9
10
(K9a
39
)
A knot diagram
1
Linearized knot diagam
7 6 9 8 1 2 3 4 5
Solving Sequence
1,7
2 6 3 8 5 4 9
c
1
c
6
c
2
c
7
c
5
c
4
c
9
c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
+ 3u
4
+ u
3
+ 2u
2
+ 2u โˆ’ 1i
I
u
2
= hu
10
โˆ’ u
9
+ 4u
8
โˆ’ 4u
7
+ 6u
6
โˆ’ 6u
5
+ 3u
4
โˆ’ 3u
3
+ 1i
* 2 irreducible components of dim
C
= 0, with total 16 representations.
1
The image of knot diagram is generated by the software โ€œDraw programmeโ€ developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
๏ฌed some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coe๏ฌƒcients of polynomials are rational numbers. But the coe๏ฌƒcients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= hu
6
+ 3u
4
+ u
3
+ 2u
2
+ 2u โˆ’ 1i
(i) Arc colorings
a
1
=
๎€’
1
0
๎€“
a
7
=
๎€’
0
u
๎€“
a
2
=
๎€’
1
u
2
๎€“
a
6
=
๎€’
u
u
3
+ u
๎€“
a
3
=
๎€’
u
2
+ 1
u
4
+ 2u
2
๎€“
a
8
=
๎€’
โˆ’u
5
โˆ’ 2u
3
โˆ’ u
u
4
+ 2u
2
๎€“
a
5
=
๎€’
u
3
+ 2u
u
3
+ u
๎€“
a
4
=
๎€’
โˆ’u
4
โˆ’ u
2
+ 1
โˆ’u
5
โˆ’ u
3
+ u
๎€“
a
9
=
๎€’
u
3
+ 2u
u
4
+ u
3
+ u
2
+ 2u โˆ’ 1
๎€“
a
9
=
๎€’
u
3
+ 2u
u
4
+ u
3
+ u
2
+ 2u โˆ’ 1
๎€“
(ii) Obstruction class = โˆ’1
(iii) Cusp Shapes = โˆ’4u
5
โˆ’ 4u
4
โˆ’ 8u
3
โˆ’ 12u
2
โˆ’ 4u โˆ’ 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
u
6
+ 3u
4
โˆ’ u
3
+ 2u
2
โˆ’ 2u โˆ’ 1
c
5
, c
7
, c
9
u
6
โˆ’ 3u
5
+ 2u
4
โˆ’ u
3
+ 5u
2
โˆ’ 3u โˆ’ 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
y
6
+ 6y
5
+ 13y
4
+ 9y
3
โˆ’ 6y
2
โˆ’ 8y + 1
c
5
, c
7
, c
9
y
6
โˆ’ 5y
5
+ 8y
4
โˆ’ 3y
3
+ 11y
2
โˆ’ 29y + 4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
โˆš
โˆ’1(vol +
โˆš
โˆ’1CS) Cusp shape
u = โˆ’0.841864
โˆ’6.52764 โˆ’14.6820
u = โˆ’0.126468 + 1.352400I
8.36373 + 3.39374I โˆ’1.63982 โˆ’ 3.51762I
u = โˆ’0.126468 โˆ’ 1.352400I
8.36373 โˆ’ 3.39374I โˆ’1.63982 + 3.51762I
u = 0.376468 + 1.319680I
1.76812 โˆ’ 8.77346I โˆ’6.43784 + 5.90094I
u = 0.376468 โˆ’ 1.319680I
1.76812 + 8.77346I โˆ’6.43784 โˆ’ 5.90094I
u = 0.341865
โˆ’0.576591 โˆ’17.1630
5
II. I
u
2
= hu
10
โˆ’ u
9
+ 4u
8
โˆ’ 4u
7
+ 6u
6
โˆ’ 6u
5
+ 3u
4
โˆ’ 3u
3
+ 1i
(i) Arc colorings
a
1
=
๎€’
1
0
๎€“
a
7
=
๎€’
0
u
๎€“
a
2
=
๎€’
1
u
2
๎€“
a
6
=
๎€’
u
u
3
+ u
๎€“
a
3
=
๎€’
u
2
+ 1
u
4
+ 2u
2
๎€“
a
8
=
๎€’
โˆ’u
5
โˆ’ 2u
3
โˆ’ u
โˆ’u
7
โˆ’ 3u
5
โˆ’ 2u
3
+ u
๎€“
a
5
=
๎€’
u
3
+ 2u
u
3
+ u
๎€“
a
4
=
๎€’
โˆ’2u
9
โˆ’ 8u
7
โˆ’ 11u
5
+ u
4
โˆ’ 2u
3
+ 3u
2
+ 5u + 3
โˆ’2u
9
โˆ’ 8u
7
+ u
6
โˆ’ 11u
5
+ 4u
4
โˆ’ 3u
3
+ 5u
2
+ 3u + 2
๎€“
a
9
=
๎€’
โˆ’u
6
โˆ’ 3u
4
โˆ’ 2u
2
+ 1
โˆ’u
6
โˆ’ 2u
4
โˆ’ u
2
๎€“
a
9
=
๎€’
โˆ’u
6
โˆ’ 3u
4
โˆ’ 2u
2
+ 1
โˆ’u
6
โˆ’ 2u
4
โˆ’ u
2
๎€“
(ii) Obstruction class = โˆ’1
(iii) Cusp Shapes = 4u
9
+ 12u
7
+ 12u
5
โˆ’ 4u
3
โˆ’ 8u โˆ’ 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
u
10
+ u
9
+ 4u
8
+ 4u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 3u
3
+ 1
c
5
, c
7
, c
9
(u
5
+ u
4
โˆ’ 2u
3
โˆ’ u
2
+ u โˆ’ 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
y
10
+ 7y
9
+ 20y
8
+ 26y
7
+ 6y
6
โˆ’ 22y
5
โˆ’ 19y
4
+ 3y
3
+ 6y
2
+ 1
c
5
, c
7
, c
9
(y
5
โˆ’ 5y
4
+ 8y
3
โˆ’ 3y
2
โˆ’ y โˆ’1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
โˆš
โˆ’1(vol +
โˆš
โˆ’1CS) Cusp shape
u = 0.839548 + 0.070481I
โˆ’2.58269 โˆ’ 4.40083I โˆ’10.74431 + 3.49859I
u = 0.839548 โˆ’ 0.070481I
โˆ’2.58269 + 4.40083I โˆ’10.74431 โˆ’ 3.49859I
u = 0.090539 + 1.215350I
2.96077 โˆ’ 1.53058I โˆ’6.51511 + 4.43065I
u = 0.090539 โˆ’ 1.215350I
2.96077 + 1.53058I โˆ’6.51511 โˆ’ 4.43065I
u = 0.383413 + 1.200420I
0.888787 โˆ’7.48114 + 0.I
u = 0.383413 โˆ’ 1.200420I
0.888787 โˆ’7.48114 + 0.I
u = โˆ’0.383851 + 1.270630I
โˆ’2.58269 + 4.40083I โˆ’10.74431 โˆ’ 3.49859I
u = โˆ’0.383851 โˆ’ 1.270630I
โˆ’2.58269 โˆ’ 4.40083I โˆ’10.74431 + 3.49859I
u = โˆ’0.429649 + 0.392970I
2.96077 + 1.53058I โˆ’6.51511 โˆ’ 4.43065I
u = โˆ’0.429649 โˆ’ 0.392970I
2.96077 โˆ’ 1.53058I โˆ’6.51511 + 4.43065I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
(u
6
+ 3u
4
โˆ’ u
3
+ 2u
2
โˆ’ 2u โˆ’ 1)
ยท (u
10
+ u
9
+ 4u
8
+ 4u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 3u
3
+ 1)
c
5
, c
7
, c
9
(u
5
+ u
4
โˆ’ 2u
3
โˆ’ u
2
+ u โˆ’ 1)
2
(u
6
โˆ’ 3u
5
+ 2u
4
โˆ’ u
3
+ 5u
2
โˆ’ 3u โˆ’ 2)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
(y
6
+ 6y
5
+ 13y
4
+ 9y
3
โˆ’ 6y
2
โˆ’ 8y + 1)
ยท (y
10
+ 7y
9
+ 20y
8
+ 26y
7
+ 6y
6
โˆ’ 22y
5
โˆ’ 19y
4
+ 3y
3
+ 6y
2
+ 1)
c
5
, c
7
, c
9
(y
5
โˆ’ 5y
4
+ 8y
3
โˆ’ 3y
2
โˆ’ y โˆ’1)
2
ยท (y
6
โˆ’ 5y
5
+ 8y
4
โˆ’ 3y
3
+ 11y
2
โˆ’ 29y + 4)
11