12n
0689
(K12n
0689
)
A knot diagram
1
Linearized knot diagam
4 5 8 2 11 10 3 12 5 6 8 9
Solving Sequence
5,11 3,6
2 4 1 10 7 8 9 12
c
5
c
2
c
4
c
1
c
10
c
6
c
7
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−249863221u
14
− 685643435u
13
+ ··· + 7149815356b + 6972584040,
− 9675904336u
14
− 21495350106u
13
+ ··· + 21449446068a + 126465591430,
u
15
+ 2u
14
+ ··· − 4u + 4i
I
u
2
= hb + 1, 2u
5
+ 4u
4
+ 7u
3
+ 8u
2
+ 3a + 6u + 5, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
I
u
3
= h−au + 3b + 2a − 3, 2a
2
− au − 2a − 2u − 1, u
2
+ 2i
I
v
1
= ha, b − v −2, v
2
+ 3v + 1i
* 4 irreducible components of dim
C
= 0, with total 27 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
= h−2.50 × 10
8
u
14
− 6.86 × 10
8
u
13
+ · · · + 7.15 × 10
9
b + 6.97 × 10
9
, −9.68 ×
10
9
u
14
−2.15×10
10
u
13
+· · ·+2.14×10
10
a+1.26×10
11
, u
15
+2u
14
+· · ·−4u+4i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
0.451103u
14
+ 1.00214u
13
+ ··· − 17.5109u − 5.89598
0.0349468u
14
+ 0.0958967u
13
+ ··· − 1.33742u − 0.975212
a
6
=
1
u
2
a
2
=
0.486050u
14
+ 1.09804u
13
+ ··· − 18.8483u − 6.87120
0.0349468u
14
+ 0.0958967u
13
+ ··· − 1.33742u − 0.975212
a
4
=
0.294788u
14
+ 0.657010u
13
+ ··· − 14.7148u − 4.23924
−0.0459078u
14
− 0.0956177u
13
+ ··· − 0.878005u − 0.288978
a
1
=
0.130153u
14
+ 0.257727u
13
+ ··· − 6.22583u − 2.21939
−0.0158172u
14
− 0.0711109u
13
+ ··· + 0.500235u − 0.0616192
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
0.201267u
14
+ 0.468199u
13
+ ··· − 7.66259u − 2.24374
0.0648205u
14
+ 0.172140u
13
+ ··· − 1.47894u − 0.348628
a
9
=
u
3
+ 2u
u
3
+ u
a
12
=
0.0813622u
14
+ 0.162342u
13
+ ··· − 5.15784u − 1.63915
−0.0550842u
14
− 0.133716u
13
+ ··· + 1.02581u + 0.255966
(ii) Obstruction class = −1
(iii) Cusp Shapes =
20309339548
16087084551
u
14
+
5027559920
1787453839
u
13
+ ··· −
319023539806
5362361517
u −
592867371580
16087084551
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
15
− 10u
14
+ ··· − 57u − 9
c
3
, c
7
u
15
− 2u
14
+ ··· − 192u + 576
c
5
, c
6
, c
10
u
15
+ 2u
14
+ ··· − 4u + 4
c
8
, c
11
, c
12
u
15
+ 4u
14
+ ··· − 37u − 19
c
9
u
15
− 2u
14
+ ··· − 6348u + 2116
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
15
+ 52y
13
+ ··· + 3177y −81
c
3
, c
7
y
15
+ 66y
14
+ ··· + 4349952y −331776
c
5
, c
6
, c
10
y
15
+ 24y
14
+ ··· + 336y −16
c
8
, c
11
, c
12
y
15
− 2y
14
+ ··· + 4561y −361
c
9
y
15
+ 144y
14
+ ··· + 53534800y −4477456
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.016193 + 1.236613I
a = 0.649765 − 0.309826I
b = −0.019483 + 0.380184I
3.11226 + 1.37153I −7.52048 − 4.70500I
u = −0.016193 − 1.236613I
a = 0.649765 + 0.309826I
b = −0.019483 − 0.380184I
3.11226 − 1.37153I −7.52048 + 4.70500I
u = −0.382234 + 0.648228I
a = 0.093614 + 0.314483I
b = −0.910726 − 0.608079I
−0.561570 − 0.602510I −12.89865 + 0.16991I
u = −0.382234 − 0.648228I
a = 0.093614 − 0.314483I
b = −0.910726 + 0.608079I
−0.561570 + 0.602510I −12.89865 − 0.16991I
u = 0.481765
a = 0.876017
b = 1.52879
−10.9150 −27.4910
u = 0.43189 + 1.48192I
a = −0.381495 − 0.081254I
b = 1.32115 − 0.59294I
−5.53981 − 3.37298I −14.3863 + 0.4326I
u = 0.43189 − 1.48192I
a = −0.381495 + 0.081254I
b = 1.32115 + 0.59294I
−5.53981 + 3.37298I −14.3863 − 0.4326I
u = −1.01904 + 1.21960I
a = 0.793015 + 0.629146I
b = 1.05096 − 1.67638I
4.94872 + 5.24403I −12.95978 − 2.95789I
u = −1.01904 − 1.21960I
a = 0.793015 − 0.629146I
b = 1.05096 + 1.67638I
4.94872 − 5.24403I −12.95978 + 2.95789I
u = −0.358485
a = 0.777590
b = −0.141539
−0.594411 −16.4650
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.269953
a = −9.24911
b = −0.853104
−2.86090 −50.4000
u = −0.39688 + 1.74630I
a = 0.064779 + 1.248157I
b = 1.80094 − 0.90555I
14.2231 + 11.2321I −13.44172 − 4.13184I
u = −0.39688 − 1.74630I
a = 0.064779 − 1.248157I
b = 1.80094 + 0.90555I
14.2231 − 11.2321I −13.44172 + 4.13184I
u = 0.18584 + 2.25780I
a = −0.088591 − 0.982699I
b = 1.49008 + 2.48794I
18.1439 − 0.8477I −12.17052 + 0.18757I
u = 0.18584 − 2.25780I
a = −0.088591 + 0.982699I
b = 1.49008 − 2.48794I
18.1439 + 0.8477I −12.17052 − 0.18757I
6
II.
I
u
2
= hb+1, 2u
5
+4u
4
+7u
3
+8u
2
+3a+6u+5, u
6
+u
5
+3u
4
+2u
3
+2u
2
+u−1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−
2
3
u
5
−
4
3
u
4
+ ··· − 2u −
5
3
−1
a
6
=
1
u
2
a
2
=
−
2
3
u
5
−
4
3
u
4
+ ··· − 2u −
8
3
−1
a
4
=
−
2
3
u
5
−
4
3
u
4
+ ··· − 2u −
5
3
−1
a
1
=
−1
0
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
u
3
+ 2u
u
3
+ u
a
12
=
−u
5
− 2u
3
− u
−u
5
− u
4
− 2u
3
− u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =
7
9
u
5
−
31
9
u
4
−
10
9
u
3
−
41
9
u
2
− 2u −
110
9
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
7
u
6
c
4
(u + 1)
6
c
5
, c
6
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
8
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
9
, c
11
, c
12
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
10
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
6
, c
10
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
c
8
, c
9
, c
11
c
12
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873214
a = −0.836730
b = −1.00000
−9.30502 −15.6070
u = 0.138835 + 1.234445I
a = −0.366605 + 0.544193I
b = −1.00000
1.31531 − 1.97241I −11.11410 + 3.48248I
u = 0.138835 − 1.234445I
a = −0.366605 − 0.544193I
b = −1.00000
1.31531 + 1.97241I −11.11410 − 3.48248I
u = −0.408802 + 1.276377I
a = 0.031424 − 0.540243I
b = −1.00000
−5.34051 + 4.59213I −13.8624 − 6.6392I
u = −0.408802 − 1.276377I
a = 0.031424 + 0.540243I
b = −1.00000
−5.34051 − 4.59213I −13.8624 + 6.6392I
u = 0.413150
a = −3.15957
b = −1.00000
−2.38379 −13.9950
10
III. I
u
3
= h−au + 3b + 2a − 3, 2a
2
− au − 2a − 2u − 1, u
2
+ 2i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
a
1
3
au −
2
3
a + 1
a
6
=
1
−2
a
2
=
1
3
au +
1
3
a + 1
1
3
au −
2
3
a + 1
a
4
=
−
1
3
au +
2
3
a +
1
2
u
−
1
3
au +
2
3
a − 2
a
1
=
1
2
u
−
1
3
au +
2
3
a − 2
a
10
=
u
−u
a
7
=
−1
0
a
8
=
1
2
u
−
1
3
au +
2
3
a − 2
a
9
=
0
−u
a
12
=
1
2
u
−
1
3
au +
2
3
a + u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u
2
+ u − 1)
2
c
3
, c
4
(u
2
− u − 1)
2
c
5
, c
6
, c
9
c
10
(u
2
+ 2)
2
c
8
(u + 1)
4
c
11
, c
12
(u − 1)
4
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
(y
2
− 3y + 1)
2
c
5
, c
6
, c
9
c
10
(y + 2)
4
c
8
, c
11
, c
12
(y −1)
4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = −0.618034 − 0.437016I
b = 1.61803
−5.59278 −16.0000
u = 1.414210I
a = 1.61803 + 1.14412I
b = −0.618034
2.30291 −16.0000
u = − 1.414210I
a = −0.618034 + 0.437016I
b = 1.61803
−5.59278 −16.0000
u = − 1.414210I
a = 1.61803 − 1.14412I
b = −0.618034
2.30291 −16.0000
14
IV. I
v
1
= ha, b − v − 2, v
2
+ 3v + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
v
0
a
3
=
0
v + 2
a
6
=
1
0
a
2
=
v + 2
v + 2
a
4
=
−v − 2
−v − 3
a
1
=
−1
−v − 3
a
10
=
v
0
a
7
=
1
0
a
8
=
1
v + 3
a
9
=
v
0
a
12
=
v − 1
−v − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
u
2
+ u − 1
c
4
, c
7
u
2
− u − 1
c
5
, c
6
, c
9
c
10
u
2
c
8
(u − 1)
2
c
11
, c
12
(u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
y
2
− 3y + 1
c
5
, c
6
, c
9
c
10
y
2
c
8
, c
11
, c
12
(y − 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.381966
a = 0
b = 1.61803
−10.5276 −6.00000
v = −2.61803
a = 0
b = −0.618034
−2.63189 −6.00000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
6
)(u
2
+ u − 1)
3
(u
15
− 10u
14
+ ··· − 57u − 9)
c
3
u
6
(u
2
− u − 1)
2
(u
2
+ u − 1)(u
15
− 2u
14
+ ··· − 192u + 576)
c
4
((u + 1)
6
)(u
2
− u − 1)
3
(u
15
− 10u
14
+ ··· − 57u − 9)
c
5
, c
6
u
2
(u
2
+ 2)
2
(u
6
+ u
5
+ ··· + u − 1)(u
15
+ 2u
14
+ ··· − 4u + 4)
c
7
u
6
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
15
− 2u
14
+ ··· − 192u + 576)
c
8
(u − 1)
2
(u + 1)
4
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
15
+ 4u
14
+ ··· − 37u − 19)
c
9
u
2
(u
2
+ 2)
2
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
15
− 2u
14
+ ··· − 6348u + 2116)
c
10
u
2
(u
2
+ 2)
2
(u
6
− u
5
+ ··· − u − 1)(u
15
+ 2u
14
+ ··· − 4u + 4)
c
11
, c
12
(u − 1)
4
(u + 1)
2
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
15
+ 4u
14
+ ··· − 37u − 19)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y − 1)
6
)(y
2
− 3y + 1)
3
(y
15
+ 52y
13
+ ··· + 3177y − 81)
c
3
, c
7
y
6
(y
2
− 3y + 1)
3
(y
15
+ 66y
14
+ ··· + 4349952y − 331776)
c
5
, c
6
, c
10
y
2
(y + 2)
4
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
15
+ 24y
14
+ ··· + 336y − 16)
c
8
, c
11
, c
12
(y − 1)
6
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
15
− 2y
14
+ ··· + 4561y − 361)
c
9
y
2
(y + 2)
4
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
15
+ 144y
14
+ ··· + 53534800y − 4477456)
20
