11n
133
(K11n
133
)
A knot diagram
1
Linearized knot diagam
5 1 10 11 2 9 4 5 1 8 7
Solving Sequence
1,5
2 3
6,9
7 10 8 11 4
c
1
c
2
c
5
c
6
c
9
c
8
c
11
c
4
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
2
+ b + u, a − 1, u
5
− 5u
4
+ 7u
3
− 2u
2
+ u − 1i
I
u
2
= hu
2
+ b + u, a + 1, u
5
+ 3u
4
+ u
3
− 2u
2
− u + 1i
I
u
3
= h−u
2
a + b + u, a
2
− au + u
2
+ 3u + 3, u
3
+ 2u
2
+ 1i
I
u
4
= h−u
5
+ 4u
4
− u
3
− 8u
2
+ 4b + 7u, −u
5
+ 3u
4
+ 3u
3
− 11u
2
+ 4a − 3u + 9,
u
6
− 4u
5
− u
4
+ 18u
3
− 9u
2
− 20u + 16i
I
u
5
= h−u
2
b + b
2
+ u
2
− u − 1, a − 1, u
3
+ 2u
2
+ 1i
I
u
6
= hb + u + 1, a − u − 1, u
2
+ u + 1i
I
u
7
= hb − a + 1, a
2
− a + 1, u − 1i
I
u
8
= hb
2
+ b + 1, a + 1, u − 1i
* 8 irreducible components of dim
C
= 0, with total 34 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
= h−u
2
+ b + u, a − 1, u
5
− 5u
4
+ 7u
3
− 2u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
−u
3
+ u
a
9
=
1
u
2
− u
a
7
=
u
2
− 2u
u
4
− 3u
3
+ u
2
+ u
a
10
=
−u
2
+ u + 1
u
2
− u
a
8
=
1
−u
a
11
=
u
3
− 3u
2
+ u + 1
−u
4
+ 2u
3
+ u
2
− u
a
4
=
u
3
− 2u
2
− u + 1
−u
3
+ 2u
2
a
4
=
u
3
− 2u
2
− u + 1
−u
3
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6u
3
− 15u
2
− 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
8
u
5
+ 5u
4
+ 7u
3
+ 2u
2
+ u + 1
c
2
u
5
+ 11u
4
+ 31u
3
− 3u + 1
c
4
, c
7
, c
11
u
5
− 3u
4
+ 5u
3
− 3u
2
+ 1
c
6
, c
9
u
5
− 6u
4
+ 9u
3
+ 4u + 1
c
10
u
5
− 4u
4
+ 6u
3
− u
2
− 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
y
5
− 11y
4
+ 31y
3
− 3y −1
c
2
y
5
− 59y
4
+ 955y
3
− 208y
2
+ 9y −1
c
4
, c
7
, c
11
y
5
+ y
4
+ 7y
3
− 3y
2
+ 6y −1
c
6
, c
9
y
5
− 18y
4
+ 89y
3
+ 84y
2
+ 16y −1
c
10
y
5
− 4y
4
+ 24y
3
− 17y
2
+ 6y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.668174
a = 1.00000
b = −0.221718
−1.20060 −7.90700
u = −0.181543 + 0.487016I
a = 1.00000
b = −0.022683 − 0.663845I
1.11858 − 1.59084I 0.80256 + 2.24828I
u = −0.181543 − 0.487016I
a = 1.00000
b = −0.022683 + 0.663845I
1.11858 + 1.59084I 0.80256 − 2.24828I
u = 2.34746 + 0.17191I
a = 1.00000
b = 3.13354 + 0.63521I
−18.6126 − 10.9920I −8.84907 + 4.91483I
u = 2.34746 − 0.17191I
a = 1.00000
b = 3.13354 − 0.63521I
−18.6126 + 10.9920I −8.84907 − 4.91483I
5
II. I
u
2
= hu
2
+ b + u, a + 1, u
5
+ 3u
4
+ u
3
− 2u
2
− u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
−u
3
+ u
a
9
=
−1
−u
2
− u
a
7
=
−u
2
− 2u
−u
4
− 3u
3
− u
2
+ u
a
10
=
u
2
+ u − 1
−u
2
− u
a
8
=
−1
−u
a
11
=
u
3
+ 3u
2
+ u − 1
u
4
+ 2u
3
− u
2
− u
a
4
=
−u
3
− 2u
2
+ u + 1
u
3
+ 2u
2
a
4
=
−u
3
− 2u
2
+ u + 1
u
3
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
3
+ 15u
2
− 15
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ 3u
4
+ u
3
− 2u
2
− u + 1
c
2
u
5
+ 7u
4
+ 11u
3
+ 12u
2
+ 5u + 1
c
3
, c
5
, c
8
u
5
− 3u
4
+ u
3
+ 2u
2
− u − 1
c
4
, c
7
, c
11
u
5
+ u
4
+ u
3
− u
2
− 1
c
6
, c
9
u
5
− 4u
4
+ 3u
3
− 1
c
10
u
5
+ 2u
4
− 5u
2
− 6u − 3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
y
5
− 7y
4
+ 11y
3
− 12y
2
+ 5y −1
c
2
y
5
− 27y
4
− 37y
3
− 48y
2
+ y −1
c
4
, c
7
, c
11
y
5
+ y
4
+ 3y
3
+ y
2
− 2y −1
c
6
, c
9
y
5
− 10y
4
+ 9y
3
− 8y
2
− 1
c
10
y
5
− 4y
4
+ 8y
3
− 13y
2
+ 6y −9
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.921567 + 0.544227I
a = −1.00000
b = 0.368464 + 0.458856I
−2.41702 + 7.42796I −6.48635 − 7.69371I
u = −0.921567 − 0.544227I
a = −1.00000
b = 0.368464 − 0.458856I
−2.41702 − 7.42796I −6.48635 + 7.69371I
u = 0.575451 + 0.217130I
a = −1.00000
b = −0.859450 − 0.467025I
−2.58971 − 1.95896I −10.08501 + 4.98123I
u = 0.575451 − 0.217130I
a = −1.00000
b = −0.859450 + 0.467025I
−2.58971 + 1.95896I −10.08501 − 4.98123I
u = −2.30777
a = −1.00000
b = −3.01803
−16.3055 −8.85730
9
III. I
u
3
= h−u
2
a + b + u, a
2
− au + u
2
+ 3u + 3, u
3
+ 2u
2
+ 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
2u
2
+ u + 1
a
9
=
a
u
2
a − u
a
7
=
u
2
+ 2u − 1
u
2
a + u
2
+ 1
a
10
=
−u
2
a + a + u
u
2
a − u
a
8
=
a
−u
a
11
=
−u
2
a − au + u
u
2
a + u
2
a
4
=
−2u
2
a − au − 2u
2
− a + 1
2u
2
a + 2u
2
+ a
a
4
=
−2u
2
a − au − 2u
2
− a + 1
2u
2
a + 2u
2
+ a
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
2
a − 3u
2
+ 3u − 11
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
3
− 2u
2
− 1)
2
c
2
(u
3
+ 4u
2
− 4u + 1)
2
c
4
u
6
− 5u
5
+ 12u
4
− 15u
3
+ 11u
2
− 4u + 4
c
6
u
6
− 4u
5
− u
4
+ 18u
3
− 9u
2
− 20u + 16
c
7
, c
11
(u
2
+ u + 1)
3
c
8
u
6
+ 4u
5
− u
4
− 18u
3
− 9u
2
+ 20u + 16
c
9
u
6
+ 7u
5
+ 18u
4
+ 27u
3
+ 38u
2
+ 11u + 1
c
10
(u
3
+ u
2
− u − 2)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
(y
3
− 4y
2
− 4y −1)
2
c
2
(y
3
− 24y
2
+ 8y −1)
2
c
4
y
6
− y
5
+ 16y
4
+ 7y
3
+ 97y
2
+ 72y + 16
c
6
, c
8
y
6
− 18y
5
+ 127y
4
− 434y
3
+ 769y
2
− 688y + 256
c
7
, c
11
(y
2
+ y + 1)
3
c
9
y
6
− 13y
5
+ 22y
4
+ 487y
3
+ 886y
2
− 45y + 1
c
10
(y
3
− 3y
2
+ 5y −4)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.102785 + 0.665457I
a = 0.62769 − 1.48834I
b = −0.170516 + 0.063771I
−2.25297 − 0.53909I −9.12391 − 1.33093I
u = 0.102785 + 0.665457I
a = −0.52491 + 2.15379I
b = −0.17052 − 1.66828I
−2.25297 − 4.59885I −9.12391 + 5.59727I
u = 0.102785 − 0.665457I
a = 0.62769 + 1.48834I
b = −0.170516 − 0.063771I
−2.25297 + 0.53909I −9.12391 + 1.33093I
u = 0.102785 − 0.665457I
a = −0.52491 − 2.15379I
b = −0.17052 + 1.66828I
−2.25297 + 4.59885I −9.12391 − 5.59727I
u = −2.20557
a = −1.102790 + 0.178028I
b = −3.15897 + 0.86603I
−16.8782 + 2.0299I −10.75217 − 3.46410I
u = −2.20557
a = −1.102790 − 0.178028I
b = −3.15897 − 0.86603I
−16.8782 − 2.0299I −10.75217 + 3.46410I
13
IV. I
u
4
= h−u
5
+ 4u
4
− u
3
− 8u
2
+ 4b + 7u, −u
5
+ 3u
4
+ 3u
3
− 11u
2
+ 4a −
3u + 9, u
6
− 4u
5
− u
4
+ 18u
3
− 9u
2
− 20u + 16i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
−u
3
+ u
a
9
=
1
4
u
5
−
3
4
u
4
+ ··· +
3
4
u −
9
4
1
4
u
5
− u
4
+
1
4
u
3
+ 2u
2
−
7
4
u
a
7
=
−
3
16
u
5
+
1
2
u
4
+ ··· −
21
16
u + 1
−
1
4
u
5
+ u
4
−
1
4
u
3
− 3u
2
+
7
4
u + 1
a
10
=
1
4
u
4
− u
3
+
3
4
u
2
+
5
2
u −
9
4
1
4
u
5
− u
4
+
1
4
u
3
+ 2u
2
−
7
4
u
a
8
=
1
4
u
5
−
3
4
u
4
+ ··· +
3
4
u −
9
4
−
1
4
u
5
+
1
2
u
4
+ ··· −
11
4
u + 4
a
11
=
1
8
u
5
−
1
4
u
4
+ ··· +
19
8
u −
11
4
1
4
u
5
− u
4
+
5
4
u
3
+ u
2
−
15
4
u + 2
a
4
=
−
3
16
u
5
+
1
2
u
4
+ ··· −
29
16
u + 4
1
4
u
5
−
1
2
u
4
+ ··· +
11
4
u − 3
a
4
=
−
3
16
u
5
+
1
2
u
4
+ ··· −
29
16
u + 4
1
4
u
5
−
1
2
u
4
+ ··· +
11
4
u − 3
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
1
2
u
5
+ 3u
4
− 3u
3
−
13
2
u
2
+ 8u − 10
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
6
+ 4u
5
− u
4
− 18u
3
− 9u
2
+ 20u + 16
c
2
u
6
+ 18u
5
+ 127u
4
+ 434u
3
+ 769u
2
+ 688u + 256
c
3
, c
8
(u
3
− 2u
2
− 1)
2
c
4
, c
7
(u
2
+ u + 1)
3
c
6
, c
9
u
6
+ 7u
5
+ 18u
4
+ 27u
3
+ 38u
2
+ 11u + 1
c
10
u
6
− 8u
5
+ 29u
4
− 54u
3
+ 51u
2
− 22u + 4
c
11
u
6
− 5u
5
+ 12u
4
− 15u
3
+ 11u
2
− 4u + 4
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
6
− 18y
5
+ 127y
4
− 434y
3
+ 769y
2
− 688y + 256
c
2
y
6
− 70y
5
+ 2043y
4
− 17286y
3
+ 59201y
2
− 79616y + 65536
c
3
, c
8
(y
3
− 4y
2
− 4y −1)
2
c
4
, c
7
(y
2
+ y + 1)
3
c
6
, c
9
y
6
− 13y
5
+ 22y
4
+ 487y
3
+ 886y
2
− 45y + 1
c
10
y
6
− 6y
5
+ 79y
4
− 302y
3
+ 457y
2
− 76y + 16
c
11
y
6
− y
5
+ 16y
4
+ 7y
3
+ 97y
2
+ 72y + 16
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.054940 + 0.264726I
a = 0.240575 + 0.570430I
b = −0.170516 + 0.063771I
−2.25297 − 0.53909I −9.12391 − 1.33093I
u = 1.054940 − 0.264726I
a = 0.240575 − 0.570430I
b = −0.170516 − 0.063771I
−2.25297 + 0.53909I −9.12391 + 1.33093I
u = −1.48721 + 0.12793I
a = −0.106812 + 0.438266I
b = −0.17052 + 1.66828I
−2.25297 + 4.59885I −9.12391 − 5.59727I
u = −1.48721 − 0.12793I
a = −0.106812 − 0.438266I
b = −0.17052 − 1.66828I
−2.25297 − 4.59885I −9.12391 + 5.59727I
u = 2.43227 + 0.39265I
a = −0.883763 + 0.142671I
b = −3.15897 − 0.86603I
−16.8782 − 2.0299I −10.75217 + 3.46410I
u = 2.43227 − 0.39265I
a = −0.883763 − 0.142671I
b = −3.15897 + 0.86603I
−16.8782 + 2.0299I −10.75217 − 3.46410I
17
V. I
u
5
= h−u
2
b + b
2
+ u
2
− u − 1, a − 1, u
3
+ 2u
2
+ 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
2u
2
+ u + 1
a
9
=
1
b
a
7
=
−bu − 2u
2
− 2u − 1
−bu − u
2
a
10
=
−b + 1
b
a
8
=
1
−u
2
+ b
a
11
=
−b − u
−bu − u
2
− 1
a
4
=
−u
2
+ b − u
u + 1
a
4
=
−u
2
+ b − u
u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4bu + 5u
2
+ 3u − 7
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
8
(u
3
− 2u
2
− 1)
2
c
2
(u
3
+ 4u
2
− 4u + 1)
2
c
3
u
6
+ 4u
5
− u
4
− 18u
3
− 9u
2
+ 20u + 16
c
4
, c
11
(u
2
+ u + 1)
3
c
6
u
6
+ 7u
5
+ 18u
4
+ 27u
3
+ 38u
2
+ 11u + 1
c
7
u
6
− 5u
5
+ 12u
4
− 15u
3
+ 11u
2
− 4u + 4
c
9
u
6
− 4u
5
− u
4
+ 18u
3
− 9u
2
− 20u + 16
c
10
(u
3
+ u
2
− u − 2)
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
8
(y
3
− 4y
2
− 4y −1)
2
c
2
(y
3
− 24y
2
+ 8y −1)
2
c
3
, c
9
y
6
− 18y
5
+ 127y
4
− 434y
3
+ 769y
2
− 688y + 256
c
4
, c
11
(y
2
+ y + 1)
3
c
6
y
6
− 13y
5
+ 22y
4
+ 487y
3
+ 886y
2
− 45y + 1
c
7
y
6
− y
5
+ 16y
4
+ 7y
3
+ 97y
2
+ 72y + 16
c
10
(y
3
− 3y
2
+ 5y −4)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.102785 + 0.665457I
a = 1.00000
b = 1.054940 + 0.264726I
−2.25297 − 4.59885I −9.12391 + 5.59727I
u = 0.102785 + 0.665457I
a = 1.00000
b = −1.48721 − 0.12793I
−2.25297 − 0.53909I −9.12391 − 1.33093I
u = 0.102785 − 0.665457I
a = 1.00000
b = 1.054940 − 0.264726I
−2.25297 + 4.59885I −9.12391 − 5.59727I
u = 0.102785 − 0.665457I
a = 1.00000
b = −1.48721 + 0.12793I
−2.25297 + 0.53909I −9.12391 + 1.33093I
u = −2.20557
a = 1.00000
b = 2.43227 + 0.39265I
−16.8782 + 2.0299I −10.75217 − 3.46410I
u = −2.20557
a = 1.00000
b = 2.43227 − 0.39265I
−16.8782 − 2.0299I −10.75217 + 3.46410I
21
VI. I
u
6
= hb + u + 1, a − u − 1, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u − 1
a
3
=
u + 2
−u − 1
a
6
=
−u
u − 1
a
9
=
u + 1
−u − 1
a
7
=
0
−1
a
10
=
2u + 2
−u − 1
a
8
=
u + 1
−1
a
11
=
1
1
a
4
=
−u
0
a
4
=
−u
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u − 7
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
2
+ u + 1
c
2
, c
4
, c
5
c
6
, c
7
, c
9
u
2
− u + 1
c
3
, c
8
, c
11
(u + 1)
2
c
10
u
2
+ 3u + 3
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
y
2
+ y + 1
c
3
, c
8
, c
11
(y −1)
2
c
10
y
2
− 3y + 9
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = −0.500000 − 0.866025I
−1.64493 − 2.02988I −9.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = −0.500000 + 0.866025I
−1.64493 + 2.02988I −9.00000 − 3.46410I
25
VII. I
u
7
= hb − a + 1, a
2
− a + 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
−1
0
a
9
=
a
a − 1
a
7
=
0
a
a
10
=
1
a − 1
a
8
=
a
−1
a
11
=
1
a − 1
a
4
=
−1
−a + 2
a
4
=
−1
−a + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a − 7
26
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
c
2
, c
3
, c
4
c
5
(u + 1)
2
c
6
, c
7
, c
8
c
9
, c
11
u
2
− u + 1
c
10
u
2
27
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
(y −1)
2
c
6
, c
7
, c
8
c
9
, c
11
y
2
+ y + 1
c
10
y
2
28
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.500000 + 0.866025I
b = −0.500000 + 0.866025I
−1.64493 + 2.02988I −9.00000 − 3.46410I
u = 1.00000
a = 0.500000 − 0.866025I
b = −0.500000 − 0.866025I
−1.64493 − 2.02988I −9.00000 + 3.46410I
29
VIII. I
u
8
= hb
2
+ b + 1, a + 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
−1
0
a
9
=
−1
b
a
7
=
b − 1
b + 1
a
10
=
−b − 1
b
a
8
=
−1
b + 1
a
11
=
−b − 1
b
a
4
=
−b
0
a
4
=
−b
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4b − 11
30
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
c
2
, c
5
, c
7
c
8
(u + 1)
2
c
3
, c
4
, c
6
c
9
, c
11
u
2
− u + 1
c
10
u
2
31
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
8
(y −1)
2
c
3
, c
4
, c
6
c
9
, c
11
y
2
+ y + 1
c
10
y
2
32
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −0.500000 + 0.866025I
−1.64493 + 2.02988I −9.00000 − 3.46410I
u = 1.00000
a = −1.00000
b = −0.500000 − 0.866025I
−1.64493 − 2.02988I −9.00000 + 3.46410I
33
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
(u
2
+ u + 1)(u
3
− 2u
2
− 1)
4
(u
5
+ 3u
4
+ u
3
− 2u
2
− u + 1)
· (u
5
+ 5u
4
+ 7u
3
+ 2u
2
+ u + 1)(u
6
+ 4u
5
+ ··· + 20u + 16)
c
2
(u + 1)
4
(u
2
− u + 1)(u
3
+ 4u
2
− 4u + 1)
4
· (u
5
+ 7u
4
+ 11u
3
+ 12u
2
+ 5u + 1)(u
5
+ 11u
4
+ 31u
3
− 3u + 1)
· (u
6
+ 18u
5
+ 127u
4
+ 434u
3
+ 769u
2
+ 688u + 256)
c
3
, c
5
, c
8
(u + 1)
4
(u
2
− u + 1)(u
3
− 2u
2
− 1)
4
(u
5
− 3u
4
+ u
3
+ 2u
2
− u − 1)
· (u
5
+ 5u
4
+ 7u
3
+ 2u
2
+ u + 1)(u
6
+ 4u
5
+ ··· + 20u + 16)
c
4
, c
7
, c
11
(u + 1)
2
(u
2
− u + 1)
2
(u
2
+ u + 1)
6
(u
5
− 3u
4
+ 5u
3
− 3u
2
+ 1)
· (u
5
+ u
4
+ u
3
− u
2
− 1)(u
6
− 5u
5
+ 12u
4
− 15u
3
+ 11u
2
− 4u + 4)
c
6
, c
9
(u
2
− u + 1)
3
(u
5
− 6u
4
+ 9u
3
+ 4u + 1)(u
5
− 4u
4
+ 3u
3
− 1)
· (u
6
− 4u
5
− u
4
+ 18u
3
− 9u
2
− 20u + 16)
· (u
6
+ 7u
5
+ 18u
4
+ 27u
3
+ 38u
2
+ 11u + 1)
2
c
10
u
4
(u
2
+ 3u + 3)(u
3
+ u
2
− u − 2)
4
(u
5
− 4u
4
+ 6u
3
− u
2
− 2u + 1)
· (u
5
+ 2u
4
− 5u
2
− 6u − 3)(u
6
− 8u
5
+ ··· − 22u + 4)
34
X. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
((y −1)
4
)(y
2
+ y + 1)(y
3
− 4y
2
− 4y −1)
4
(y
5
− 11y
4
+ ··· − 3y −1)
· (y
5
− 7y
4
+ 11y
3
− 12y
2
+ 5y −1)
· (y
6
− 18y
5
+ 127y
4
− 434y
3
+ 769y
2
− 688y + 256)
c
2
(y −1)
4
(y
2
+ y + 1)(y
3
− 24y
2
+ 8y −1)
4
· (y
5
− 59y
4
+ ··· + 9y −1)(y
5
− 27y
4
− 37y
3
− 48y
2
+ y −1)
· (y
6
− 70y
5
+ 2043y
4
− 17286y
3
+ 59201y
2
− 79616y + 65536)
c
4
, c
7
, c
11
(y −1)
2
(y
2
+ y + 1)
8
(y
5
+ y
4
+ 3y
3
+ y
2
− 2y −1)
· (y
5
+ y
4
+ 7y
3
− 3y
2
+ 6y −1)(y
6
− y
5
+ ··· + 72y + 16)
c
6
, c
9
(y
2
+ y + 1)
3
(y
5
− 18y
4
+ 89y
3
+ 84y
2
+ 16y −1)
· (y
5
− 10y
4
+ 9y
3
− 8y
2
− 1)
· (y
6
− 18y
5
+ 127y
4
− 434y
3
+ 769y
2
− 688y + 256)
· (y
6
− 13y
5
+ 22y
4
+ 487y
3
+ 886y
2
− 45y + 1)
2
c
10
y
4
(y
2
− 3y + 9)(y
3
− 3y
2
+ 5y −4)
4
(y
5
− 4y
4
+ ··· + 6y −9)
· (y
5
− 4y
4
+ 24y
3
− 17y
2
+ 6y −1)
· (y
6
− 6y
5
+ 79y
4
− 302y
3
+ 457y
2
− 76y + 16)
35
