9
44
(K9n
1
)
A knot diagram
1
Linearized knot diagam
6 9 1 7 1 8 5 3 8
Solving Sequence
5,8
7
1,4
3 6 2 9
c
7
c
4
c
3
c
6
c
1
c
9
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
9
+ 2u
8
− 5u
6
+ 3u
5
+ 3u
4
− 4u
3
− 2u
2
+ 2b − u − 1,
− u
9
+ 3u
8
− 3u
7
− 3u
6
+ 7u
5
− 3u
4
− 4u
3
+ 2u
2
+ a − u + 1,
u
10
− 3u
9
+ 4u
8
+ u
7
− 6u
6
+ 6u
5
+ u
4
− 2u
3
+ 3u
2
− 2u + 1i
I
u
2
= hb
2
− b + 1, a, u + 1i
* 2 irreducible components of dim
C
= 0, with total 12 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
9
+2u
8
+· · ·+2b −1, −u
9
+3u
8
+· · ·+a +1, u
10
−3u
9
+· · ·−2u +1i
(i) Arc colorings
a
5
=
0
u
a
8
=
1
0
a
7
=
1
−u
2
a
1
=
u
9
− 3u
8
+ 3u
7
+ 3u
6
− 7u
5
+ 3u
4
+ 4u
3
− 2u
2
+ u − 1
1
2
u
9
− u
8
+ ··· +
1
2
u +
1
2
a
4
=
u
−u
3
+ u
a
3
=
u
4
− u
2
+ 2u + 1
−
1
2
u
9
+ u
8
+ ··· −
1
2
u +
1
2
a
6
=
−u
2
+ 1
−u
2
a
2
=
u
8
− u
7
+ u
6
+ 2u
5
− u
4
+ 3u
3
+ 2u
2
− 2u + 1
−
1
2
u
9
+ 2u
8
+ ··· −
3
2
u +
3
2
a
9
=
3
2
u
9
− 4u
8
+ ··· +
3
2
u −
1
2
1
2
u
9
− u
8
+ ··· +
1
2
u +
1
2
a
9
=
3
2
u
9
− 4u
8
+ ··· +
3
2
u −
1
2
1
2
u
9
− u
8
+ ··· +
1
2
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
9
− 7u
8
+ 5u
7
+ 12u
6
− 15u
5
+ 2u
4
+ 16u
3
− 2u
2
+ 5u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
10
+ u
9
− 7u
8
− 8u
7
+ 13u
6
+ 14u
5
− 2u
4
+ 2u
3
+ 13u
2
+ 12u + 4
c
2
, c
8
u
10
+ 2u
9
+ 3u
8
+ 2u
7
+ 4u
6
+ 3u
5
+ 3u
4
+ 3u
2
+ u + 1
c
3
u
10
− 2u
9
+ ··· + 21u + 17
c
4
, c
7
u
10
− 3u
9
+ 4u
8
+ u
7
− 6u
6
+ 6u
5
+ u
4
− 2u
3
+ 3u
2
− 2u + 1
c
6
u
10
+ u
9
+ 10u
8
+ 11u
7
+ 26u
6
+ 30u
5
+ u
4
− 14u
3
+ 3u
2
− 2u + 1
c
9
u
10
+ 2u
9
+ 9u
8
+ 14u
7
+ 28u
6
+ 31u
5
+ 35u
4
+ 20u
3
+ 15u
2
+ 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
10
− 15y
9
+ ··· − 40y + 16
c
2
, c
8
y
10
+ 2y
9
+ 9y
8
+ 14y
7
+ 28y
6
+ 31y
5
+ 35y
4
+ 20y
3
+ 15y
2
+ 5y + 1
c
3
y
10
+ 26y
9
+ ··· + 2925y + 289
c
4
, c
7
y
10
− y
9
+ 10y
8
− 11y
7
+ 26y
6
− 30y
5
+ y
4
+ 14y
3
+ 3y
2
+ 2y + 1
c
6
y
10
+ 19y
9
+ ··· + 2y + 1
c
9
y
10
+ 14y
9
+ ··· + 5y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.016000 + 0.211624I
a = 0.493565 − 0.413430I
b = 0.230117 − 0.236010I
−1.89922 + 0.79591I −4.77960 + 0.81155I
u = −1.016000 − 0.211624I
a = 0.493565 + 0.413430I
b = 0.230117 + 0.236010I
−1.89922 − 0.79591I −4.77960 − 0.81155I
u = −0.076965 + 0.657059I
a = 1.01532 − 1.06291I
b = −0.110515 − 0.762837I
1.14579 + 1.46073I 2.65931 − 3.28644I
u = −0.076965 − 0.657059I
a = 1.01532 + 1.06291I
b = −0.110515 + 0.762837I
1.14579 − 1.46073I 2.65931 + 3.28644I
u = 0.482659 + 0.410726I
a = −1.077630 + 0.665030I
b = 0.826051 + 0.890915I
−0.41291 − 2.81207I 0.88002 + 4.64391I
u = 0.482659 − 0.410726I
a = −1.077630 − 0.665030I
b = 0.826051 − 0.890915I
−0.41291 + 2.81207I 0.88002 − 4.64391I
u = 0.98889 + 1.13481I
a = 0.766166 − 1.067440I
b = −0.18099 − 1.73332I
9.86147 − 0.50253I 1.49701 − 0.08773I
u = 0.98889 − 1.13481I
a = 0.766166 + 1.067440I
b = −0.18099 + 1.73332I
9.86147 + 0.50253I 1.49701 + 0.08773I
u = 1.12142 + 1.03617I
a = −0.697426 + 1.061500I
b = 0.23534 + 1.84389I
9.39914 − 7.40677I 0.74326 + 4.41038I
u = 1.12142 − 1.03617I
a = −0.697426 − 1.061500I
b = 0.23534 − 1.84389I
9.39914 + 7.40677I 0.74326 − 4.41038I
5
II. I
u
2
= hb
2
− b + 1, a, u + 1i
(i) Arc colorings
a
5
=
0
−1
a
8
=
1
0
a
7
=
1
−1
a
1
=
0
b
a
4
=
−1
0
a
3
=
−1
b − 1
a
6
=
0
−1
a
2
=
0
b
a
9
=
b
b
a
9
=
b
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b − 5
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
2
c
2
u
2
− u + 1
c
3
, c
8
, c
9
u
2
+ u + 1
c
4
, c
6
(u − 1)
2
c
7
(u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
2
c
2
, c
3
, c
8
c
9
y
2
+ y + 1
c
4
, c
6
, c
7
(y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 0.500000 + 0.866025I
−1.64493 − 2.02988I −3.00000 + 3.46410I
u = −1.00000
a = 0
b = 0.500000 − 0.866025I
−1.64493 + 2.02988I −3.00000 − 3.46410I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
2
(u
10
+ u
9
− 7u
8
− 8u
7
+ 13u
6
+ 14u
5
− 2u
4
+ 2u
3
+ 13u
2
+ 12u + 4)
c
2
(u
2
− u + 1)(u
10
+ 2u
9
+ 3u
8
+ 2u
7
+ 4u
6
+ 3u
5
+ 3u
4
+ 3u
2
+ u + 1)
c
3
(u
2
+ u + 1)(u
10
− 2u
9
+ ··· + 21u + 17)
c
4
(u − 1)
2
(u
10
− 3u
9
+ 4u
8
+ u
7
− 6u
6
+ 6u
5
+ u
4
− 2u
3
+ 3u
2
− 2u + 1)
c
6
(u − 1)
2
· (u
10
+ u
9
+ 10u
8
+ 11u
7
+ 26u
6
+ 30u
5
+ u
4
− 14u
3
+ 3u
2
− 2u + 1)
c
7
(u + 1)
2
(u
10
− 3u
9
+ 4u
8
+ u
7
− 6u
6
+ 6u
5
+ u
4
− 2u
3
+ 3u
2
− 2u + 1)
c
8
(u
2
+ u + 1)(u
10
+ 2u
9
+ 3u
8
+ 2u
7
+ 4u
6
+ 3u
5
+ 3u
4
+ 3u
2
+ u + 1)
c
9
(u
2
+ u + 1)
· (u
10
+ 2u
9
+ 9u
8
+ 14u
7
+ 28u
6
+ 31u
5
+ 35u
4
+ 20u
3
+ 15u
2
+ 5u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
2
(y
10
− 15y
9
+ ··· − 40y + 16)
c
2
, c
8
(y
2
+ y + 1)
· (y
10
+ 2y
9
+ 9y
8
+ 14y
7
+ 28y
6
+ 31y
5
+ 35y
4
+ 20y
3
+ 15y
2
+ 5y + 1)
c
3
(y
2
+ y + 1)(y
10
+ 26y
9
+ ··· + 2925y + 289)
c
4
, c
7
(y −1)
2
· (y
10
− y
9
+ 10y
8
− 11y
7
+ 26y
6
− 30y
5
+ y
4
+ 14y
3
+ 3y
2
+ 2y + 1)
c
6
((y −1)
2
)(y
10
+ 19y
9
+ ··· + 2y + 1)
c
9
(y
2
+ y + 1)(y
10
+ 14y
9
+ ··· + 5y + 1)
11
