8
14
(K8a
1
)
A knot diagram
1
Linearized knot diagam
6 4 8 7 1 5 2 3
Solving Sequence
1,5
6 2 7 4 3 8
c
5
c
1
c
6
c
4
c
2
c
8
c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
15
− u
14
− 2u
13
+ 3u
12
+ 4u
11
− 6u
10
− 4u
9
+ 9u
8
+ 2u
7
− 8u
6
+ 6u
4
− 2u
3
− 2u
2
+ 2u − 1i
* 1 irreducible components of dim
C
= 0, with total 15 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
15
− u
14
− 2u
13
+ 3u
12
+ 4u
11
− 6u
10
− 4u
9
+ 9u
8
+ 2u
7
− 8u
6
+
6u
4
− 2u
3
− 2u
2
+ 2u − 1i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
2
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
u
2
a
4
=
u
4
− u
2
+ 1
−u
4
a
3
=
u
11
− 2u
9
+ 4u
7
− 4u
5
+ 3u
3
− 2u
−u
11
+ u
9
− 2u
7
+ u
5
− u
3
+ u
a
8
=
−u
6
+ u
4
− 2u
2
+ 1
−u
8
+ 2u
6
− 2u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
14
+12u
12
−4u
11
−24u
10
+8u
9
+32u
8
−20u
7
−28u
6
+24u
5
+16u
4
−20u
3
−4u
2
+12u−10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
15
+ u
14
+ ··· + 2u + 1
c
2
u
15
+ 7u
14
+ ··· + 4u
2
− 1
c
3
, c
8
u
15
+ u
14
+ ··· + 2u + 1
c
4
, c
6
u
15
+ 5u
14
+ ··· + 12u
3
+ 1
c
7
u
15
− u
14
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
15
− 5y
14
+ ··· + 12y
3
− 1
c
2
y
15
+ 3y
14
+ ··· + 8y −1
c
3
, c
8
y
15
+ 7y
14
+ ··· + 4y
2
− 1
c
4
, c
6
y
15
+ 11y
14
+ ··· − 84y
2
− 1
c
7
y
15
− y
14
+ ··· + 16y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.035190 + 0.117787I
−4.20816 + 3.60340I −10.16372 − 4.47672I
u = −1.035190 − 0.117787I
−4.20816 − 3.60340I −10.16372 + 4.47672I
u = 0.690784 + 0.795701I
1.98305 + 3.51852I −2.28698 − 2.59027I
u = 0.690784 − 0.795701I
1.98305 − 3.51852I −2.28698 + 2.59027I
u = 0.928223 + 0.554966I
−1.82075 − 2.07402I −7.82822 + 2.67122I
u = 0.928223 − 0.554966I
−1.82075 + 2.07402I −7.82822 − 2.67122I
u = −0.778519 + 0.756850I
3.53338 + 1.50523I 0.15133 − 2.74048I
u = −0.778519 − 0.756850I
3.53338 − 1.50523I 0.15133 + 2.74048I
u = 0.860077
−1.42428 −6.56340
u = −0.946375 + 0.717051I
3.01689 + 4.09199I −0.95573 − 3.15094I
u = −0.946375 − 0.717051I
3.01689 − 4.09199I −0.95573 + 3.15094I
u = 1.006640 + 0.715109I
1.02630 − 9.21780I −4.14540 + 7.39135I
u = 1.006640 − 0.715109I
1.02630 + 9.21780I −4.14540 − 7.39135I
u = 0.204399 + 0.532644I
−0.35117 − 1.66084I −2.48958 + 3.96405I
u = 0.204399 − 0.532644I
−0.35117 + 1.66084I −2.48958 − 3.96405I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
15
+ u
14
+ ··· + 2u + 1
c
2
u
15
+ 7u
14
+ ··· + 4u
2
− 1
c
3
, c
8
u
15
+ u
14
+ ··· + 2u + 1
c
4
, c
6
u
15
+ 5u
14
+ ··· + 12u
3
+ 1
c
7
u
15
− u
14
+ ··· − 4u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
15
− 5y
14
+ ··· + 12y
3
− 1
c
2
y
15
+ 3y
14
+ ··· + 8y −1
c
3
, c
8
y
15
+ 7y
14
+ ··· + 4y
2
− 1
c
4
, c
6
y
15
+ 11y
14
+ ··· − 84y
2
− 1
c
7
y
15
− y
14
+ ··· + 16y −1
7
