9
45
(K9n
2
)
A knot diagram
1
Linearized knot diagam
6 9 1 7 1 4 6 3 8
Solving Sequence
4,6
7
1,8
3 5 9 2
c
6
c
7
c
3
c
5
c
9
c
2
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
+ 2u
10
− 5u
8
− 4u
7
+ 3u
6
+ 4u
5
− u
4
− u
3
+ 4u
2
+ b − 1,
3u
12
+ 8u
11
+ 7u
10
− 15u
9
− 27u
8
− 8u
7
+ 32u
6
+ 13u
5
− 8u
4
− 8u
3
+ 21u
2
+ 2a − 3u − 7,
u
13
+ 3u
12
+ 3u
11
− 4u
10
− 10u
9
− 5u
8
+ 8u
7
+ 7u
6
− u
5
− 2u
4
+ 5u
3
+ 2u
2
− 2u − 1i
I
u
2
= hb, a
2
− a + 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 15 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
= hu
11
+2u
10
+· · ·+b−1, 3u
12
+8u
11
+· · ·+2a−7, u
13
+3u
12
+· · ·−2u−1i
(i) Arc colorings
a
4
=
0
u
a
6
=
1
0
a
7
=
1
−u
2
a
1
=
−
3
2
u
12
− 4u
11
+ ··· +
3
2
u +
7
2
−u
11
− 2u
10
+ 5u
8
+ 4u
7
− 3u
6
− 4u
5
+ u
4
+ u
3
− 4u
2
+ 1
a
8
=
−u
2
+ 1
−u
2
a
3
=
1
2
u
12
+ u
11
+ ··· −
5
2
u +
1
2
u
2
a
5
=
u
−u
3
+ u
a
9
=
−
7
2
u
12
− 8u
11
+ ··· +
7
2
u +
11
2
−
1
2
u
12
− u
11
+ ··· +
1
2
u +
1
2
a
2
=
3
2
u
12
+ 5u
11
+ ··· −
3
2
u −
9
2
u
11
+ 2u
10
− 5u
8
− 4u
7
+ 3u
6
+ 4u
5
− u
4
− u
3
+ 4u
2
− 1
a
2
=
3
2
u
12
+ 5u
11
+ ··· −
3
2
u −
9
2
u
11
+ 2u
10
− 5u
8
− 4u
7
+ 3u
6
+ 4u
5
− u
4
− u
3
+ 4u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
12
− 3u
11
+ u
10
+ 14u
9
+ 9u
8
− 10u
7
− 26u
6
+ 4u
5
+ 13u
4
+ 5u
3
− 15u
2
+ 8u + 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
13
+ u
12
+ ··· + 4u − 4
c
2
, c
8
u
13
+ 2u
12
+ ··· + u − 1
c
3
u
13
− 2u
12
+ ··· + 3u − 1
c
4
, c
6
u
13
+ 3u
12
+ ··· − 2u − 1
c
7
u
13
− 3u
12
+ ··· + 8u − 1
c
9
u
13
+ 8u
12
+ ··· + 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
13
+ 15y
12
+ ··· − 56y −16
c
2
, c
8
y
13
+ 8y
12
+ ··· + 5y −1
c
3
y
13
− 16y
12
+ ··· + 5y −1
c
4
, c
6
y
13
− 3y
12
+ ··· + 8y −1
c
7
y
13
+ 17y
12
+ ··· + 8y −1
c
9
y
13
− 4y
12
+ ··· + 85y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.139730 + 0.201820I
a = 0.107110 − 0.295303I
b = −0.452299 + 0.637242I
0.965349 + 0.999086I 3.54362 + 0.58191I
u = 1.139730 − 0.201820I
a = 0.107110 + 0.295303I
b = −0.452299 − 0.637242I
0.965349 − 0.999086I 3.54362 − 0.58191I
u = 0.431606 + 0.658497I
a = −0.349504 + 0.760906I
b = 0.997974 + 0.288600I
−1.60812 + 2.52293I 1.64572 − 4.38707I
u = 0.431606 − 0.658497I
a = −0.349504 − 0.760906I
b = 0.997974 − 0.288600I
−1.60812 − 2.52293I 1.64572 + 4.38707I
u = −0.946506 + 0.889214I
a = 0.759526 + 1.128230I
b = −0.25689 + 1.55234I
−4.36446 − 3.30324I 4.83610 + 2.39821I
u = −0.946506 − 0.889214I
a = 0.759526 − 1.128230I
b = −0.25689 − 1.55234I
−4.36446 + 3.30324I 4.83610 − 2.39821I
u = 0.650994
a = 0.569843
b = −0.612460
1.00303 10.1180
u = −0.831561 + 1.070510I
a = −0.593619 − 1.044890I
b = −0.02169 − 1.76519I
−8.78028 + 1.38297I 1.065751 − 0.716223I
u = −0.831561 − 1.070510I
a = −0.593619 + 1.044890I
b = −0.02169 + 1.76519I
−8.78028 − 1.38297I 1.065751 + 0.716223I
u = −1.10810 + 0.91291I
a = −0.854060 − 1.005570I
b = 0.50699 − 1.66583I
−7.87584 − 8.60203I 2.41458 + 5.32797I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.10810 − 0.91291I
a = −0.854060 + 1.005570I
b = 0.50699 + 1.66583I
−7.87584 + 8.60203I 2.41458 − 5.32797I
u = −0.510670 + 0.169591I
a = 0.14563 + 2.33106I
b = 0.032142 + 0.650070I
0.60016 − 2.36301I 1.43513 + 4.19898I
u = −0.510670 − 0.169591I
a = 0.14563 − 2.33106I
b = 0.032142 − 0.650070I
0.60016 + 2.36301I 1.43513 − 4.19898I
6
II. I
u
2
= hb, a
2
− a + 1, u − 1i
(i) Arc colorings
a
4
=
0
1
a
6
=
1
0
a
7
=
1
−1
a
1
=
a
0
a
8
=
0
−1
a
3
=
a − 1
1
a
5
=
1
0
a
9
=
a
−a
a
2
=
a
0
a
2
=
a
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a + 7
7
(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
(u + 1)
2
c
6
, c
7
(u − 1)
2
8
(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
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.500000 + 0.866025I
b = 0
1.64493 − 2.02988I 9.00000 + 3.46410I
u = 1.00000
a = 0.500000 − 0.866025I
b = 0
1.64493 + 2.02988I 9.00000 − 3.46410I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
2
(u
13
+ u
12
+ ··· + 4u − 4)
c
2
(u
2
− u + 1)(u
13
+ 2u
12
+ ··· + u − 1)
c
3
(u
2
+ u + 1)(u
13
− 2u
12
+ ··· + 3u − 1)
c
4
((u + 1)
2
)(u
13
+ 3u
12
+ ··· − 2u − 1)
c
6
((u − 1)
2
)(u
13
+ 3u
12
+ ··· − 2u − 1)
c
7
((u − 1)
2
)(u
13
− 3u
12
+ ··· + 8u − 1)
c
8
(u
2
+ u + 1)(u
13
+ 2u
12
+ ··· + u − 1)
c
9
(u
2
+ u + 1)(u
13
+ 8u
12
+ ··· + 5u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
2
(y
13
+ 15y
12
+ ··· − 56y − 16)
c
2
, c
8
(y
2
+ y + 1)(y
13
+ 8y
12
+ ··· + 5y − 1)
c
3
(y
2
+ y + 1)(y
13
− 16y
12
+ ··· + 5y − 1)
c
4
, c
6
((y − 1)
2
)(y
13
− 3y
12
+ ··· + 8y − 1)
c
7
((y − 1)
2
)(y
13
+ 17y
12
+ ··· + 8y − 1)
c
9
(y
2
+ y + 1)(y
13
− 4y
12
+ ··· + 85y − 1)
12
