8
21
(K8n
2
)
A knot diagram
1
Linearized knot diagam
4 7 5 2 7 1 2 6
Solving Sequence
2,4
5 1
3,7
6 8
c
4
c
1
c
3
c
6
c
8
c
2
, c
5
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
2
+ b + u + 1, −u
2
+ a + u, u
4
− u
3
+ u + 1i
I
u
2
= hb − 1, u
3
+ a + 1, u
4
− u
3
+ 2u − 1i
I
u
3
= hb − 1, a, u + 1i
* 3 irreducible components of dim
C
= 0, with total 9 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 + 1, −u
2
+ a + u, u
4
− u
3
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
1
=
u
u
a
3
=
−u
2
+ 1
−u
3
+ u + 1
a
7
=
u
2
− u
u
2
− u − 1
a
6
=
−u
−u − 1
a
8
=
−u
2
+ u
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
3
+ 6u
2
− 2u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
u
4
− u
3
+ u + 1
c
2
, c
7
u
4
+ 3u
3
+ 3u
2
+ 2u + 2
c
3
, c
5
u
4
+ u
3
+ 4u
2
+ u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
y
4
− y
3
+ 4y
2
− y + 1
c
2
, c
7
y
4
− 3y
3
+ y
2
+ 8y + 4
c
3
, c
5
y
4
+ 7y
3
+ 16y
2
+ 7y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.566121 + 0.458821I
a = 0.676097 − 0.978318I
b = −0.323903 − 0.978318I
−0.66070 + 1.45022I −4.56010 − 4.72374I
u = −0.566121 − 0.458821I
a = 0.676097 + 0.978318I
b = −0.323903 + 0.978318I
−0.66070 − 1.45022I −4.56010 + 4.72374I
u = 1.066120 + 0.864054I
a = −0.676097 + 0.978318I
b = −1.67610 + 0.97832I
4.77303 − 6.78371I −3.43990 + 4.72374I
u = 1.066120 − 0.864054I
a = −0.676097 − 0.978318I
b = −1.67610 − 0.97832I
4.77303 + 6.78371I −3.43990 − 4.72374I
5
II. I
u
2
= hb − 1, u
3
+ a + 1, u
4
− u
3
+ 2u − 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
1
=
u
u
a
3
=
−u
2
+ 1
−u
3
+ 2u − 1
a
7
=
−u
3
− 1
1
a
6
=
−u
u
3
− u + 2
a
8
=
u
3
+ 1
−u
3
+ u
2
+ u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
u
4
− u
3
+ 2u − 1
c
2
, c
7
(u
2
− u − 1)
2
c
3
, c
5
u
4
+ u
3
+ 2u
2
+ 4u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
y
4
− y
3
+ 2y
2
− 4y + 1
c
2
, c
7
(y
2
− 3y + 1)
2
c
3
, c
5
y
4
+ 3y
3
− 2y
2
− 12y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.15372
a = 0.535687
b = 1.00000
−2.30291 −2.00000
u = 0.809017 + 0.981593I
a = 0.809017 − 0.981593I
b = 1.00000
5.59278 −2.00000
u = 0.809017 − 0.981593I
a = 0.809017 + 0.981593I
b = 1.00000
5.59278 −2.00000
u = 0.535687
a = −1.15372
b = 1.00000
−2.30291 −2.00000
9
III. I
u
3
= hb − 1, a, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
4
=
1
0
a
5
=
1
1
a
1
=
−1
−1
a
3
=
0
−1
a
7
=
0
1
a
6
=
1
2
a
8
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
6
u − 1
c
2
, c
7
u
c
4
, c
8
u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
8
y −1
c
2
, c
7
y
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u − 1)(u
4
− u
3
+ u + 1)(u
4
− u
3
+ 2u − 1)
c
2
, c
7
u(u
2
− u − 1)
2
(u
4
+ 3u
3
+ 3u
2
+ 2u + 2)
c
3
, c
5
(u − 1)(u
4
+ u
3
+ 2u
2
+ 4u + 1)(u
4
+ u
3
+ 4u
2
+ u + 1)
c
4
, c
8
(u + 1)(u
4
− u
3
+ u + 1)(u
4
− u
3
+ 2u − 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
(y −1)(y
4
− y
3
+ 2y
2
− 4y + 1)(y
4
− y
3
+ 4y
2
− y + 1)
c
2
, c
7
y(y
2
− 3y + 1)
2
(y
4
− 3y
3
+ y
2
+ 8y + 4)
c
3
, c
5
(y −1)(y
4
+ 3y
3
− 2y
2
− 12y + 1)(y
4
+ 7y
3
+ 16y
2
+ 7y + 1)
15
