10
75
(K10a
27
)
A knot diagram
1
Linearized knot diagam
8 9 10 3 1 5 2 7 4 6
Solving Sequence
4,9
10 3 5
2,7
6 8 1
c
9
c
3
c
4
c
2
c
6
c
8
c
1
c
5
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
2
+ b, −u
5
− u
4
− u
3
− u
2
+ a − u − 1, u
6
+ u
5
+ 2u
4
+ u
3
+ 2u
2
+ 2u + 1i
I
u
2
= h−u
2
+ b, u
9
− 2u
8
+ 3u
7
− 4u
6
+ 5u
5
− 6u
4
+ 4u
3
− 3u
2
+ a + 3u − 2,
u
10
− u
9
+ 3u
8
− 3u
7
+ 5u
6
− 5u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1i
I
u
3
= h−u
9
− 2u
8
− 4u
7
− 4u
6
− 4u
5
− 2u
4
− 2u
3
− u
2
+ b − 2u − 1,
− u
9
− 4u
8
− 5u
7
− 8u
6
− 5u
5
− 4u
4
− 4u
3
− 2u
2
+ 2a − 5u − 3,
u
10
+ 2u
9
+ 5u
8
+ 6u
7
+ 7u
6
+ 6u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 3u + 2i
I
u
4
= hu
9
+ 2u
7
+ 2u
5
+ b + 1, u
9
+ u
7
− 2u
3
+ a + 1, u
10
− u
9
+ 3u
8
− 3u
7
+ 5u
6
− 5u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1i
I
u
5
= hb + 1, a − u + 1, u
2
+ 1i
I
u
6
= h2u
2
a − au + 2u
2
+ 3b + a − u + 4, u
2
a + a
2
+ a − 2u, u
3
+ u + 1i
* 6 irreducible components of dim
C
= 0, with total 44 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
5
−u
4
−u
3
−u
2
+a−u−1, u
6
+u
5
+2u
4
+u
3
+2u
2
+2u+1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
−u
u
3
+ u
a
5
=
−u
3
u
5
+ u
3
+ u
a
2
=
u
3
u
3
+ u
a
7
=
u
5
+ u
4
+ u
3
+ u
2
+ u + 1
u
2
a
6
=
u
4
+ u
3
+ u
2
+ u + 1
u
4
+ u
2
+ u + 1
a
8
=
u
5
+ u
3
+ u
2
+ u + 1
−u
4
a
1
=
u
5
+ u
4
+ u
3
+ u
2
+ u + 1
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6u
5
+ 6u
4
+ 6u
3
+ 6u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
9
, c
10
u
6
− u
5
+ 2u
4
− u
3
+ 2u
2
− 2u + 1
c
2
u
6
+ u
5
− u
4
+ 3u
3
+ 4u
2
− 4u + 4
c
4
, c
6
, c
8
u
6
+ 3u
5
+ 6u
4
+ 5u
3
+ 4u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
9
, c
10
y
6
+ 3y
5
+ 6y
4
+ 5y
3
+ 4y
2
+ 1
c
2
y
6
− 3y
5
+ 3y
4
− y
3
+ 32y
2
+ 16y + 16
c
4
, c
6
, c
8
y
6
+ 3y
5
+ 14y
4
+ 25y
3
+ 28y
2
+ 8y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.601492 + 0.919611I
a = −0.791230 − 0.378440I
b = −0.483891 + 1.106280I
0.69113 + 7.13350I 2.15597 − 8.90831I
u = 0.601492 − 0.919611I
a = −0.791230 + 0.378440I
b = −0.483891 − 1.106280I
0.69113 − 7.13350I 2.15597 + 8.90831I
u = −0.560586 + 0.395699I
a = 0.664051 + 0.133626I
b = 0.157679 − 0.443647I
1.168610 − 0.699600I 7.03823 + 3.46364I
u = −0.560586 − 0.395699I
a = 0.664051 − 0.133626I
b = 0.157679 + 0.443647I
1.168610 + 0.699600I 7.03823 − 3.46364I
u = −0.540906 + 1.210940I
a = −2.37282 + 0.19030I
b = −1.17379 − 1.31001I
−6.7946 − 13.4307I −3.19420 + 9.00183I
u = −0.540906 − 1.210940I
a = −2.37282 − 0.19030I
b = −1.17379 + 1.31001I
−6.7946 + 13.4307I −3.19420 − 9.00183I
5
II. I
u
2
= h−u
2
+ b, u
9
− 2u
8
+ · · · + a − 2, u
10
− u
9
+ · · · − 2u + 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
−u
u
3
+ u
a
5
=
−u
3
u
5
+ u
3
+ u
a
2
=
u
3
u
3
+ u
a
7
=
−u
9
+ 2u
8
− 3u
7
+ 4u
6
− 5u
5
+ 6u
4
− 4u
3
+ 3u
2
− 3u + 2
u
2
a
6
=
−u
9
+ 2u
8
− 3u
7
+ 5u
6
− 5u
5
+ 7u
4
− 4u
3
+ 4u
2
− 4u + 2
−u
9
− 3u
7
+ u
6
− 5u
5
+ 2u
4
− 3u
3
+ 3u
2
− 2u + 1
a
8
=
−u
9
+ 2u
8
− 3u
7
+ 4u
6
− 5u
5
+ 5u
4
− 4u
3
+ 3u
2
− 3u + 2
−u
4
a
1
=
−u
9
− u
7
+ 2u
3
− 1
u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
9
+ 4u
8
− 8u
7
+ 8u
6
− 8u
5
+ 12u
4
+ 4u
2
− 4u + 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
9
u
10
+ u
9
+ 3u
8
+ 3u
7
+ 5u
6
+ 5u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 2u + 1
c
2
u
10
+ 2u
9
− u
8
− 5u
7
− 3u
6
+ 4u
5
+ 12u
4
+ 13u
3
+ 5u
2
+ u + 2
c
4
, c
8
u
10
+ 5u
9
+ 13u
8
+ 19u
7
+ 17u
6
+ 7u
5
− 2u
3
+ u
2
+ 2u + 1
c
5
, c
10
u
10
− 2u
9
+ 5u
8
− 6u
7
+ 7u
6
− 6u
5
+ 4u
4
− 4u
3
+ 3u
2
− 3u + 2
c
6
u
10
+ 6u
9
+ 15u
8
+ 18u
7
+ 7u
6
− 6u
5
− 6u
4
+ u
2
+ 3u + 4
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
c
9
y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
− 2y
3
+ y
2
+ 2y + 1
c
2
y
10
− 6y
9
+ ··· + 19y + 4
c
4
, c
8
y
10
+ y
9
+ 13y
8
+ 11y
7
+ 45y
6
+ 35y
5
+ 12y
4
+ 2y
3
+ 9y
2
− 2y + 1
c
5
, c
10
y
10
+ 6y
9
+ 15y
8
+ 18y
7
+ 7y
6
− 6y
5
− 6y
4
+ y
2
+ 3y + 4
c
6
y
10
− 6y
9
+ ··· − y + 16
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.584958 + 0.771492I
a = −0.164635 + 0.412534I
b = −0.253024 − 0.902582I
1.64732 − 2.31006I 4.86369 + 3.52133I
u = −0.584958 − 0.771492I
a = −0.164635 − 0.412534I
b = −0.253024 + 0.902582I
1.64732 + 2.31006I 4.86369 − 3.52133I
u = 0.248527 + 0.782547I
a = 0.99372 − 1.81329I
b = −0.550614 + 0.388968I
−3.73792 + 1.23169I −0.90177 − 5.44908I
u = 0.248527 − 0.782547I
a = 0.99372 + 1.81329I
b = −0.550614 − 0.388968I
−3.73792 − 1.23169I −0.90177 + 5.44908I
u = 0.761643 + 0.208049I
a = 0.785123 + 0.059495I
b = 0.536815 + 0.316918I
−0.87626 − 3.47839I 3.19503 + 2.79515I
u = 0.761643 − 0.208049I
a = 0.785123 − 0.059495I
b = 0.536815 − 0.316918I
−0.87626 + 3.47839I 3.19503 − 2.79515I
u = −0.449566 + 1.164790I
a = −2.43053 + 0.82165I
b = −1.15461 − 1.04730I
−8.16652 − 4.14585I −4.98134 + 3.97600I
u = −0.449566 − 1.164790I
a = −2.43053 − 0.82165I
b = −1.15461 + 1.04730I
−8.16652 + 4.14585I −4.98134 − 3.97600I
u = 0.524355 + 1.163410I
a = −2.18368 − 0.41240I
b = −1.07856 + 1.22007I
−3.67102 + 8.28632I −0.17560 − 6.14881I
u = 0.524355 − 1.163410I
a = −2.18368 + 0.41240I
b = −1.07856 − 1.22007I
−3.67102 − 8.28632I −0.17560 + 6.14881I
9
III.
I
u
3
= h−u
9
−2u
8
+· · ·+b −1, −u
9
−4u
8
+· · ·+2a −3, u
10
+2u
9
+· · ·+3u +2i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
−u
u
3
+ u
a
5
=
−u
3
u
5
+ u
3
+ u
a
2
=
u
3
u
3
+ u
a
7
=
1
2
u
9
+ 2u
8
+ ··· +
5
2
u +
3
2
u
9
+ 2u
8
+ 4u
7
+ 4u
6
+ 4u
5
+ 2u
4
+ 2u
3
+ u
2
+ 2u + 1
a
6
=
1
2
u
9
+
1
2
u
7
+ ··· −
1
2
u −
1
2
−u
9
− 2u
8
− 5u
7
− 5u
6
− 7u
5
− 4u
4
− 4u
3
− 3u
2
− 3u − 3
a
8
=
−
1
2
u
9
−
1
2
u
7
+ ··· +
1
2
u +
3
2
u
9
+ 2u
8
+ 5u
7
+ 4u
6
+ 6u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ 2u + 3
a
1
=
1
2
u
9
+ 2u
8
+ ··· +
5
2
u +
3
2
u
9
+ 2u
8
+ 3u
7
+ 4u
6
+ 2u
5
+ 3u
4
+ u
3
+ 2u
2
+ 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
7
− 8u
5
− 4u
3
+ 4u − 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
c
10
u
10
+ u
9
+ 3u
8
+ 3u
7
+ 5u
6
+ 5u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 2u + 1
c
2
u
10
+ 2u
9
− u
8
− 5u
7
− 3u
6
+ 4u
5
+ 12u
4
+ 13u
3
+ 5u
2
+ u + 2
c
3
, c
9
u
10
− 2u
9
+ 5u
8
− 6u
7
+ 7u
6
− 6u
5
+ 4u
4
− 4u
3
+ 3u
2
− 3u + 2
c
4
u
10
+ 6u
9
+ 15u
8
+ 18u
7
+ 7u
6
− 6u
5
− 6u
4
+ u
2
+ 3u + 4
c
6
, c
8
u
10
+ 5u
9
+ 13u
8
+ 19u
7
+ 17u
6
+ 7u
5
− 2u
3
+ u
2
+ 2u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
10
y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
− 2y
3
+ y
2
+ 2y + 1
c
2
y
10
− 6y
9
+ ··· + 19y + 4
c
3
, c
9
y
10
+ 6y
9
+ 15y
8
+ 18y
7
+ 7y
6
− 6y
5
− 6y
4
+ y
2
+ 3y + 4
c
4
y
10
− 6y
9
+ ··· − y + 16
c
6
, c
8
y
10
+ y
9
+ 13y
8
+ 11y
7
+ 45y
6
+ 35y
5
+ 12y
4
+ 2y
3
+ 9y
2
− 2y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.871979 + 0.168588I
a = −0.409574 − 0.178135I
b = −1.07856 + 1.22007I
−3.67102 + 8.28632I −0.17560 − 6.14881I
u = −0.871979 − 0.168588I
a = −0.409574 + 0.178135I
b = −1.07856 − 1.22007I
−3.67102 − 8.28632I −0.17560 + 6.14881I
u = 0.642886 + 0.580182I
a = 0.842379 + 0.211365I
b = −0.253024 − 0.902582I
1.64732 − 2.31006I 4.86369 + 3.52133I
u = 0.642886 − 0.580182I
a = 0.842379 − 0.211365I
b = −0.253024 + 0.902582I
1.64732 + 2.31006I 4.86369 − 3.52133I
u = 0.060791 + 1.179490I
a = −0.201487 − 0.633222I
b = −0.550614 − 0.388968I
−3.73792 − 1.23169I −0.90177 + 5.44908I
u = 0.060791 − 1.179490I
a = −0.201487 + 0.633222I
b = −0.550614 + 0.388968I
−3.73792 + 1.23169I −0.90177 − 5.44908I
u = −0.480814 + 1.084510I
a = 1.43693 − 0.34109I
b = 0.536815 + 0.316918I
−0.87626 − 3.47839I 3.19503 + 2.79515I
u = −0.480814 − 1.084510I
a = 1.43693 + 0.34109I
b = 0.536815 − 0.316918I
−0.87626 + 3.47839I 3.19503 − 2.79515I
u = −0.350885 + 1.264620I
a = −0.91824 + 1.61467I
b = −1.15461 + 1.04730I
−8.16652 + 4.14585I −4.98134 − 3.97600I
u = −0.350885 − 1.264620I
a = −0.91824 − 1.61467I
b = −1.15461 − 1.04730I
−8.16652 − 4.14585I −4.98134 + 3.97600I
13
IV.
I
u
4
= hu
9
+ 2u
7
+ 2u
5
+ b + 1, u
9
+ u
7
− 2u
3
+ a + 1, u
10
− u
9
+ · · · − 2u + 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
−u
u
3
+ u
a
5
=
−u
3
u
5
+ u
3
+ u
a
2
=
u
3
u
3
+ u
a
7
=
−u
9
− u
7
+ 2u
3
− 1
−u
9
− 2u
7
− 2u
5
− 1
a
6
=
−u
9
− u
7
− u
5
+ 2u
3
− 1
−u
9
− u
7
− u
5
+ u
3
− 1
a
8
=
−u
9
− u
7
− u
5
+ u
4
+ u
3
+ u
2
− u
−2u
9
− 4u
7
− 5u
5
+ u
4
− u
3
+ 2u
2
− u
a
1
=
−u
9
+ 2u
8
− 3u
7
+ 4u
6
− 5u
5
+ 6u
4
− 4u
3
+ 3u
2
− 3u + 2
−u
9
+ 2u
8
− 3u
7
+ 4u
6
− 5u
5
+ 5u
4
− 4u
3
+ 2u
2
− 3u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
9
+ 4u
8
− 8u
7
+ 8u
6
− 8u
5
+ 12u
4
+ 4u
2
− 4u + 2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
10
− 2u
9
+ 5u
8
− 6u
7
+ 7u
6
− 6u
5
+ 4u
4
− 4u
3
+ 3u
2
− 3u + 2
c
2
u
10
+ 2u
9
− u
8
− 5u
7
− 3u
6
+ 4u
5
+ 12u
4
+ 13u
3
+ 5u
2
+ u + 2
c
3
, c
5
, c
9
c
10
u
10
+ u
9
+ 3u
8
+ 3u
7
+ 5u
6
+ 5u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 2u + 1
c
4
, c
6
u
10
+ 5u
9
+ 13u
8
+ 19u
7
+ 17u
6
+ 7u
5
− 2u
3
+ u
2
+ 2u + 1
c
8
u
10
+ 6u
9
+ 15u
8
+ 18u
7
+ 7u
6
− 6u
5
− 6u
4
+ u
2
+ 3u + 4
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
10
+ 6y
9
+ 15y
8
+ 18y
7
+ 7y
6
− 6y
5
− 6y
4
+ y
2
+ 3y + 4
c
2
y
10
− 6y
9
+ ··· + 19y + 4
c
3
, c
5
, c
9
c
10
y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
− 2y
3
+ y
2
+ 2y + 1
c
4
, c
6
y
10
+ y
9
+ 13y
8
+ 11y
7
+ 45y
6
+ 35y
5
+ 12y
4
+ 2y
3
+ 9y
2
− 2y + 1
c
8
y
10
− 6y
9
+ ··· − y + 16
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.584958 + 0.771492I
a = 1.153020 − 0.145190I
b = 0.076692 + 0.745982I
1.64732 − 2.31006I 4.86369 + 3.52133I
u = −0.584958 − 0.771492I
a = 1.153020 + 0.145190I
b = 0.076692 − 0.745982I
1.64732 + 2.31006I 4.86369 − 3.52133I
u = 0.248527 + 0.782547I
a = −1.73424 − 0.64880I
b = −1.387500 − 0.143405I
−3.73792 + 1.23169I −0.90177 − 5.44908I
u = 0.248527 − 0.782547I
a = −1.73424 + 0.64880I
b = −1.387500 + 0.143405I
−3.73792 − 1.23169I −0.90177 + 5.44908I
u = 0.761643 + 0.208049I
a = −0.170482 + 0.442613I
b = −0.944976 − 1.042890I
−0.87626 − 3.47839I 3.19503 + 2.79515I
u = 0.761643 − 0.208049I
a = −0.170482 − 0.442613I
b = −0.944976 + 1.042890I
−0.87626 + 3.47839I 3.19503 − 2.79515I
u = −0.449566 + 1.164790I
a = −1.31989 + 1.51437I
b = −1.47614 + 0.88747I
−8.16652 − 4.14585I −4.98134 + 3.97600I
u = −0.449566 − 1.164790I
a = −1.31989 − 1.51437I
b = −1.47614 − 0.88747I
−8.16652 + 4.14585I −4.98134 − 3.97600I
u = 0.524355 + 1.163410I
a = 1.57160 + 0.38323I
b = 0.731926 − 0.294010I
−3.67102 + 8.28632I −0.17560 − 6.14881I
u = 0.524355 − 1.163410I
a = 1.57160 − 0.38323I
b = 0.731926 + 0.294010I
−3.67102 − 8.28632I −0.17560 + 6.14881I
17
V. I
u
5
= hb + 1, a − u + 1, u
2
+ 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
1
a
3
=
−u
0
a
5
=
u
u
a
2
=
−u
0
a
7
=
u − 1
−1
a
6
=
2u − 1
u − 1
a
8
=
u
−1
a
1
=
−u − 1
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
9
, c
10
u
2
+ 1
c
2
u
2
c
4
, c
6
, c
8
(u + 1)
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
9
, c
10
(y + 1)
2
c
2
y
2
c
4
, c
6
, c
8
(y −1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −1.00000 + 1.00000I
b = −1.00000
−4.93480 −8.00000
u = − 1.000000I
a = −1.00000 − 1.00000I
b = −1.00000
−4.93480 −8.00000
21
VI. I
u
6
= h2u
2
a − au + 2u
2
+ 3b + a − u + 4, u
2
a + a
2
+ a − 2u, u
3
+ u + 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
−u
−1
a
5
=
u + 1
−u
2
+ u
a
2
=
−u − 1
−1
a
7
=
a
−
2
3
u
2
a −
2
3
u
2
+ ··· −
1
3
a −
4
3
a
6
=
1
3
u
2
a −
2
3
u
2
+ ··· +
2
3
a −
1
3
−
2
3
u
2
a −
2
3
u
2
+ ··· −
4
3
a −
4
3
a
8
=
1
3
u
2
a −
2
3
u
2
+ ··· +
2
3
a −
1
3
−u
2
a + au − u
2
− a − 2
a
1
=
−u
2
− a − 1
1
3
u
2
a +
1
3
u
2
+ ··· −
1
3
a +
2
3
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
9
, c
10
(u
3
+ u − 1)
2
c
2
(u − 1)
6
c
4
, c
6
, c
8
(u
3
+ 2u
2
+ u − 1)
2
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
9
, c
10
(y
3
+ 2y
2
+ y − 1)
2
c
2
(y − 1)
6
c
4
, c
6
, c
8
(y
3
− 2y
2
+ 5y − 1)
2
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 0.341164 + 1.161540I
a = 1.30674 + 0.54078I
b = 0.465571
−4.93480 −2.00000
u = 0.341164 + 1.161540I
a = −1.07395 − 1.33333I
b = −1.23279 − 0.79255I
−4.93480 −2.00000
u = 0.341164 − 1.161540I
a = 1.30674 − 0.54078I
b = 0.465571
−4.93480 −2.00000
u = 0.341164 − 1.161540I
a = −1.07395 + 1.33333I
b = −1.23279 + 0.79255I
−4.93480 −2.00000
u = −0.682328
a = −0.732786 + 0.909770I
b = −1.23279 − 0.79255I
−4.93480 −2.00000
u = −0.682328
a = −0.732786 − 0.909770I
b = −1.23279 + 0.79255I
−4.93480 −2.00000
25
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
9
, c
10
(u
2
+ 1)(u
3
+ u − 1)
2
(u
6
− u
5
+ 2u
4
− u
3
+ 2u
2
− 2u + 1)
· (u
10
− 2u
9
+ 5u
8
− 6u
7
+ 7u
6
− 6u
5
+ 4u
4
− 4u
3
+ 3u
2
− 3u + 2)
· (u
10
+ u
9
+ 3u
8
+ 3u
7
+ 5u
6
+ 5u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 2u + 1)
2
c
2
u
2
(u − 1)
6
(u
6
+ u
5
− u
4
+ 3u
3
+ 4u
2
− 4u + 4)
· (u
10
+ 2u
9
− u
8
− 5u
7
− 3u
6
+ 4u
5
+ 12u
4
+ 13u
3
+ 5u
2
+ u + 2)
3
c
4
, c
6
, c
8
(u + 1)
2
(u
3
+ 2u
2
+ u − 1)
2
(u
6
+ 3u
5
+ 6u
4
+ 5u
3
+ 4u
2
+ 1)
· (u
10
+ 5u
9
+ 13u
8
+ 19u
7
+ 17u
6
+ 7u
5
− 2u
3
+ u
2
+ 2u + 1)
2
· (u
10
+ 6u
9
+ 15u
8
+ 18u
7
+ 7u
6
− 6u
5
− 6u
4
+ u
2
+ 3u + 4)
26
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
7
, c
9
, c
10
(y + 1)
2
(y
3
+ 2y
2
+ y − 1)
2
(y
6
+ 3y
5
+ 6y
4
+ 5y
3
+ 4y
2
+ 1)
· (y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
− 2y
3
+ y
2
+ 2y + 1)
2
· (y
10
+ 6y
9
+ 15y
8
+ 18y
7
+ 7y
6
− 6y
5
− 6y
4
+ y
2
+ 3y + 4)
c
2
y
2
(y − 1)
6
(y
6
− 3y
5
+ 3y
4
− y
3
+ 32y
2
+ 16y + 16)
· (y
10
− 6y
9
+ ··· + 19y + 4)
3
c
4
, c
6
, c
8
((y − 1)
2
)(y
3
− 2y
2
+ 5y − 1)
2
(y
6
+ 3y
5
+ ··· + 8y + 1)
· (y
10
− 6y
9
+ ··· − y + 16)
· (y
10
+ y
9
+ 13y
8
+ 11y
7
+ 45y
6
+ 35y
5
+ 12y
4
+ 2y
3
+ 9y
2
− 2y + 1)
2
27
