12n
0159
(K12n
0159
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 9 11 3 5 12 6 10 8
Solving Sequence
6,10
11 7 12 9
3,5
2 1 4 8
c
10
c
6
c
11
c
9
c
5
c
2
c
1
c
4
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
34
+ 2u
33
+ ··· + b − 1, −u
34
+ 5u
32
+ ··· + a + 3u, u
35
+ 2u
34
+ ··· − 2u − 1i
I
u
2
= h−u
5
+ u
3
− u
2
+ b − u, −u
7
+ u
5
− u
4
− u
3
+ a − 1, u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 43 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
34
+2u
33
+· · ·+b−1, −u
34
+5u
32
+· · ·+a+3u, u
35
+2u
34
+· · ·−2u−1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
2
a
9
=
u
4
− u
2
+ 1
u
4
a
3
=
u
34
− 5u
32
+ ··· + 3u
2
− 3u
−u
34
− 2u
33
+ ··· + 2u + 1
a
5
=
−u
9
+ 2u
7
− 3u
5
+ 2u
3
− u
−u
9
+ u
7
− u
5
+ u
a
2
=
u
34
+ u
33
+ ··· − 3u − 1
−u
29
+ 5u
27
+ ··· − u
3
+ 3u
2
a
1
=
u
26
− 5u
24
+ ··· + 3u
2
− 1
u
26
− 4u
24
+ ··· + 2u
4
+ u
2
a
4
=
u
34
+ u
33
+ ··· − 4u − 1
u
34
+ u
33
+ ··· − u − 1
a
8
=
−u
14
+ 3u
12
− 6u
10
+ 7u
8
− 6u
6
+ 4u
4
− 2u
2
+ 1
−u
14
+ 2u
12
− 3u
10
+ 2u
8
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −7u
34
− 6u
33
+ 39u
32
+ 43u
31
− 141u
30
− 173u
29
+ 355u
28
+ 506u
27
− 686u
26
−
1134u
25
+ 1034u
24
+ 2071u
23
− 1204u
22
− 3115u
21
+ 1010u
20
+ 3917u
19
− 424u
18
−
4134u
17
− 365u
16
+ 3614u
15
+ 1028u
14
− 2600u
13
− 1314u
12
+ 1451u
11
+ 1204u
10
−
562u
9
− 821u
8
+ 77u
7
+ 436u
6
+ 85u
5
− 158u
4
− 75u
3
+ 20u
2
+ 26u + 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
35
+ 49u
34
+ ··· + 102u + 1
c
2
, c
4
u
35
− 9u
34
+ ··· − 14u + 1
c
3
, c
7
u
35
− u
34
+ ··· − 640u + 256
c
5
, c
8
u
35
− 2u
34
+ ··· + 108u − 36
c
6
, c
10
u
35
+ 2u
34
+ ··· − 2u − 1
c
9
, c
11
u
35
− 12u
34
+ ··· + 2u − 1
c
12
u
35
+ 36u
33
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
35
− 117y
34
+ ··· + 6150y −1
c
2
, c
4
y
35
− 49y
34
+ ··· + 102y −1
c
3
, c
7
y
35
+ 51y
34
+ ··· + 835584y −65536
c
5
, c
8
y
35
− 12y
34
+ ··· + 6840y −1296
c
6
, c
10
y
35
− 12y
34
+ ··· + 2y −1
c
9
, c
11
y
35
+ 24y
34
+ ··· + 2y −1
c
12
y
35
+ 72y
34
+ ··· + 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.003180 + 0.076770I
a = −0.427602 + 0.732202I
b = 0.040765 − 1.055450I
1.87836 + 2.29361I 9.35864 − 3.99437I
u = 1.003180 − 0.076770I
a = −0.427602 − 0.732202I
b = 0.040765 + 1.055450I
1.87836 − 2.29361I 9.35864 + 3.99437I
u = −0.792898 + 0.645336I
a = −0.232938 − 0.928895I
b = 0.112888 − 0.720959I
−1.72435 − 2.14542I 5.01876 + 4.63119I
u = −0.792898 − 0.645336I
a = −0.232938 + 0.928895I
b = 0.112888 + 0.720959I
−1.72435 + 2.14542I 5.01876 − 4.63119I
u = −0.698567 + 0.764106I
a = 0.18575 + 2.20654I
b = −1.02864 + 2.02071I
−3.83985 + 2.10941I 1.06203 − 1.84479I
u = −0.698567 − 0.764106I
a = 0.18575 − 2.20654I
b = −1.02864 − 2.02071I
−3.83985 − 2.10941I 1.06203 + 1.84479I
u = 0.753011 + 0.738009I
a = 1.37904 − 1.86948I
b = −0.93662 − 2.24841I
−4.73261 + 0.86629I 0.579778 − 0.147183I
u = 0.753011 − 0.738009I
a = 1.37904 + 1.86948I
b = −0.93662 + 2.24841I
−4.73261 − 0.86629I 0.579778 + 0.147183I
u = −0.650584 + 0.839948I
a = −0.26514 − 2.94245I
b = 1.70386 − 2.51843I
−13.3967 + 6.2863I 1.00799 − 1.99078I
u = −0.650584 − 0.839948I
a = −0.26514 + 2.94245I
b = 1.70386 + 2.51843I
−13.3967 − 6.2863I 1.00799 + 1.99078I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.597910 + 0.716545I
a = −0.474108 + 0.553918I
b = 0.273509 + 0.526339I
0.466250 − 0.951396I 10.00295 + 0.38249I
u = 0.597910 − 0.716545I
a = −0.474108 − 0.553918I
b = 0.273509 − 0.526339I
0.466250 + 0.951396I 10.00295 − 0.38249I
u = −0.922847
a = −1.31041
b = −1.21035
0.182011 10.8590
u = −1.08201
a = 0.437613
b = 0.472974
5.82108 17.0620
u = 1.111560 + 0.128216I
a = 1.138050 − 0.133066I
b = 0.300334 + 1.303030I
−6.72567 + 5.75996I 7.32314 − 3.54445I
u = 1.111560 − 0.128216I
a = 1.138050 + 0.133066I
b = 0.300334 − 1.303030I
−6.72567 − 5.75996I 7.32314 + 3.54445I
u = −0.934946 + 0.641378I
a = −0.922641 − 0.641680I
b = 0.090797 − 0.681520I
−1.26680 − 2.86899I 5.76020 + 1.94310I
u = −0.934946 − 0.641378I
a = −0.922641 + 0.641680I
b = 0.090797 + 0.681520I
−1.26680 + 2.86899I 5.76020 − 1.94310I
u = −1.036700 + 0.513296I
a = 0.468566 + 0.238758I
b = 0.355938 − 0.949026I
−9.06115 − 1.11837I 4.93303 + 2.48933I
u = −1.036700 − 0.513296I
a = 0.468566 − 0.238758I
b = 0.355938 + 0.949026I
−9.06115 + 1.11837I 4.93303 − 2.48933I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.958291 + 0.699514I
a = 1.42433 − 1.41318I
b = −0.04365 − 2.87306I
−4.10408 + 4.62202I 2.53243 − 5.37025I
u = 0.958291 − 0.699514I
a = 1.42433 + 1.41318I
b = −0.04365 + 2.87306I
−4.10408 − 4.62202I 2.53243 + 5.37025I
u = 0.878474 + 0.799434I
a = −2.09357 + 2.58300I
b = 0.51980 + 3.81456I
−17.4916 + 2.9871I −0.26712 − 2.67515I
u = 0.878474 − 0.799434I
a = −2.09357 − 2.58300I
b = 0.51980 − 3.81456I
−17.4916 − 2.9871I −0.26712 + 2.67515I
u = 1.021530 + 0.658784I
a = −0.388047 + 0.431455I
b = −0.012993 + 0.931220I
1.70279 + 6.25040I 12.13050 − 4.97456I
u = 1.021530 − 0.658784I
a = −0.388047 − 0.431455I
b = −0.012993 − 0.931220I
1.70279 − 6.25040I 12.13050 + 4.97456I
u = −0.993451 + 0.702180I
a = 2.14762 + 0.79394I
b = 0.45485 + 2.36980I
−2.94898 − 7.68050I 3.12136 + 6.96771I
u = −0.993451 − 0.702180I
a = 2.14762 − 0.79394I
b = 0.45485 − 2.36980I
−2.94898 + 7.68050I 3.12136 − 6.96771I
u = −0.263163 + 0.716876I
a = −1.150580 + 0.479086I
b = 0.686293 − 0.231426I
−11.29280 − 3.30354I 1.02998 + 2.25929I
u = −0.263163 − 0.716876I
a = −1.150580 − 0.479086I
b = 0.686293 + 0.231426I
−11.29280 + 3.30354I 1.02998 − 2.25929I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.039590 + 0.718934I
a = −2.70067 − 0.80992I
b = −1.22822 − 3.37949I
−12.2131 − 12.1134I 2.82361 + 6.65221I
u = −1.039590 − 0.718934I
a = −2.70067 + 0.80992I
b = −1.22822 + 3.37949I
−12.2131 + 12.1134I 2.82361 − 6.65221I
u = 0.515516
a = −0.450972
b = 0.317465
0.694754 14.6380
u = −0.169379 + 0.388841I
a = 0.57383 − 1.55781I
b = −0.578952 − 0.351526I
−1.66775 − 0.90576I −1.19698 + 2.88649I
u = −0.169379 − 0.388841I
a = 0.57383 + 1.55781I
b = −0.578952 + 0.351526I
−1.66775 + 0.90576I −1.19698 − 2.88649I
8
II. I
u
2
= h−u
5
+ u
3
− u
2
+ b − u, −u
7
+ u
5
− u
4
− u
3
+ a − 1, u
8
− u
7
− u
6
+
2u
5
+ u
4
− 2u
3
+ 2u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
2
a
9
=
u
4
− u
2
+ 1
u
4
a
3
=
u
7
− u
5
+ u
4
+ u
3
+ 1
u
5
− u
3
+ u
2
+ u
a
5
=
u
6
− u
4
+ 2u
2
− 1
−u
7
+ u
6
+ 2u
5
− u
4
− 2u
3
+ 2u
2
+ 2u − 1
a
2
=
u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
+ 2
u
7
− u
6
− u
5
+ u
4
+ u
3
− u
2
− u + 1
a
1
=
−u
6
+ u
4
− 2u
2
+ 1
u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u
2
− 2u + 1
a
4
=
u
7
− u
5
+ u
4
+ u
3
+ 1
u
5
− u
3
+ u
2
+ u
a
8
=
u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
7
+ u
6
− 5u
5
+ 5u
3
− u
2
− 4u + 5
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
7
u
8
c
4
(u + 1)
8
c
5
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
6
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
8
, c
12
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
9
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
10
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
11
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
7
y
8
c
5
, c
8
, c
12
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
6
, c
10
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
9
, c
11
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570868 + 0.730671I
a = 0.325934 + 0.693334I
b = 0.972127 + 0.565636I
−0.604279 − 1.131230I 1.47926 + 0.84929I
u = 0.570868 − 0.730671I
a = 0.325934 − 0.693334I
b = 0.972127 − 0.565636I
−0.604279 + 1.131230I 1.47926 − 0.84929I
u = −0.855237 + 0.665892I
a = −1.03462 − 0.99451I
b = 0.39611 − 1.88650I
−3.80435 − 2.57849I 2.50535 + 3.23297I
u = −0.855237 − 0.665892I
a = −1.03462 + 0.99451I
b = 0.39611 + 1.88650I
−3.80435 + 2.57849I 2.50535 − 3.23297I
u = −1.09818
a = 0.801005
b = −0.165005
4.85780 7.45240
u = 1.031810 + 0.655470I
a = −0.842429 − 0.289836I
b = −0.699541 + 1.033710I
0.73474 + 6.44354I 3.27544 − 5.90525I
u = 1.031810 − 0.655470I
a = −0.842429 + 0.289836I
b = −0.699541 − 1.033710I
0.73474 − 6.44354I 3.27544 + 5.90525I
u = 0.603304
a = 1.30123
b = 0.827616
−0.799899 3.02750
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
35
+ 49u
34
+ ··· + 102u + 1)
c
2
((u − 1)
8
)(u
35
− 9u
34
+ ··· − 14u + 1)
c
3
, c
7
u
8
(u
35
− u
34
+ ··· − 640u + 256)
c
4
((u + 1)
8
)(u
35
− 9u
34
+ ··· − 14u + 1)
c
5
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
35
− 2u
34
+ ··· + 108u − 36)
c
6
(u
8
+ u
7
+ ··· − 2u − 1)(u
35
+ 2u
34
+ ··· − 2u − 1)
c
8
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
35
− 2u
34
+ ··· + 108u − 36)
c
9
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
35
− 12u
34
+ ··· + 2u − 1)
c
10
(u
8
− u
7
+ ··· + 2u − 1)(u
35
+ 2u
34
+ ··· − 2u − 1)
c
11
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
35
− 12u
34
+ ··· + 2u − 1)
c
12
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
35
+ 36u
33
+ ··· − 4u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
35
− 117y
34
+ ··· + 6150y −1)
c
2
, c
4
((y −1)
8
)(y
35
− 49y
34
+ ··· + 102y −1)
c
3
, c
7
y
8
(y
35
+ 51y
34
+ ··· + 835584y −65536)
c
5
, c
8
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
35
− 12y
34
+ ··· + 6840y −1296)
c
6
, c
10
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
35
− 12y
34
+ ··· + 2y −1)
c
9
, c
11
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
35
+ 24y
34
+ ··· + 2y −1)
c
12
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
35
+ 72y
34
+ ··· + 2y −1)
14
