10
129
(K10n
18
)
A knot diagram
1
Linearized knot diagam
4 9 5 2 10 9 5 3 6 7
Solving Sequence
5,10 2,6
4 1 3 9 7 8
c
5
c
4
c
1
c
3
c
9
c
6
c
7
c
2
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
12
+ u
11
+ 5u
10
+ 4u
9
+ 9u
8
+ 6u
7
+ 5u
6
+ 4u
5
− 2u
4
+ u
3
− 2u
2
+ b + 1,
− u
14
− 2u
13
− 8u
12
− 11u
11
− 23u
10
− 23u
9
− 28u
8
− 20u
7
− 9u
6
− 5u
5
+ 7u
4
+ u
3
+ u
2
+ a − u − 3,
u
15
+ 2u
14
+ 8u
13
+ 12u
12
+ 24u
11
+ 28u
10
+ 32u
9
+ 29u
8
+ 14u
7
+ 9u
6
− 6u
5
− 5u
4
− 2u
3
− 2u
2
+ 4u + 1i
I
u
2
= hb + 1, −u
2
+ a − u − 1, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 18 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
12
+u
11
+· · ·+b+1, −u
14
−2u
13
+· · ·+a−3, u
15
+2u
14
+· · ·+4u+1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
14
+ 2u
13
+ ··· + u + 3
−u
12
− u
11
− 5u
10
− 4u
9
− 9u
8
− 6u
7
− 5u
6
− 4u
5
+ 2u
4
− u
3
+ 2u
2
− 1
a
6
=
1
−u
2
a
4
=
u
14
+ 2u
13
+ ··· − u + 3
−u
13
− u
12
+ ··· − u − 1
a
1
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
3
=
u
14
+ u
13
+ ··· − 2u + 2
−u
13
− u
12
+ ··· − u − 1
a
9
=
−u
u
3
+ u
a
7
=
u
2
+ 1
−u
4
− 2u
2
a
8
=
−u
4
− u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
14
− 2u
13
− 11u
12
− 16u
11
− 42u
10
− 47u
9
− 71u
8
− 62u
7
−
44u
6
− 31u
5
+ 12u
4
+ 4u
3
+ 14u
2
+ 5u − 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
15
− 4u
14
+ ··· − 3u + 1
c
2
, c
8
u
15
+ u
14
+ ··· + 12u + 8
c
3
u
15
+ 2u
14
+ ··· − 3u + 1
c
5
, c
6
, c
9
u
15
+ 2u
14
+ ··· + 4u + 1
c
7
u
15
+ 8u
14
+ ··· + 280u − 49
c
10
u
15
− 2u
14
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
15
− 2y
14
+ ··· − 3y −1
c
2
, c
8
y
15
+ 21y
14
+ ··· − 48y −64
c
3
y
15
+ 26y
14
+ ··· − 3y −1
c
5
, c
6
, c
9
y
15
+ 12y
14
+ ··· + 20y −1
c
7
y
15
− 32y
14
+ ··· + 220108y −2401
c
10
y
15
− 20y
14
+ ··· + 20y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.946822 + 0.058779I
a = −0.57292 − 1.67757I
b = 1.05231 + 1.07767I
10.46560 − 3.92970I 3.25200 + 2.37642I
u = −0.946822 − 0.058779I
a = −0.57292 + 1.67757I
b = 1.05231 − 1.07767I
10.46560 + 3.92970I 3.25200 − 2.37642I
u = −0.078813 + 1.147950I
a = −0.99527 + 1.21138I
b = −1.148380 − 0.278021I
−4.30318 − 1.14653I −3.69630 − 0.14216I
u = −0.078813 − 1.147950I
a = −0.99527 − 1.21138I
b = −1.148380 + 0.278021I
−4.30318 + 1.14653I −3.69630 + 0.14216I
u = 0.271249 + 1.119280I
a = 0.070766 − 0.823663I
b = −0.282237 + 0.716387I
−1.32042 + 2.58137I 0.00443 − 4.00241I
u = 0.271249 − 1.119280I
a = 0.070766 + 0.823663I
b = −0.282237 − 0.716387I
−1.32042 − 2.58137I 0.00443 + 4.00241I
u = −0.488190 + 1.251290I
a = −0.601814 + 0.190541I
b = 0.92821 − 1.13080I
6.78648 − 1.17157I 0.521469 + 0.840506I
u = −0.488190 − 1.251290I
a = −0.601814 − 0.190541I
b = 0.92821 + 1.13080I
6.78648 + 1.17157I 0.521469 − 0.840506I
u = 0.604547 + 0.198361I
a = 0.727011 + 0.890995I
b = 0.195944 − 0.500014I
1.37013 + 0.70150I 5.29100 − 2.23884I
u = 0.604547 − 0.198361I
a = 0.727011 − 0.890995I
b = 0.195944 + 0.500014I
1.37013 − 0.70150I 5.29100 + 2.23884I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.197329 + 1.368030I
a = 1.103000 + 0.360621I
b = 0.560305 − 0.345696I
−3.65536 + 3.51330I 0.20706 − 4.67402I
u = 0.197329 − 1.368030I
a = 1.103000 − 0.360621I
b = 0.560305 + 0.345696I
−3.65536 − 3.51330I 0.20706 + 4.67402I
u = −0.445416 + 1.338930I
a = 0.91370 − 1.42147I
b = 1.13244 + 0.99333I
6.09422 − 8.90152I −0.37309 + 5.02376I
u = −0.445416 − 1.338930I
a = 0.91370 + 1.42147I
b = 1.13244 − 0.99333I
6.09422 + 8.90152I −0.37309 − 5.02376I
u = −0.227769
a = 2.71106
b = −0.877160
−1.26612 −9.41310
6
II. I
u
2
= hb + 1, −u
2
+ a − u − 1, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
2
+ u + 1
−1
a
6
=
1
−u
2
a
4
=
u
2
+ u + 2
−1
a
1
=
−1
0
a
3
=
u
2
+ u + 1
−1
a
9
=
−u
−u
2
− u − 1
a
7
=
u
2
+ 1
−u
2
− u − 1
a
8
=
−u
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
2
+ 4u + 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u − 1)
3
c
2
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
u
3
+ u
2
+ 2u + 1
c
7
, c
10
u
3
+ u
2
− 1
c
9
u
3
− u
2
+ 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y −1)
3
c
2
, c
8
y
3
c
5
, c
6
, c
9
y
3
+ 3y
2
+ 2y −1
c
7
, c
10
y
3
− y
2
+ 2y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.877439 + 0.744862I
b = −1.00000
−4.66906 − 2.82812I −5.17211 + 2.41717I
u = −0.215080 − 1.307140I
a = −0.877439 − 0.744862I
b = −1.00000
−4.66906 + 2.82812I −5.17211 − 2.41717I
u = −0.569840
a = 0.754878
b = −1.00000
−0.531480 3.34420
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
15
− 4u
14
+ ··· − 3u + 1)
c
2
, c
8
u
3
(u
15
+ u
14
+ ··· + 12u + 8)
c
3
((u − 1)
3
)(u
15
+ 2u
14
+ ··· − 3u + 1)
c
4
((u + 1)
3
)(u
15
− 4u
14
+ ··· − 3u + 1)
c
5
, c
6
(u
3
+ u
2
+ 2u + 1)(u
15
+ 2u
14
+ ··· + 4u + 1)
c
7
(u
3
+ u
2
− 1)(u
15
+ 8u
14
+ ··· + 280u − 49)
c
9
(u
3
− u
2
+ 2u − 1)(u
15
+ 2u
14
+ ··· + 4u + 1)
c
10
(u
3
+ u
2
− 1)(u
15
− 2u
14
+ ··· + 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
3
)(y
15
− 2y
14
+ ··· − 3y −1)
c
2
, c
8
y
3
(y
15
+ 21y
14
+ ··· − 48y −64)
c
3
((y −1)
3
)(y
15
+ 26y
14
+ ··· − 3y −1)
c
5
, c
6
, c
9
(y
3
+ 3y
2
+ 2y −1)(y
15
+ 12y
14
+ ··· + 20y −1)
c
7
(y
3
− y
2
+ 2y −1)(y
15
− 32y
14
+ ··· + 220108y −2401)
c
10
(y
3
− y
2
+ 2y −1)(y
15
− 20y
14
+ ··· + 20y −1)
12
