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