u − 1)
2
· (u
11
− 2u
10
+ u
9
+ 2u
8
− 5u
6
+ 7u
5
+ 6u
4
− 13u
3
+ 3u
2
+ 8u − 4)
c
2
u(u + 1)
2
(u
2
+ 3u + 1)(u
4
+ u
3
+ 2u
2
+ 4u + 1)
2
· ((u
4
+ u
3
+ 4u
2
+ u + 1)
2
)(u
11
+ 2u
10
+ ··· + 88u + 16)
c
3
, c
7
u
3
(u
2
+ u − 1)
5
(u
4
− 3u
3
+ 3u
2
− 2u + 2)
2
· (u
11
+ 2u
10
− u
9
− 8u
8
− 11u
7
+ 46u
5
+ 76u
4
+ 32u
3
− 12u
2
− 16u − 8)
c
4
u(u + 1)
2
(u
2
+ u − 1)(u
4
− u
3
+ u + 1)
2
(u
4
− u
3
+ 2u − 1)
2
· (u
11
− 2u
10
+ u
9
+ 2u
8
− 5u
6
+ 7u
5
+ 6u
4
− 13u
3
+ 3u
2
+ 8u − 4)
c
5
, c
11
u(u − 1)(u + 1)(u
2
+ u − 1)
3
(u
4
− u
3
+ 2u − 1)
· (u
8
+ u
7
+ ··· + 4u + 4)(u
11
+ 2u
10
+ ··· − u − 1)
c
6
u(u + 1)
2
(u
2
+ u − 1)
3
(u
4
− u
3
+ 2u − 1)
· (u
8
+ u
7
+ ··· + 4u + 4)(u
11
+ 2u
10
+ ··· − u − 1)
c
8
, c
9
u(u − 1)
2
(u
2
+ u − 1)
3
(u
4
− u
3
+ 2u − 1)
· (u
8
+ u
7
+ ··· + 4u + 4)(u
11
+ 2u
10
+ ··· − u − 1)
c
10
u(u + 1)
2
(u
2
− 3u + 1)
3
(u
4
− u
3
+ 2u
2
− 4u + 1)
· (u
8
− 7u
7
+ 17u
6
− 17u
5
+ 19u
4
− 50u
3
+ 65u
2
− 40u + 16)
· (u
11
− 16u
10
+ ··· − 5u − 1)
38
XI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
y(y − 1)
2
(y
2
− 3y + 1)(y
4
− y
3
+ 2y
2
− 4y + 1)
2
· ((y
4
− y
3
+ 4y
2
− y + 1)
2
)(y
11
− 2y
10
+ ··· + 88y − 16)
c
2
y(y − 1)
2
(y
2
− 7y + 1)(y
4
+ 3y
3
− 2y
2
− 12y + 1)
2
· ((y
4
+ 7y
3
+ 16y
2
+ 7y + 1)
2
)(y
11
+ 14y
10
+ ··· + 2336y − 256)
c
3
, c
7
y
3
(y
2
− 3y + 1)
5
(y
4
− 3y
3
+ y
2
+ 8y + 4)
2
· (y
11
− 6y
10
+ ··· + 64y − 64)
c
5
, c
6
, c
8
c
9
, c
11
y(y − 1)
2
(y
2
− 3y + 1)
3
(y
4
− y
3
+ 2y
2
− 4y + 1)
· (y
8
− 7y
7
+ 17y
6
− 17y
5
+ 19y
4
− 50y
3
+ 65y
2
− 40y + 16)
· (y
11
− 16y
10
+ ··· − 5y − 1)
c
10
y(y − 1)
2
(y
2
− 7y + 1)
3
(y
4
+ 3y
3
− 2y
2
− 12y + 1)
· (y
8
− 15y
7
+ 89y
6
− 213y
5
+ 343y
4
− 846y
3
+ 833y
2
+ 480y + 256)
· (y
11
− 36y
10
+ ··· − 93y − 1)
39
