11n
52
(K11n
52
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 10 11 4 5 6 9 8
Solving Sequence
5,10 2,6
4 1 3 9 11 8 7
c
5
c
4
c
1
c
2
c
9
c
10
c
8
c
7
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
32
− u
31
+ ··· + b + u, −u
32
− u
31
+ ··· + a − 1, u
34
+ 2u
33
+ ··· − 2u − 1i
I
u
2
= hb + 1, −u
3
+ u
2
+ a − u + 2, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 39 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
32
−u
31
+· · ·+b+u, −u
32
−u
31
+· · ·+a−1, u
34
+2u
33
+· · ·−2u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
32
+ u
31
+ ··· − 5u
3
+ 1
u
32
+ u
31
+ ··· − u
2
− u
a
6
=
1
−u
2
a
4
=
−2u
32
− 2u
31
+ ··· + u
2
+ u
−u
32
− u
31
+ ··· + u
2
+ 2u
a
1
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
11
− 3u
9
− 4u
7
− u
5
+ u
3
+ u
a
3
=
4u
32
+ 4u
31
+ ··· − 3u − 1
2u
32
+ u
31
+ ··· − u
2
− 3u
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
8
=
u
3
u
3
+ u
a
7
=
−u
6
− u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
7
=
−u
6
− u
4
+ 1
u
8
+ 2u
6
+ 2u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
33
− 11u
32
− 46u
31
− 97u
30
− 238u
29
− 421u
28
− 758u
27
−
1154u
26
− 1651u
25
− 2191u
24
− 2586u
23
− 2978u
22
− 2933u
21
− 2856u
20
− 2304u
19
−
1728u
18
− 973u
17
− 263u
16
+ 258u
15
+ 646u
14
+ 764u
13
+ 710u
12
+ 560u
11
+ 314u
10
+
160u
9
+ 16u
8
− 52u
7
− 64u
6
− 51u
5
− 4u
4
+ 16u
3
+ 18u
2
+ 15u + 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
34
− 6u
33
+ ··· + 4u − 1
c
2
u
34
+ 10u
33
+ ··· + 2u
2
+ 1
c
3
, c
7
u
34
− u
33
+ ··· + 32u + 32
c
5
, c
9
u
34
− 2u
33
+ ··· + 2u − 1
c
6
, c
8
u
34
+ 2u
33
+ ··· − 58u − 17
c
10
u
34
+ 18u
33
+ ··· + 2u + 1
c
11
u
34
− 2u
33
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
34
− 10y
33
+ ··· + 2y
2
+ 1
c
2
y
34
+ 34y
33
+ ··· + 4y + 1
c
3
, c
7
y
34
− 33y
33
+ ··· − 11776y + 1024
c
5
, c
9
y
34
+ 18y
33
+ ··· + 2y + 1
c
6
, c
8
y
34
− 22y
33
+ ··· + 682y + 289
c
10
y
34
− 2y
33
+ ··· − 18y + 1
c
11
y
34
+ 38y
33
+ ··· + 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.392064 + 0.911772I
a = −1.220400 + 0.149546I
b = −0.151137 − 0.378084I
−0.33123 − 1.99737I −1.04171 + 3.94659I
u = −0.392064 − 0.911772I
a = −1.220400 − 0.149546I
b = −0.151137 + 0.378084I
−0.33123 + 1.99737I −1.04171 − 3.94659I
u = 0.643857 + 0.740919I
a = 0.444510 + 0.058747I
b = −0.919309 − 0.963610I
7.32303 − 1.01150I 0.462803 − 0.538404I
u = 0.643857 − 0.740919I
a = 0.444510 − 0.058747I
b = −0.919309 + 0.963610I
7.32303 + 1.01150I 0.462803 + 0.538404I
u = 0.631061 + 0.814143I
a = −1.09866 − 1.38992I
b = −0.986984 + 0.934448I
7.11131 + 5.93371I −0.19300 − 5.69756I
u = 0.631061 − 0.814143I
a = −1.09866 + 1.38992I
b = −0.986984 − 0.934448I
7.11131 − 5.93371I −0.19300 + 5.69756I
u = −0.820098 + 0.217356I
a = −1.021230 − 0.783883I
b = −1.088070 + 0.833628I
3.98565 + 7.54944I −1.73478 − 4.55602I
u = −0.820098 − 0.217356I
a = −1.021230 + 0.783883I
b = −1.088070 − 0.833628I
3.98565 − 7.54944I −1.73478 + 4.55602I
u = −0.775445 + 0.276843I
a = −0.000113 + 0.147223I
b = −0.749373 − 0.980750I
5.05465 + 0.88184I −0.0341976 + 0.1167760I
u = −0.775445 − 0.276843I
a = −0.000113 − 0.147223I
b = −0.749373 + 0.980750I
5.05465 − 0.88184I −0.0341976 − 0.1167760I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.255241 + 1.154760I
a = −1.63194 − 0.61364I
b = −0.719802 − 0.838712I
0.59034 − 2.20193I −5.39762 + 2.89255I
u = −0.255241 − 1.154760I
a = −1.63194 + 0.61364I
b = −0.719802 + 0.838712I
0.59034 + 2.20193I −5.39762 − 2.89255I
u = 0.401589 + 1.121620I
a = 0.740490 + 0.229483I
b = 0.867707 + 0.523486I
−4.35229 + 1.60461I −8.26502 − 1.09622I
u = 0.401589 − 1.121620I
a = 0.740490 − 0.229483I
b = 0.867707 − 0.523486I
−4.35229 − 1.60461I −8.26502 + 1.09622I
u = 0.803313
a = −1.12207
b = −0.598522
−3.18504 0.914630
u = −0.421626 + 0.667896I
a = −0.457995 − 0.776702I
b = 0.257061 + 0.435953I
0.37642 − 1.53920I 0.52977 + 5.14051I
u = −0.421626 − 0.667896I
a = −0.457995 + 0.776702I
b = 0.257061 − 0.435953I
0.37642 + 1.53920I 0.52977 − 5.14051I
u = −0.449017 + 1.136970I
a = 2.83048 − 1.35162I
b = 1.307530 + 0.065436I
−5.63834 − 3.94702I −6.39479 + 3.36113I
u = −0.449017 − 1.136970I
a = 2.83048 + 1.35162I
b = 1.307530 − 0.065436I
−5.63834 + 3.94702I −6.39479 − 3.36113I
u = 0.490186 + 1.136270I
a = 1.31533 + 1.01208I
b = 0.706452 − 0.661902I
−3.70744 + 6.19607I −6.22304 − 6.67245I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.490186 − 1.136270I
a = 1.31533 − 1.01208I
b = 0.706452 + 0.661902I
−3.70744 − 6.19607I −6.22304 + 6.67245I
u = −0.316626 + 1.211300I
a = −1.82873 + 0.86114I
b = −1.058350 + 0.773720I
−0.44863 + 3.88868I −6.63703 − 2.26154I
u = −0.316626 − 1.211300I
a = −1.82873 − 0.86114I
b = −1.058350 − 0.773720I
−0.44863 − 3.88868I −6.63703 + 2.26154I
u = −0.548880 + 1.145350I
a = 0.412467 + 1.244140I
b = −0.692953 + 1.024120I
2.49823 − 5.83735I −3.10039 + 3.72465I
u = −0.548880 − 1.145350I
a = 0.412467 − 1.244140I
b = −0.692953 − 1.024120I
2.49823 + 5.83735I −3.10039 − 3.72465I
u = 0.456179 + 1.214870I
a = −1.73132 − 0.47368I
b = −0.649218 + 0.049959I
−6.76337 + 4.50518I −1.87945 − 4.07859I
u = 0.456179 − 1.214870I
a = −1.73132 + 0.47368I
b = −0.649218 − 0.049959I
−6.76337 − 4.50518I −1.87945 + 4.07859I
u = −0.544284 + 1.179770I
a = −2.56097 + 1.10413I
b = −1.127080 − 0.824983I
1.13195 − 12.58770I −4.92167 + 7.87699I
u = −0.544284 − 1.179770I
a = −2.56097 − 1.10413I
b = −1.127080 + 0.824983I
1.13195 + 12.58770I −4.92167 − 7.87699I
u = 0.157297 + 0.676103I
a = 1.45465 + 1.84751I
b = 1.034700 − 0.173788I
−2.09473 + 0.78471I −6.87662 + 2.65408I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.157297 − 0.676103I
a = 1.45465 − 1.84751I
b = 1.034700 + 0.173788I
−2.09473 − 0.78471I −6.87662 − 2.65408I
u = 0.637571 + 0.171045I
a = −0.049058 − 1.073020I
b = 0.658221 + 0.529258I
−0.99883 − 1.83078I −2.95289 + 3.76618I
u = 0.637571 − 0.171045I
a = −0.049058 + 1.073020I
b = 0.658221 − 0.529258I
−0.99883 + 1.83078I −2.95289 − 3.76618I
u = −0.592232
a = 1.92705
b = 1.21971
−2.64346 −1.59530
8
II. I
u
2
= hb + 1, −u
3
+ u
2
+ a − u + 2, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
3
− u
2
+ u − 2
−1
a
6
=
1
−u
2
a
4
=
u
3
− u
2
+ u − 1
−1
a
1
=
−1
0
a
3
=
u
3
− u
2
+ u − 1
−1
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
4
− u
3
+ u
2
+ 1
a
8
=
u
3
u
3
+ u
a
7
=
u
3
u
3
+ u
a
7
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
4
+ 7u
3
− 8u
2
+ 6u − 12
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
4
(u + 1)
5
c
3
, c
7
u
5
c
5
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
6
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
8
, c
11
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
9
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
10
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
7
y
5
c
5
, c
9
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
6
, c
8
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
10
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339110 + 0.822375I
a = −1.12878 + 1.10766I
b = −1.00000
−1.97403 − 1.53058I −5.00899 + 6.23673I
u = −0.339110 − 0.822375I
a = −1.12878 − 1.10766I
b = −1.00000
−1.97403 + 1.53058I −5.00899 − 6.23673I
u = 0.766826
a = −1.37029
b = −1.00000
−4.04602 −9.63840
u = 0.455697 + 1.200150I
a = −2.18608 − 0.87465I
b = −1.00000
−7.51750 + 4.40083I −13.17182 − 3.02310I
u = 0.455697 − 1.200150I
a = −2.18608 + 0.87465I
b = −1.00000
−7.51750 − 4.40083I −13.17182 + 3.02310I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
34
− 6u
33
+ ··· + 4u − 1)
c
2
((u + 1)
5
)(u
34
+ 10u
33
+ ··· + 2u
2
+ 1)
c
3
, c
7
u
5
(u
34
− u
33
+ ··· + 32u + 32)
c
4
((u + 1)
5
)(u
34
− 6u
33
+ ··· + 4u − 1)
c
5
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
34
− 2u
33
+ ··· + 2u − 1)
c
6
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
34
+ 2u
33
+ ··· − 58u − 17)
c
8
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
34
+ 2u
33
+ ··· − 58u − 17)
c
9
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
34
− 2u
33
+ ··· + 2u − 1)
c
10
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)(u
34
+ 18u
33
+ ··· + 2u + 1)
c
11
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
34
− 2u
33
+ ··· + 2u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
34
− 10y
33
+ ··· + 2y
2
+ 1)
c
2
((y −1)
5
)(y
34
+ 34y
33
+ ··· + 4y + 1)
c
3
, c
7
y
5
(y
34
− 33y
33
+ ··· − 11776y + 1024)
c
5
, c
9
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
34
+ 18y
33
+ ··· + 2y + 1)
c
6
, c
8
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
34
− 22y
33
+ ··· + 682y + 289)
c
10
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
34
− 2y
33
+ ··· − 18y + 1)
c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
34
+ 38y
33
+ ··· + 2y + 1)
14
