12n
0166
(K12n
0166
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 11 3 5 12 6 7 9
Solving Sequence
6,10
11
3,7
8 12 5 2 1 4 9
c
10
c
6
c
7
c
11
c
5
c
2
c
1
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
25
− 14u
23
+ ··· + b − 1, 2u
25
− u
24
+ ··· + a − 5u, u
26
− 2u
25
+ ··· + 3u + 1i
I
u
2
= hu
4
− 2u
2
+ b + u, −u
5
+ 3u
3
+ a − u + 1, u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 32 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
25
−14u
23
+· · ·+b−1, 2u
25
−u
24
+· · ·+a−5u, u
26
−2u
25
+· · ·+3u+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
3
=
−2u
25
+ u
24
+ ··· + 9u
2
+ 5u
−u
25
+ 14u
23
+ ··· + 4u + 1
a
7
=
−u
−u
3
+ u
a
8
=
−u
10
+ 5u
8
− 8u
6
+ 5u
4
− 3u
2
+ 1
−u
10
+ 4u
8
− 3u
6
− 2u
4
− u
2
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
5
=
u
u
a
2
=
−u
25
+ u
24
+ ··· + u − 1
u
22
− 12u
20
+ ··· − 4u
3
− 3u
2
a
1
=
u
10
− 5u
8
+ 8u
6
− 5u
4
+ 3u
2
− 1
u
12
− 6u
10
+ 12u
8
− 8u
6
+ u
4
− 2u
2
a
4
=
u
24
− 13u
22
+ ··· − u − 1
u
25
− 14u
23
+ ··· − 3u − 1
a
9
=
−u
6
+ 3u
4
− 2u
2
+ 1
−u
8
+ 4u
6
− 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
25
+ 5u
24
+ 51u
23
− 62u
22
− 269u
21
+ 312u
20
+ 760u
19
−
813u
18
− 1264u
17
+ 1176u
16
+ 1341u
15
− 1013u
14
− 1049u
13
+ 682u
12
+ 692u
11
−
399u
10
− 430u
9
+ 65u
8
+ 311u
7
− 6u
6
− 115u
5
− 18u
4
+ 50u
3
+ 9u
2
− 6u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
+ 37u
25
+ ··· + 76u + 1
c
2
, c
4
u
26
− 7u
25
+ ··· − 2u − 1
c
3
, c
7
u
26
− u
25
+ ··· − 128u − 64
c
5
, c
6
, c
10
c
11
u
26
− 2u
25
+ ··· + 3u + 1
c
8
u
26
+ 2u
25
+ ··· + 3u + 1
c
9
, c
12
u
26
− 6u
25
+ ··· − 21u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
− 89y
25
+ ··· − 2892y + 1
c
2
, c
4
y
26
− 37y
25
+ ··· − 76y + 1
c
3
, c
7
y
26
− 39y
25
+ ··· − 12288y + 4096
c
5
, c
6
, c
10
c
11
y
26
− 30y
25
+ ··· − 11y + 1
c
8
y
26
− 54y
25
+ ··· − 11y + 1
c
9
, c
12
y
26
+ 6y
25
+ ··· − 531y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.681707 + 0.582351I
a = −0.132311 + 0.107455I
b = −1.80048 − 0.98897I
−10.93830 + 7.21103I −16.0759 − 5.4438I
u = −0.681707 − 0.582351I
a = −0.132311 − 0.107455I
b = −1.80048 + 0.98897I
−10.93830 − 7.21103I −16.0759 + 5.4438I
u = 1.175060 + 0.078346I
a = 0.222046 + 0.015295I
b = 1.71753 − 0.10327I
−14.2898 + 0.0080I −18.3523 + 0.3239I
u = 1.175060 − 0.078346I
a = 0.222046 − 0.015295I
b = 1.71753 + 0.10327I
−14.2898 − 0.0080I −18.3523 − 0.3239I
u = −0.615423 + 0.435220I
a = 0.006046 + 0.650453I
b = 1.43706 + 0.60644I
−1.64268 + 3.44770I −15.9366 − 6.5929I
u = −0.615423 − 0.435220I
a = 0.006046 − 0.650453I
b = 1.43706 − 0.60644I
−1.64268 − 3.44770I −15.9366 + 6.5929I
u = 0.492369 + 0.545154I
a = 0.033687 + 0.462693I
b = 0.322628 + 0.025417I
2.35945 − 1.88336I −5.73263 + 3.81073I
u = 0.492369 − 0.545154I
a = 0.033687 − 0.462693I
b = 0.322628 − 0.025417I
2.35945 + 1.88336I −5.73263 − 3.81073I
u = −0.265310 + 0.672765I
a = 0.80138 + 1.96514I
b = 0.117100 − 0.374073I
−9.70379 − 2.98173I −13.78370 + 0.17341I
u = −0.265310 − 0.672765I
a = 0.80138 − 1.96514I
b = 0.117100 + 0.374073I
−9.70379 + 2.98173I −13.78370 − 0.17341I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.589835 + 0.287549I
a = 0.399854 − 1.018310I
b = −1.126490 + 0.098071I
−2.66891 − 0.88385I −16.9206 + 6.0063I
u = 0.589835 − 0.287549I
a = 0.399854 + 1.018310I
b = −1.126490 − 0.098071I
−2.66891 + 0.88385I −16.9206 − 6.0063I
u = −0.277498 + 0.391559I
a = −0.71760 − 1.38397I
b = −0.619638 + 0.253799I
−0.683753 − 0.414385I −12.43905 − 0.47517I
u = −0.277498 − 0.391559I
a = −0.71760 + 1.38397I
b = −0.619638 − 0.253799I
−0.683753 + 0.414385I −12.43905 + 0.47517I
u = 1.52883 + 0.05644I
a = 0.949715 − 0.290539I
b = 0.830011 + 0.081100I
−6.94574 − 0.62089I −15.5634 − 0.9743I
u = 1.52883 − 0.05644I
a = 0.949715 + 0.290539I
b = 0.830011 − 0.081100I
−6.94574 + 0.62089I −15.5634 + 0.9743I
u = −1.52725 + 0.15077I
a = −0.741655 − 0.245504I
b = −1.002680 − 0.376527I
−4.34854 + 4.33683I −10.06939 − 2.72465I
u = −1.52725 − 0.15077I
a = −0.741655 + 0.245504I
b = −1.002680 + 0.376527I
−4.34854 − 4.33683I −10.06939 + 2.72465I
u = −1.57781 + 0.08698I
a = 2.47671 + 0.53175I
b = 3.33781 + 0.79397I
−10.09930 + 2.28663I −19.0760 − 2.3439I
u = −1.57781 − 0.08698I
a = 2.47671 − 0.53175I
b = 3.33781 − 0.79397I
−10.09930 − 2.28663I −19.0760 + 2.3439I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.57863 + 0.12294I
a = −2.06274 + 1.32680I
b = −2.70573 + 0.88516I
−9.08260 − 5.47988I −18.5451 + 4.2333I
u = 1.57863 − 0.12294I
a = −2.06274 − 1.32680I
b = −2.70573 − 0.88516I
−9.08260 + 5.47988I −18.5451 − 4.2333I
u = 1.59730 + 0.17727I
a = 2.38789 − 2.06908I
b = 3.52206 − 2.02315I
−18.6017 − 10.0445I −18.6964 + 4.4096I
u = 1.59730 − 0.17727I
a = 2.38789 + 2.06908I
b = 3.52206 + 2.02315I
−18.6017 + 10.0445I −18.6964 − 4.4096I
u = −0.383361
a = −0.709996
b = −0.351806
−0.582197 −16.9580
u = −1.65067
a = −3.53607
b = −4.70658
15.9598 −20.6600
7
II.
I
u
2
= hu
4
−2u
2
+b+u, −u
5
+3u
3
+a−u+1, u
6
+u
5
−3u
4
−2u
3
+2u
2
−u−1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
3
=
u
5
− 3u
3
+ u − 1
−u
4
+ 2u
2
− u
a
7
=
−u
−u
3
+ u
a
8
=
−u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
5
=
u
u
a
2
=
u
5
− 3u
3
− 1
−u
4
+ 2u
2
− 2u
a
1
=
−u
−u
a
4
=
u
5
− 3u
3
+ u − 1
−u
4
+ 2u
2
− u
a
9
=
u
5
− 2u
3
− u
u
5
− 3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
5
− u
4
+ 6u
3
+ u
2
+ 2u − 14
8
(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
, c
12
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
10
, c
11
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
9
(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
c
11
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
8
, c
9
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.493180 + 0.575288I
a = 0.504580 − 0.342767I
b = −0.354346 + 0.659157I
1.31531 − 1.97241I −14.7121 + 3.8836I
u = 0.493180 − 0.575288I
a = 0.504580 + 0.342767I
b = −0.354346 − 0.659157I
1.31531 + 1.97241I −14.7121 − 3.8836I
u = −0.483672
a = −1.17069
b = 0.896823
−2.38379 −15.3880
u = −1.52087 + 0.16310I
a = 0.462019 + 1.043570I
b = 1.11206 + 1.11328I
−5.34051 + 4.59213I −18.4963 − 3.9250I
u = −1.52087 − 0.16310I
a = 0.462019 − 1.043570I
b = 1.11206 − 1.11328I
−5.34051 − 4.59213I −18.4963 + 3.9250I
u = 1.53904
a = −1.76250
b = −2.41226
−9.30502 −18.1960
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
26
+ 37u
25
+ ··· + 76u + 1)
c
2
((u − 1)
6
)(u
26
− 7u
25
+ ··· − 2u − 1)
c
3
, c
7
u
6
(u
26
− u
25
+ ··· − 128u − 64)
c
4
((u + 1)
6
)(u
26
− 7u
25
+ ··· − 2u − 1)
c
5
, c
6
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
26
− 2u
25
+ ··· + 3u + 1)
c
8
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
26
+ 2u
25
+ ··· + 3u + 1)
c
9
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
26
− 6u
25
+ ··· − 21u − 9)
c
10
, c
11
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
26
− 2u
25
+ ··· + 3u + 1)
c
12
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
26
− 6u
25
+ ··· − 21u − 9)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
26
− 89y
25
+ ··· − 2892y + 1)
c
2
, c
4
((y −1)
6
)(y
26
− 37y
25
+ ··· − 76y + 1)
c
3
, c
7
y
6
(y
26
− 39y
25
+ ··· − 12288y + 4096)
c
5
, c
6
, c
10
c
11
(y
6
− 7y
5
+ ··· − 5y + 1)(y
26
− 30y
25
+ ··· − 11y + 1)
c
8
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
26
− 54y
25
+ ··· − 11y + 1)
c
9
, c
12
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
26
+ 6y
25
+ ··· − 531y + 81)
13
