8
12
(K8a
5
)
A knot diagram
1
Linearized knot diagam
5 1 7 2 4 8 3 6
Solving Sequence
3,8
7 4 6 1 2 5
c
7
c
3
c
6
c
8
c
2
c
5
c
1
, c
4
Ideals for irreducible components
2
of X
par
I
u
1
= hu
14
− u
13
+ 3u
12
− 2u
11
+ 6u
10
− 3u
9
+ 7u
8
− 2u
7
+ 6u
6
+ 4u
4
+ 2u
2
+ u + 1i
* 1 irreducible components of dim
C
= 0, with total 14 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
14
−u
13
+3u
12
−2u
11
+6u
10
−3u
9
+7u
8
−2u
7
+6u
6
+4u
4
+2u
2
+u +1i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
7
=
1
−u
2
a
4
=
−u
u
3
+ u
a
6
=
u
2
+ 1
−u
2
a
1
=
u
4
+ u
2
+ 1
−u
4
a
2
=
u
9
+ 2u
7
+ 3u
5
+ 2u
3
+ u
−u
9
− u
7
− u
5
+ u
a
5
=
u
6
+ u
4
+ 2u
2
+ 1
−u
8
− 2u
6
− 2u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
12
− 4u
11
+ 8u
10
− 8u
9
+ 16u
8
− 12u
7
+ 12u
6
− 12u
5
+ 8u
4
− 4u
3
+ 4u
2
− 8u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
14
+ u
13
+ ··· − u + 1
c
2
, c
5
u
14
+ 5u
13
+ ··· + 3u + 1
c
3
, c
7
u
14
− u
13
+ ··· + u + 1
c
6
, c
8
u
14
− 5u
13
+ ··· − 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
y
14
+ 5y
13
+ ··· + 3y + 1
c
2
, c
5
, c
6
c
8
y
14
+ 9y
13
+ ··· + 15y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.772300 + 0.626535I
−1.19029 + 3.41271I −2.10600 − 2.62516I
u = 0.772300 − 0.626535I
−1.19029 − 3.41271I −2.10600 + 2.62516I
u = −0.050221 + 1.076790I
4.64273 + 2.76747I 5.41762 − 3.21377I
u = −0.050221 − 1.076790I
4.64273 − 2.76747I 5.41762 + 3.21377I
u = 0.727524 + 0.860849I
−4.64273 − 2.76747I −5.41762 + 3.21377I
u = 0.727524 − 0.860849I
−4.64273 + 2.76747I −5.41762 − 3.21377I
u = −0.494052 + 0.663856I
0.022819 + 1.377700I 0.88590 − 4.12207I
u = −0.494052 − 0.663856I
0.022819 − 1.377700I 0.88590 + 4.12207I
u = −0.622207 + 1.001070I
1.19029 + 3.41271I 2.10600 − 2.62516I
u = −0.622207 − 1.001070I
1.19029 − 3.41271I 2.10600 + 2.62516I
u = 0.683715 + 1.025590I
− 8.93586I 0. + 7.26077I
u = 0.683715 − 1.025590I
8.93586I 0. − 7.26077I
u = −0.517057 + 0.454483I
−0.022819 + 1.377700I −0.88590 − 4.12207I
u = −0.517057 − 0.454483I
−0.022819 − 1.377700I −0.88590 + 4.12207I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
u
14
+ u
13
+ ··· − u + 1
c
2
, c
5
u
14
+ 5u
13
+ ··· + 3u + 1
c
3
, c
7
u
14
− u
13
+ ··· + u + 1
c
6
, c
8
u
14
− 5u
13
+ ··· − 3u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
y
14
+ 5y
13
+ ··· + 3y + 1
c
2
, c
5
, c
6
c
8
y
14
+ 9y
13
+ ··· + 15y + 1
7
