11n
8
(K11n
8
)
A knot diagram
1
Linearized knot diagam
5 1 8 2 3 10 3 11 7 1 9
Solving Sequence
1,5
2 3 6
4,9
11 8 7 10
c
1
c
2
c
5
c
4
c
11
c
8
c
7
c
10
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h17327884311u
31
− 51370177786u
30
+ ··· + 50776700428b + 45898407811,
65060365722u
31
− 300114124597u
30
+ ··· + 50776700428a + 117492282989,
u
32
− 4u
31
+ ··· + 4u + 1i
I
u
2
= h−au + b − a + u + 1, a
3
− a
2
u − 3a
2
+ 2au + 3a − u, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 38 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
= h1.73 × 10
10
u
31
− 5.14 × 10
10
u
30
+ · · · + 5.08 × 10
10
b + 4.59 × 10
10
, 6.51 ×
10
10
u
31
−3.00×10
11
u
30
+· · ·+5.08×10
10
a+1.17×10
11
, u
32
−4u
31
+· · ·+4u+1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
6
=
−u
5
− 2u
3
− u
u
5
+ u
3
+ u
a
4
=
−u
u
3
+ u
a
9
=
−1.28130u
31
+ 5.91047u
30
+ ··· − 2.94899u − 2.31390
−0.341257u
31
+ 1.01169u
30
+ ··· − 2.34798u − 0.903927
a
11
=
0.466042u
31
− 2.36981u
30
+ ··· + 6.05202u + 2.89832
−0.115026u
31
+ 0.296752u
30
+ ··· + 0.358069u + 0.384294
a
8
=
−0.710734u
31
+ 3.43037u
30
+ ··· − 0.0540527u − 1.07276
−0.587434u
31
+ 1.13312u
30
+ ··· − 1.77017u − 0.710734
a
7
=
−1.23870u
31
+ 3.83973u
30
+ ··· − 2.78410u − 1.05967
0.352028u
31
− 0.853772u
30
+ ··· − 1.08485u − 0.642279
a
10
=
0.581069u
31
− 2.66656u
30
+ ··· + 5.69396u + 2.51403
−0.115026u
31
+ 0.296752u
30
+ ··· + 0.358069u + 0.384294
a
10
=
0.581069u
31
− 2.66656u
30
+ ··· + 5.69396u + 2.51403
−0.115026u
31
+ 0.296752u
30
+ ··· + 0.358069u + 0.384294
(ii) Obstruction class = −1
(iii) Cusp Shapes =
19161957911
12694175107
u
31
−
109525992335
25388350214
u
30
+ ···+
73849369350
12694175107
u +
129286515817
12694175107
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
32
+ 4u
31
+ ··· − 4u + 1
c
2
u
32
+ 8u
31
+ ··· − 4u + 1
c
3
, c
7
u
32
+ 3u
31
+ ··· − 32u + 64
c
5
u
32
− 4u
31
+ ··· − 5956u + 3137
c
6
, c
9
u
32
+ 3u
31
+ ··· − 3u − 1
c
8
, c
11
u
32
+ 3u
31
+ ··· + 5u − 1
c
10
u
32
− 21u
31
+ ··· − 11u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
32
+ 8y
31
+ ··· − 4y + 1
c
2
y
32
+ 36y
31
+ ··· − 400y + 1
c
3
, c
7
y
32
− 35y
31
+ ··· − 50176y + 4096
c
5
y
32
+ 64y
31
+ ··· + 37894220y + 9840769
c
6
, c
9
y
32
+ 3y
31
+ ··· − 11y + 1
c
8
, c
11
y
32
− 21y
31
+ ··· − 11y + 1
c
10
y
32
− 17y
31
+ ··· + 225y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.438343 + 0.910233I
a = −1.25471 + 2.63396I
b = −0.961368 + 0.135359I
1.31776 − 2.35125I 21.8207 − 1.6456I
u = −0.438343 − 0.910233I
a = −1.25471 − 2.63396I
b = −0.961368 − 0.135359I
1.31776 + 2.35125I 21.8207 + 1.6456I
u = −0.607222 + 0.839985I
a = −0.488594 + 0.148224I
b = −0.225556 + 0.193839I
0.60688 − 2.35983I 1.68069 + 4.72936I
u = −0.607222 − 0.839985I
a = −0.488594 − 0.148224I
b = −0.225556 − 0.193839I
0.60688 + 2.35983I 1.68069 − 4.72936I
u = −0.246944 + 1.020470I
a = −0.297055 + 0.703928I
b = 0.195786 + 0.475797I
−1.60404 − 2.42369I 1.66627 + 4.26671I
u = −0.246944 − 1.020470I
a = −0.297055 − 0.703928I
b = 0.195786 − 0.475797I
−1.60404 + 2.42369I 1.66627 − 4.26671I
u = −1.14802
a = −1.35165
b = 1.22763
5.55891 17.6890
u = −0.512800 + 0.618429I
a = 2.43693 − 0.85304I
b = −1.158390 + 0.012405I
2.28712 − 1.38183I 3.82254 + 3.38886I
u = −0.512800 − 0.618429I
a = 2.43693 + 0.85304I
b = −1.158390 − 0.012405I
2.28712 + 1.38183I 3.82254 − 3.38886I
u = 0.237631 + 0.764192I
a = 0.549640 + 0.448330I
b = 0.754657 − 0.723151I
−3.57711 − 1.46097I 0.39348 + 5.16672I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.237631 − 0.764192I
a = 0.549640 − 0.448330I
b = 0.754657 + 0.723151I
−3.57711 + 1.46097I 0.39348 − 5.16672I
u = 0.901533 + 0.826115I
a = −0.534327 − 0.197105I
b = −0.091685 + 1.072880I
6.30379 − 0.82960I 8.30496 + 0.12180I
u = 0.901533 − 0.826115I
a = −0.534327 + 0.197105I
b = −0.091685 − 1.072880I
6.30379 + 0.82960I 8.30496 − 0.12180I
u = 0.411691 + 0.642553I
a = −1.68228 + 1.31166I
b = 0.965567 + 0.732630I
−2.97947 + 4.11215I 5.06412 + 0.81363I
u = 0.411691 − 0.642553I
a = −1.68228 − 1.31166I
b = 0.965567 − 0.732630I
−2.97947 − 4.11215I 5.06412 − 0.81363I
u = 1.030380 + 0.755077I
a = −1.378020 + 0.265871I
b = 1.38664 − 0.47203I
11.00120 − 6.27983I 10.83996 + 2.99292I
u = 1.030380 − 0.755077I
a = −1.378020 − 0.265871I
b = 1.38664 + 0.47203I
11.00120 + 6.27983I 10.83996 − 2.99292I
u = 0.907454 + 0.905430I
a = 1.190940 − 0.400729I
b = −1.40337 + 0.49810I
10.34720 + 1.52704I 10.46079 − 1.65098I
u = 0.907454 − 0.905430I
a = 1.190940 + 0.400729I
b = −1.40337 − 0.49810I
10.34720 − 1.52704I 10.46079 + 1.65098I
u = 0.829030 + 0.999225I
a = 0.630834 + 0.065885I
b = 0.039106 − 1.099540I
5.75444 + 7.24046I 7.29722 − 4.74884I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.829030 − 0.999225I
a = 0.630834 − 0.065885I
b = 0.039106 + 1.099540I
5.75444 − 7.24046I 7.29722 + 4.74884I
u = 0.883634 + 0.956116I
a = 1.60874 − 1.14616I
b = −1.34427 − 0.57687I
10.18450 + 5.08130I 10.26358 − 3.28255I
u = 0.883634 − 0.956116I
a = 1.60874 + 1.14616I
b = −1.34427 + 0.57687I
10.18450 − 5.08130I 10.26358 + 3.28255I
u = −0.407352 + 1.294030I
a = −0.463231 − 0.830063I
b = 1.120700 − 0.274244I
1.05045 − 5.46747I 8.44967 + 8.57452I
u = −0.407352 − 1.294030I
a = −0.463231 + 0.830063I
b = 1.120700 + 0.274244I
1.05045 + 5.46747I 8.44967 − 8.57452I
u = −0.988672 + 0.933470I
a = −1.47255 − 0.51973I
b = 1.198750 − 0.108112I
4.49287 − 3.58059I 15.1628 + 6.1458I
u = −0.988672 − 0.933470I
a = −1.47255 + 0.51973I
b = 1.198750 + 0.108112I
4.49287 + 3.58059I 15.1628 − 6.1458I
u = 0.842421 + 1.093730I
a = −1.53412 + 1.22066I
b = 1.35984 + 0.55004I
9.8984 + 13.0980I 9.25603 − 7.22038I
u = 0.842421 − 1.093730I
a = −1.53412 − 1.22066I
b = 1.35984 − 0.55004I
9.8984 − 13.0980I 9.25603 + 7.22038I
u = −0.157159 + 0.395434I
a = −0.59313 − 2.58536I
b = −0.687544 − 0.278246I
0.876896 + 0.039424I 8.12095 − 0.03456I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.157159 − 0.395434I
a = −0.59313 + 2.58536I
b = −0.687544 + 0.278246I
0.876896 − 0.039424I 8.12095 + 0.03456I
u = −0.222533
a = −3.08647
b = −0.525361
0.954521 10.1030
8
II. I
u
2
= h−au + b − a + u + 1, a
3
− a
2
u − 3a
2
+ 2au + 3a − u, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
6
=
−1
0
a
4
=
−u
u + 1
a
9
=
a
au + a − u − 1
a
11
=
a
2
u + a
2
− au − a + 1
a
2
u − 2au + u
a
8
=
a
2
− au − 2a + 2u + 2
a
2
u − au + a − 2
a
7
=
a
2
− au − 2a + 2u + 2
a
2
u − au + a − 2
a
10
=
a
2
+ au − a − u + 1
a
2
u − 2au + u
a
10
=
a
2
+ au − a − u + 1
a
2
u − 2au + u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3a
2
u + 5a
2
− 3au − a + 2u + 8
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
3
c
3
, c
7
u
6
c
4
(u
2
− u + 1)
3
c
6
, c
10
(u
3
+ u
2
+ 2u + 1)
2
c
8
(u
3
− u
2
+ 1)
2
c
9
(u
3
− u
2
+ 2u − 1)
2
c
11
(u
3
+ u
2
− 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
3
c
3
, c
7
y
6
c
6
, c
9
, c
10
(y
3
+ 3y
2
+ 2y −1)
2
c
8
, c
11
(y
3
− y
2
+ 2y −1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 1.37744 − 0.65374I
b = 0.754878
1.11345 − 2.02988I 15.8142 − 4.6579I
u = −0.500000 + 0.866025I
a = −0.083789 + 0.387453I
b = −0.877439 − 0.744862I
−3.02413 + 0.79824I 7.63258 + 1.54443I
u = −0.500000 + 0.866025I
a = 1.20635 + 1.13232I
b = −0.877439 + 0.744862I
−3.02413 − 4.85801I 4.05323 + 9.17563I
u = −0.500000 − 0.866025I
a = 1.37744 + 0.65374I
b = 0.754878
1.11345 + 2.02988I 15.8142 + 4.6579I
u = −0.500000 − 0.866025I
a = −0.083789 − 0.387453I
b = −0.877439 + 0.744862I
−3.02413 − 0.79824I 7.63258 − 1.54443I
u = −0.500000 − 0.866025I
a = 1.20635 − 1.13232I
b = −0.877439 − 0.744862I
−3.02413 + 4.85801I 4.05323 − 9.17563I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
32
+ 4u
31
+ ··· − 4u + 1)
c
2
((u
2
+ u + 1)
3
)(u
32
+ 8u
31
+ ··· − 4u + 1)
c
3
, c
7
u
6
(u
32
+ 3u
31
+ ··· − 32u + 64)
c
4
((u
2
− u + 1)
3
)(u
32
+ 4u
31
+ ··· − 4u + 1)
c
5
((u
2
+ u + 1)
3
)(u
32
− 4u
31
+ ··· − 5956u + 3137)
c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
32
+ 3u
31
+ ··· − 3u − 1)
c
8
((u
3
− u
2
+ 1)
2
)(u
32
+ 3u
31
+ ··· + 5u − 1)
c
9
((u
3
− u
2
+ 2u − 1)
2
)(u
32
+ 3u
31
+ ··· − 3u − 1)
c
10
((u
3
+ u
2
+ 2u + 1)
2
)(u
32
− 21u
31
+ ··· − 11u + 1)
c
11
((u
3
+ u
2
− 1)
2
)(u
32
+ 3u
31
+ ··· + 5u − 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
32
+ 8y
31
+ ··· − 4y + 1)
c
2
((y
2
+ y + 1)
3
)(y
32
+ 36y
31
+ ··· − 400y + 1)
c
3
, c
7
y
6
(y
32
− 35y
31
+ ··· − 50176y + 4096)
c
5
((y
2
+ y + 1)
3
)(y
32
+ 64y
31
+ ··· + 3.78942 × 10
7
y + 9840769)
c
6
, c
9
((y
3
+ 3y
2
+ 2y −1)
2
)(y
32
+ 3y
31
+ ··· − 11y + 1)
c
8
, c
11
((y
3
− y
2
+ 2y −1)
2
)(y
32
− 21y
31
+ ··· − 11y + 1)
c
10
((y
3
+ 3y
2
+ 2y −1)
2
)(y
32
− 17y
31
+ ··· + 225y + 1)
14
