10
47
(K10a
15
)
A knot diagram
1
Linearized knot diagam
5 6 9 1 2 3 10 7 4 8
Solving Sequence
2,5
6 3 7 1
4,9
10 8
c
5
c
2
c
6
c
1
c
4
c
9
c
8
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h3u
21
− 41u
19
+ ··· + b − 2, −2u
21
+ u
20
+ ··· + a + 1, u
22
− 2u
21
+ ··· − u + 1i
I
u
2
= hb − u, a − 1, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 24 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
= h3u
21
−41u
19
+· · ·+b−2, −2u
21
+u
20
+· · ·+a+1, u
22
−2u
21
+· · ·−u+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
3
=
u
−u
3
+ u
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
−u
u
a
4
=
−u
2
+ 1
u
2
a
9
=
2u
21
− u
20
+ ··· + 4u − 1
−3u
21
+ 41u
19
+ ··· − u + 2
a
10
=
−u
20
+ u
19
+ ··· − 9u
2
+ 4u
−u
16
+ 10u
14
+ ··· + 6u
3
− 4u
2
a
8
=
u
21
− u
20
+ ··· − 10u
2
+ 4u
−u
21
+ 14u
19
+ ··· − 3u
2
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
21
+5u
20
+53u
19
−63u
18
−292u
17
+333u
16
+862u
15
−974u
14
−1448u
13
+1758u
12
+
1295u
11
−2051u
10
−338u
9
+1521u
8
−426u
7
−628u
6
+422u
5
+65u
4
−139u
3
+35u
2
+14u+7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
u
22
− 2u
21
+ ··· − u + 1
c
3
, c
9
u
22
+ u
21
+ ··· − 21u
2
+ 4
c
7
, c
10
u
22
− 3u
21
+ ··· − 8u + 1
c
8
u
22
+ 9u
21
+ ··· + 40u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
y
22
− 30y
21
+ ··· + 3y + 1
c
3
, c
9
y
22
− 15y
21
+ ··· − 168y + 16
c
7
, c
10
y
22
− 9y
21
+ ··· − 40y + 1
c
8
y
22
+ 11y
21
+ ··· − 1080y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.964383 + 0.128666I
a = 0.184838 − 0.945621I
b = −0.299924 + 0.888159I
1.70640 + 2.06027I 8.35016 − 3.76643I
u = 0.964383 − 0.128666I
a = 0.184838 + 0.945621I
b = −0.299924 − 0.888159I
1.70640 − 2.06027I 8.35016 + 3.76643I
u = −0.889732
a = 1.34588
b = 1.19747
0.304299 11.0200
u = −1.059960 + 0.353222I
a = 0.600368 + 0.550351I
b = 0.830761 + 0.371286I
5.17561 − 7.52719I 9.40693 + 6.57102I
u = −1.059960 − 0.353222I
a = 0.600368 − 0.550351I
b = 0.830761 − 0.371286I
5.17561 + 7.52719I 9.40693 − 6.57102I
u = −1.128070 + 0.227245I
a = −0.642240 − 0.353090I
b = −0.804731 − 0.252365I
6.69355 − 1.82013I 11.99179 + 1.37946I
u = −1.128070 − 0.227245I
a = −0.642240 + 0.353090I
b = −0.804731 + 0.252365I
6.69355 + 1.82013I 11.99179 − 1.37946I
u = 0.459979 + 0.506822I
a = 0.614823 − 0.850759I
b = −0.713989 + 0.079726I
1.65851 − 0.59540I 8.05700 − 0.40058I
u = 0.459979 − 0.506822I
a = 0.614823 + 0.850759I
b = −0.713989 − 0.079726I
1.65851 + 0.59540I 8.05700 + 0.40058I
u = 0.269941 + 0.602986I
a = −0.589034 + 0.985256I
b = 0.753100 + 0.089219I
1.04219 + 4.27368I 5.66030 − 6.14849I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.269941 − 0.602986I
a = −0.589034 − 0.985256I
b = 0.753100 − 0.089219I
1.04219 − 4.27368I 5.66030 + 6.14849I
u = 0.485575
a = 0.677603
b = −0.329027
0.739737 13.5160
u = −1.58393
a = −0.123818
b = −0.196119
7.92361 15.4350
u = −0.138359 + 0.279214I
a = −0.56251 + 1.90545I
b = 0.454198 + 0.420697I
−1.65381 − 0.64556I −2.86526 + 1.77412I
u = −0.138359 − 0.279214I
a = −0.56251 − 1.90545I
b = 0.454198 − 0.420697I
−1.65381 + 0.64556I −2.86526 − 1.77412I
u = 1.70733
a = 3.19649
b = −5.45746
9.66174 9.65860
u = −1.71885 + 0.02850I
a = −0.023243 − 0.195497I
b = −0.045523 − 0.335367I
11.33560 − 2.65945I 9.22485 + 2.49660I
u = −1.71885 − 0.02850I
a = −0.023243 + 0.195497I
b = −0.045523 + 0.335367I
11.33560 + 2.65945I 9.22485 − 2.49660I
u = 1.73830 + 0.09444I
a = 2.32463 − 0.78043I
b = −4.11462 + 1.13708I
15.1237 + 9.3852I 10.11771 − 5.10224I
u = 1.73830 − 0.09444I
a = 2.32463 + 0.78043I
b = −4.11462 − 1.13708I
15.1237 − 9.3852I 10.11771 + 5.10224I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.75301 + 0.05790I
a = −2.45571 + 0.49075I
b = 4.33329 − 0.71811I
17.0458 + 3.0253I 12.24161 − 0.83109I
u = 1.75301 − 0.05790I
a = −2.45571 − 0.49075I
b = 4.33329 + 0.71811I
17.0458 − 3.0253I 12.24161 + 0.83109I
7
II. I
u
2
= hb − u, a − 1, u
2
+ u − 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u − 1
a
3
=
u
−u + 1
a
7
=
u
−u
a
1
=
−u
u
a
4
=
u
−u + 1
a
9
=
1
u
a
10
=
1
u
a
8
=
u + 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
u
2
− u − 1
c
3
, c
9
u
2
c
4
, c
5
, c
6
u
2
+ u − 1
c
7
(u − 1)
2
c
8
, c
10
(u + 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
y
2
− 3y + 1
c
3
, c
9
y
2
c
7
, c
8
, c
10
(y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.00000
b = 0.618034
−0.657974 3.00000
u = −1.61803
a = 1.00000
b = −1.61803
7.23771 3.00000
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
(u
2
− u − 1)(u
22
− 2u
21
+ ··· − u + 1)
c
3
, c
9
u
2
(u
22
+ u
21
+ ··· − 21u
2
+ 4)
c
4
, c
5
, c
6
(u
2
+ u − 1)(u
22
− 2u
21
+ ··· − u + 1)
c
7
((u − 1)
2
)(u
22
− 3u
21
+ ··· − 8u + 1)
c
8
((u + 1)
2
)(u
22
+ 9u
21
+ ··· + 40u + 1)
c
10
((u + 1)
2
)(u
22
− 3u
21
+ ··· − 8u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
(y
2
− 3y + 1)(y
22
− 30y
21
+ ··· + 3y + 1)
c
3
, c
9
y
2
(y
22
− 15y
21
+ ··· − 168y + 16)
c
7
, c
10
((y −1)
2
)(y
22
− 9y
21
+ ··· − 40y + 1)
c
8
((y −1)
2
)(y
22
+ 11y
21
+ ··· − 1080y + 1)
13
