8
13
(K8a
7
)
A knot diagram
1
Linearized knot diagam
5 8 7 6 2 1 3 4
Solving Sequence
2,5
6 1 7 4 3 8
c
5
c
1
c
6
c
4
c
3
c
8
c
2
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
14
+ u
13
− 3u
12
− 4u
11
+ 4u
10
+ 7u
9
− u
8
− 6u
7
− 2u
6
+ 2u
5
+ 2u
4
+ 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
−4u
11
+4u
10
+7u
9
−u
8
−6u
7
−2u
6
+2u
5
+2u
4
+u +1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
1
=
u
u
a
7
=
u
4
− u
2
+ 1
u
4
a
4
=
−u
2
+ 1
−u
4
a
3
=
u
12
− 3u
10
+ 5u
8
− 4u
6
+ 2u
4
− u
2
+ 1
u
12
− 2u
10
+ 2u
8
− u
4
a
8
=
u
7
− 2u
5
+ 2u
3
u
9
− u
7
+ u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
13
+ 16u
11
+ 4u
10
−28u
9
−12u
8
+ 20u
7
+ 16u
6
−8u
4
−8u
3
−2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
14
+ u
13
+ ··· + u + 1
c
2
, c
3
, c
7
u
14
+ u
13
+ ··· + u + 1
c
4
u
14
+ 7u
13
+ ··· + u + 1
c
6
u
14
+ 3u
13
+ ··· + 7u + 3
c
8
u
14
− u
13
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
14
− 7y
13
+ ··· − y + 1
c
2
, c
3
, c
7
y
14
+ 13y
13
+ ··· − y + 1
c
4
y
14
+ y
13
+ ··· + 7y + 1
c
6
y
14
+ 5y
13
+ ··· + 23y + 9
c
8
y
14
+ y
13
+ ··· − y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.989783 + 0.381937I
−1.69471 + 1.40484I −1.50927 − 0.52948I
u = −0.989783 − 0.381937I
−1.69471 − 1.40484I −1.50927 + 0.52948I
u = −0.728347 + 0.560551I
−1.44038 + 2.19128I 1.23919 − 3.85718I
u = −0.728347 − 0.560551I
−1.44038 − 2.19128I 1.23919 + 3.85718I
u = 1.068410 + 0.522447I
−0.56380 − 5.07185I 1.67153 + 6.33126I
u = 1.068410 − 0.522447I
−0.56380 + 5.07185I 1.67153 − 6.33126I
u = 1.157220 + 0.286866I
−7.82627 + 0.47055I −5.32829 + 0.18349I
u = 1.157220 − 0.286866I
−7.82627 − 0.47055I −5.32829 − 0.18349I
u = −0.268039 + 0.757899I
−3.51248 − 3.62879I 0.33383 + 2.63226I
u = −0.268039 − 0.757899I
−3.51248 + 3.62879I 0.33383 − 2.63226I
u = −1.142590 + 0.546762I
−6.06421 + 8.53123I −2.72348 − 6.18031I
u = −1.142590 − 0.546762I
−6.06421 − 8.53123I −2.72348 + 6.18031I
u = 0.403136 + 0.584808I
1.36265 + 0.62859I 6.31651 − 1.42251I
u = 0.403136 − 0.584808I
1.36265 − 0.62859I 6.31651 + 1.42251I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
14
+ u
13
+ ··· + u + 1
c
2
, c
3
, c
7
u
14
+ u
13
+ ··· + u + 1
c
4
u
14
+ 7u
13
+ ··· + u + 1
c
6
u
14
+ 3u
13
+ ··· + 7u + 3
c
8
u
14
− u
13
+ ··· + 3u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
14
− 7y
13
+ ··· − y + 1
c
2
, c
3
, c
7
y
14
+ 13y
13
+ ··· − y + 1
c
4
y
14
+ y
13
+ ··· + 7y + 1
c
6
y
14
+ 5y
13
+ ··· + 23y + 9
c
8
y
14
+ y
13
+ ··· − y + 1
7
