8
5
(K8a
13
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 8 1 3 6
Solving Sequence
1,4
2
5,7
3 6 8
c
1
c
4
c
3
c
6
c
8
c
2
, c
5
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, u
4
+ u
3
− 2u
2
+ a − 2u, u
5
+ u
4
− 3u
3
− 2u
2
+ 2u − 1i
I
u
2
= h−u
5
+ 2u
3
− u
2
+ b − u + 1, −u
4
+ u
2
+ a − u + 1, u
6
+ u
5
− 2u
4
+ 2u
2
− 2u − 1i
I
u
3
= hb + 1, a, u − 1i
* 3 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
= hb + u, u
4
+ u
3
− 2u
2
+ a − 2u, u
5
+ u
4
− 3u
3
− 2u
2
+ 2u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
7
=
−u
4
− u
3
+ 2u
2
+ 2u
−u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
−u
4
− u
3
+ 2u
2
+ u
−u
a
8
=
−u
3
− u
2
+ 2u
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
4
− 6u
3
+ 4u
2
+ 14u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
c
3
, c
7
u
5
− 3u
4
+ 6u
3
− 7u
2
+ 4u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
y
5
− 7y
4
+ 17y
3
− 14y
2
− 1
c
3
, c
7
y
5
+ 3y
4
+ 2y
3
− 13y
2
− 12y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.331409 + 0.386277I
a = 0.76001 + 1.23514I
b = −0.331409 − 0.386277I
−0.373181 − 1.138820I −4.71808 + 6.05450I
u = 0.331409 − 0.386277I
a = 0.76001 − 1.23514I
b = −0.331409 + 0.386277I
−0.373181 + 1.138820I −4.71808 − 6.05450I
u = 1.49784
a = −0.911163
b = −1.49784
−8.51482 −10.2860
u = −1.58033 + 0.28256I
a = 0.195567 + 1.002700I
b = 1.58033 − 0.28256I
−13.4637 + 6.9972I −11.13904 − 3.54683I
u = −1.58033 − 0.28256I
a = 0.195567 − 1.002700I
b = 1.58033 + 0.28256I
−13.4637 − 6.9972I −11.13904 + 3.54683I
5
II.
I
u
2
= h−u
5
+2u
3
−u
2
+b−u+1, −u
4
+u
2
+a−u+1, u
6
+u
5
−2u
4
+2u
2
−2u−1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
7
=
u
4
− u
2
+ u − 1
u
5
− 2u
3
+ u
2
+ u − 1
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
6
=
u
5
+ u
4
− 2u
3
+ 2u − 2
u
5
− 2u
3
+ u
2
+ u − 1
a
8
=
u
5
− 2u
3
+ 2u
2
+ u − 2
u
5
− u
3
+ 2u
2
− u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
5
+ 8u
3
− 4u
2
− 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
u
6
− u
5
− 2u
4
+ 2u
2
+ 2u − 1
c
3
, c
7
(u
3
+ u
2
+ 2u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
y
6
− 5y
5
+ 8y
4
− 6y
3
+ 8y
2
− 8y + 1
c
3
, c
7
(y
3
+ 3y
2
+ 2y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.592989 + 0.847544I
a = −0.916215 − 0.894804I
b = 1.47043 + 0.10268I
−6.31400 − 2.82812I −9.50976 + 2.97945I
u = 0.592989 − 0.847544I
a = −0.916215 + 0.894804I
b = 1.47043 − 0.10268I
−6.31400 + 2.82812I −9.50976 − 2.97945I
u = 1.13416
a = 0.502436
b = 0.379278
−2.17641 −2.98050
u = −1.47043 + 0.10268I
a = −0.083785 − 0.894804I
b = −0.592989 + 0.847544I
−6.31400 + 2.82812I −9.50976 − 2.97945I
u = −1.47043 − 0.10268I
a = −0.083785 + 0.894804I
b = −0.592989 − 0.847544I
−6.31400 − 2.82812I −9.50976 + 2.97945I
u = −0.379278
a = −1.50244
b = −1.13416
−2.17641 −2.98050
9
III. I
u
3
= hb + 1, a, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
7
=
0
−1
a
3
=
0
1
a
6
=
−1
−1
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
2
, c
5
c
6
u − 1
c
3
, 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
2
, c
4
c
5
, c
6
, c
8
y −1
c
3
, 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
2
, c
5
c
6
(u − 1)(u
5
− u
4
+ ··· + 2u + 1)(u
6
− u
5
+ ··· + 2u − 1)
c
3
, c
7
u(u
3
+ u
2
+ 2u + 1)
2
(u
5
− 3u
4
+ 6u
3
− 7u
2
+ 4u − 2)
c
4
, c
8
(u + 1)(u
5
− u
4
+ ··· + 2u + 1)(u
6
− u
5
+ ··· + 2u − 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
(y −1)(y
5
− 7y
4
+ ··· − 14y
2
− 1)(y
6
− 5y
5
+ ··· − 8y + 1)
c
3
, c
7
y(y
3
+ 3y
2
+ 2y −1)
2
(y
5
+ 3y
4
+ 2y
3
− 13y
2
− 12y −4)
15
