6
3
(K6a
1
)
A knot diagram
1
Linearized knot diagam
4 1 6 2 3 5
Solving Sequence
2,5
4 1 3 6
c
4
c
1
c
2
c
6
c
3
, c
5
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
+ u
5
− u
4
− 2u
3
+ u + 1i
* 1 irreducible components of dim
C
= 0, with total 6 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
6
+ u
5
− u
4
− 2u
3
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
4
=
1
−u
2
a
1
=
u
−u
3
+ u
a
3
=
−u
3
u
5
− u
3
+ u
a
6
=
u
3
−u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
− 4u
2
− 4u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
2
u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1
c
3
, c
5
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
6
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
2
, c
6
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
−1.89061 − 0.92430I −3.71672 + 0.79423I
u = 1.002190 − 0.295542I
−1.89061 + 0.92430I −3.71672 − 0.79423I
u = −0.428243 + 0.664531I
1.89061 − 0.92430I 3.71672 + 0.79423I
u = −0.428243 − 0.664531I
1.89061 + 0.92430I 3.71672 − 0.79423I
u = −1.073950 + 0.558752I
5.69302I 0. − 5.51057I
u = −1.073950 − 0.558752I
− 5.69302I 0. + 5.51057I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
2
u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1
c
3
, c
5
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
6
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
2
, c
6
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
7
