8
7
(K8a
6
)
A knot diagram
1
Linearized knot diagam
5 7 8 1 3 2 6 4
Solving Sequence
2,7
3 6 8 4 5 1
c
2
c
6
c
7
c
3
c
5
c
1
c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
+ u
10
− 2u
9
− 3u
8
+ 2u
7
+ 4u
6
− 3u
4
− u
3
+ u
2
− 1i
* 1 irreducible components of dim
C
= 0, with total 11 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
11
+ u
10
− 2u
9
− 3u
8
+ 2u
7
+ 4u
6
− 3u
4
− u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
6
=
u
u
a
8
=
−u
3
−u
3
+ u
a
4
=
u
8
− u
6
+ u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
5
=
u
3
u
5
− u
3
+ u
a
1
=
u
8
− u
6
+ u
4
+ 1
u
10
− 2u
8
+ 3u
6
− 2u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
10
− 12u
8
− 4u
7
+ 16u
6
+ 8u
5
− 8u
4
− 8u
3
+ 4u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
8
u
11
− u
10
− 6u
9
+ 5u
8
+ 12u
7
− 6u
6
− 10u
5
− u
4
+ 5u
3
+ u
2
− 1
c
2
, c
6
u
11
+ u
10
− 2u
9
− 3u
8
+ 2u
7
+ 4u
6
− 3u
4
− u
3
+ u
2
− 1
c
5
u
11
+ 3u
10
+ 4u
9
+ u
8
+ 2u
7
+ 8u
6
+ 8u
5
− 5u
4
− 3u
3
+ u
2
+ 4u + 1
c
7
u
11
+ 5u
10
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
8
y
11
− 13y
10
+ ··· + 2y −1
c
2
, c
6
y
11
− 5y
10
+ ··· + 2y −1
c
5
y
11
− y
10
+ ··· + 14y −1
c
7
y
11
+ 3y
10
+ ··· − 10y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.959860 + 0.351396I
−1.63627 + 1.27541I −1.47945 − 0.80097I
u = −0.959860 − 0.351396I
−1.63627 − 1.27541I −1.47945 + 0.80097I
u = −0.488025 + 0.800566I
9.03866 − 1.64593I 8.04988 + 0.24481I
u = −0.488025 − 0.800566I
9.03866 + 1.64593I 8.04988 − 0.24481I
u = 1.11640
3.38257 2.18570
u = 1.031510 + 0.521913I
−0.37669 − 4.75030I 2.64109 + 6.77690I
u = 1.031510 − 0.521913I
−0.37669 + 4.75030I 2.64109 − 6.77690I
u = −1.081080 + 0.631709I
7.26485 + 7.02220I 5.50054 − 4.88619I
u = −1.081080 − 0.631709I
7.26485 − 7.02220I 5.50054 + 4.88619I
u = 0.439259 + 0.522038I
1.289960 + 0.454766I 7.19508 − 1.36957I
u = 0.439259 − 0.522038I
1.289960 − 0.454766I 7.19508 + 1.36957I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
8
u
11
− u
10
− 6u
9
+ 5u
8
+ 12u
7
− 6u
6
− 10u
5
− u
4
+ 5u
3
+ u
2
− 1
c
2
, c
6
u
11
+ u
10
− 2u
9
− 3u
8
+ 2u
7
+ 4u
6
− 3u
4
− u
3
+ u
2
− 1
c
5
u
11
+ 3u
10
+ 4u
9
+ u
8
+ 2u
7
+ 8u
6
+ 8u
5
− 5u
4
− 3u
3
+ u
2
+ 4u + 1
c
7
u
11
+ 5u
10
+ ··· + 2u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
8
y
11
− 13y
10
+ ··· + 2y −1
c
2
, c
6
y
11
− 5y
10
+ ··· + 2y −1
c
5
y
11
− y
10
+ ··· + 14y −1
c
7
y
11
+ 3y
10
+ ··· − 10y −1
7
