4
1
(K4a
1
)
A knot diagram
1
Linearized knot diagam
4 1 2 3
Solving Sequence
2,4
1 3
c
1
c
3
c
2
, c
4
Ideals for irreducible components
2
of X
par
I
u
1
= hu
2
+ u + 1i
* 1 irreducible components of dim
C
= 0, with total 2 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
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
1
=
1
−u − 1
a
3
=
−u
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
2
− u + 1
c
2
, c
4
u
2
+ u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
y
2
+ y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
− 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
2.02988I 0. − 3.46410I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
u
2
− u + 1
c
2
, c
4
u
2
+ u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
y
2
+ y + 1
7
