12n
0239
(K12n
0239
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 12 11 10 3 5 6 7 9
Solving Sequence
6,11
7 12
3,5
2 1 4 10 8 9
c
6
c
11
c
5
c
2
c
1
c
4
c
10
c
7
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
28
− u
27
+ ··· + b + u, 3u
28
+ 3u
27
+ ··· + a + 1, u
29
+ 2u
28
+ ··· − u + 1i
I
u
2
= h−u
3
+ b + u + 1, −u
6
+ 3u
4
+ u
3
− 2u
2
+ a − u − 2, u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 37 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−u
28
−u
27
+· · ·+b+u, 3u
28
+3u
27
+· · ·+a+1, u
29
+2u
28
+· · ·−u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
−u
3
+ u
a
3
=
−3u
28
− 3u
27
+ ··· − u − 1
u
28
+ u
27
+ ··· + 6u
2
− u
a
5
=
−u
4
+ u
2
+ 1
u
6
− 2u
4
+ u
2
a
2
=
−2u
28
− 2u
27
+ ··· − 3u − 1
u
28
+ u
27
+ ··· + 5u
2
− u
a
1
=
−u
21
+ 8u
19
+ ··· − 6u
3
− u
u
23
− 9u
21
+ ··· + 4u
3
− u
a
4
=
−u
28
− u
27
+ ··· − 5u
2
− 3u
u
28
+ u
27
+ ··· + 8u
3
+ 5u
2
a
10
=
−u
u
a
8
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
a
9
=
u
11
− 4u
9
+ 4u
7
+ 2u
5
− 3u
3
− 2u
−u
13
+ 5u
11
− 9u
9
+ 6u
7
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −5u
28
− 4u
27
+ 53u
26
+ 33u
25
− 247u
24
− 102u
23
+ 634u
22
+ 96u
21
− 879u
20
+
214u
19
+ 362u
18
− 658u
17
+ 771u
16
+ 486u
15
− 1175u
14
+ 371u
13
+ 217u
12
− 670u
11
+
671u
10
− 12u
9
− 364u
8
+ 380u
7
− 164u
6
− 26u
5
+ 109u
4
− 112u
3
+ 24u
2
− 15u − 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
29
+ 45u
28
+ ··· + 63u + 1
c
2
, c
4
u
29
− 9u
28
+ ··· − 15u + 1
c
3
, c
8
u
29
+ u
28
+ ··· − 640u + 256
c
5
, c
7
u
29
+ 6u
28
+ ··· − 27u − 7
c
6
, c
10
, c
11
u
29
− 2u
28
+ ··· − u − 1
c
9
u
29
+ 2u
28
+ ··· + 847u − 505
c
12
u
29
+ 30u
27
+ ··· − u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
29
− 113y
28
+ ··· + 3195y −1
c
2
, c
4
y
29
− 45y
28
+ ··· + 63y −1
c
3
, c
8
y
29
+ 51y
28
+ ··· + 835584y −65536
c
5
, c
7
y
29
+ 24y
28
+ ··· + 239y −49
c
6
, c
10
, c
11
y
29
− 24y
28
+ ··· − y −1
c
9
y
29
+ 24y
28
+ ··· − 2572161y −255025
c
12
y
29
+ 60y
28
+ ··· − y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.116211 + 0.866176I
a = 0.518979 − 0.390879I
b = 0.83718 − 3.00348I
−18.1545 + 6.4457I −0.28264 − 3.13315I
u = 0.116211 − 0.866176I
a = 0.518979 + 0.390879I
b = 0.83718 + 3.00348I
−18.1545 − 6.4457I −0.28264 + 3.13315I
u = 0.025688 + 0.825088I
a = −0.594569 + 0.459352I
b = −0.23014 + 2.32002I
−6.93251 + 1.75256I −0.94314 − 1.34488I
u = 0.025688 − 0.825088I
a = −0.594569 − 0.459352I
b = −0.23014 − 2.32002I
−6.93251 − 1.75256I −0.94314 + 1.34488I
u = 1.19941
a = 2.02803
b = −1.96066
0.978770 8.30090
u = 1.149060 + 0.431164I
a = −2.80477 − 0.74741I
b = 1.17842 + 2.47370I
−14.9885 − 1.7991I 2.59660 − 0.52377I
u = 1.149060 − 0.431164I
a = −2.80477 + 0.74741I
b = 1.17842 − 2.47370I
−14.9885 + 1.7991I 2.59660 + 0.52377I
u = −0.078085 + 0.755576I
a = 0.230600 − 0.278548I
b = −0.073065 − 0.703159I
−2.53613 − 2.06791I 5.52324 + 3.00073I
u = −0.078085 − 0.755576I
a = 0.230600 + 0.278548I
b = −0.073065 + 0.703159I
−2.53613 + 2.06791I 5.52324 − 3.00073I
u = −1.215610 + 0.296484I
a = 0.992404 − 0.146386I
b = −0.306342 + 0.431513I
0.91968 − 1.73508I 8.51016 + 0.61510I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.215610 − 0.296484I
a = 0.992404 + 0.146386I
b = −0.306342 − 0.431513I
0.91968 + 1.73508I 8.51016 − 0.61510I
u = 0.497421 + 0.544735I
a = −1.52270 + 0.49021I
b = 0.327073 − 0.572505I
−12.40890 + 1.96182I 2.44090 − 3.22875I
u = 0.497421 − 0.544735I
a = −1.52270 − 0.49021I
b = 0.327073 + 0.572505I
−12.40890 − 1.96182I 2.44090 + 3.22875I
u = −1.274370 + 0.082675I
a = 0.269484 + 1.345140I
b = 0.022151 − 0.326489I
2.70011 − 2.01756I 7.88389 + 3.97312I
u = −1.274370 − 0.082675I
a = 0.269484 − 1.345140I
b = 0.022151 + 0.326489I
2.70011 + 2.01756I 7.88389 − 3.97312I
u = 1.248070 + 0.369947I
a = 1.89588 + 1.19040I
b = −0.78649 − 2.39370I
−3.15292 + 2.54446I 2.89231 − 2.35754I
u = 1.248070 − 0.369947I
a = 1.89588 − 1.19040I
b = −0.78649 + 2.39370I
−3.15292 − 2.54446I 2.89231 + 2.35754I
u = −1.289030 + 0.369909I
a = −2.15007 + 1.65647I
b = 0.23955 − 2.15882I
−2.83662 − 6.04967I 3.35930 + 4.53104I
u = −1.289030 − 0.369909I
a = −2.15007 − 1.65647I
b = 0.23955 + 2.15882I
−2.83662 + 6.04967I 3.35930 − 4.53104I
u = 1.35324
a = −0.560180
b = 0.713936
5.98164 16.6760
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.321100 + 0.324348I
a = −0.492619 − 0.546231I
b = 0.144058 + 0.921802I
1.85790 + 5.97624I 10.99426 − 4.76972I
u = 1.321100 − 0.324348I
a = −0.492619 + 0.546231I
b = 0.144058 − 0.921802I
1.85790 − 5.97624I 10.99426 + 4.76972I
u = −1.350820 + 0.384218I
a = 1.84401 − 2.84917I
b = 0.46306 + 3.28611I
−13.5440 − 10.9372I 3.78628 + 5.28747I
u = −1.350820 − 0.384218I
a = 1.84401 + 2.84917I
b = 0.46306 − 3.28611I
−13.5440 + 10.9372I 3.78628 − 5.28747I
u = −1.41186 + 0.13721I
a = −0.04816 − 1.43092I
b = −0.645077 + 1.183340I
−6.30066 − 4.18073I 6.94767 + 2.86022I
u = −1.41186 − 0.13721I
a = −0.04816 + 1.43092I
b = −0.645077 − 1.183340I
−6.30066 + 4.18073I 6.94767 − 2.86022I
u = −0.378524
a = 0.615391
b = 0.176038
0.641421 15.6290
u = 0.175172 + 0.291603I
a = 0.31991 − 2.02255I
b = −0.635045 + 0.442420I
−1.62334 + 0.70173I −1.51173 − 3.13517I
u = 0.175172 − 0.291603I
a = 0.31991 + 2.02255I
b = −0.635045 − 0.442420I
−1.62334 − 0.70173I −1.51173 + 3.13517I
7
II. I
u
2
=
h−u
3
+b+u+1, −u
6
+3u
4
+u
3
−2u
2
+a−u−2, u
8
−u
7
−3u
6
+2u
5
+3u
4
−2u−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
−u
3
+ u
a
3
=
u
6
− 3u
4
− u
3
+ 2u
2
+ u + 2
u
3
− u − 1
a
5
=
−u
4
+ u
2
+ 1
u
6
− 2u
4
+ u
2
a
2
=
u
6
− 2u
4
− u
3
+ u
2
+ u + 1
−u
6
+ 2u
4
+ u
3
− u
2
− u − 1
a
1
=
u
4
− u
2
− 1
−u
6
+ 2u
4
− u
2
a
4
=
u
6
− 3u
4
− u
3
+ 2u
2
+ u + 2
u
3
− u − 1
a
10
=
−u
u
a
8
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
a
9
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
7
+ 2u
6
− 2u
5
− 8u
4
− 3u
3
+ 7u
2
+ 8u + 7
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
8
u
8
c
4
(u + 1)
8
c
5
, c
7
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
6
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
9
, c
12
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
10
, c
11
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
8
y
8
c
5
, c
7
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
6
, c
10
, c
11
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
9
, c
12
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.180120 + 0.268597I
a = 1.53392 − 0.14090I
b = −1.20799 + 0.83423I
−0.604279 − 1.131230I 3.90459 + 0.80511I
u = −1.180120 − 0.268597I
a = 1.53392 + 0.14090I
b = −1.20799 − 0.83423I
−0.604279 + 1.131230I 3.90459 − 0.80511I
u = −0.108090 + 0.747508I
a = −0.322641 + 0.144481I
b = −0.711982 − 1.138990I
−3.80435 − 2.57849I −0.21961 + 3.88175I
u = −0.108090 − 0.747508I
a = −0.322641 − 0.144481I
b = −0.711982 + 1.138990I
−3.80435 + 2.57849I −0.21961 − 3.88175I
u = 1.37100
a = 0.595007
b = 0.205997
4.85780 7.82890
u = 1.334530 + 0.318930I
a = −0.47742 − 1.64247I
b = −0.365014 + 1.352640I
0.73474 + 6.44354I 4.50908 − 6.04101I
u = 1.334530 − 0.318930I
a = −0.47742 + 1.64247I
b = −0.365014 − 1.352640I
0.73474 − 6.44354I 4.50908 + 6.04101I
u = −0.463640
a = 1.93726
b = −0.636025
−0.799899 4.78300
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
29
+ 45u
28
+ ··· + 63u + 1)
c
2
((u − 1)
8
)(u
29
− 9u
28
+ ··· − 15u + 1)
c
3
, c
8
u
8
(u
29
+ u
28
+ ··· − 640u + 256)
c
4
((u + 1)
8
)(u
29
− 9u
28
+ ··· − 15u + 1)
c
5
, c
7
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
29
+ 6u
28
+ ··· − 27u − 7)
c
6
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
29
− 2u
28
+ ··· − u − 1)
c
9
(u
8
− u
7
+ ··· + 2u − 1)(u
29
+ 2u
28
+ ··· + 847u − 505)
c
10
, c
11
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
29
− 2u
28
+ ··· − u − 1)
c
12
(u
8
− u
7
+ ··· + 2u − 1)(u
29
+ 30u
27
+ ··· − u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
29
− 113y
28
+ ··· + 3195y −1)
c
2
, c
4
((y −1)
8
)(y
29
− 45y
28
+ ··· + 63y −1)
c
3
, c
8
y
8
(y
29
+ 51y
28
+ ··· + 835584y −65536)
c
5
, c
7
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
29
+ 24y
28
+ ··· + 239y −49)
c
6
, c
10
, c
11
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
29
− 24y
28
+ ··· − y −1)
c
9
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
29
+ 24y
28
+ ··· − 2572161y −255025)
c
12
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
29
+ 60y
28
+ ··· − y −1)
13
