11n
8
(K11n
8
)
1
Arc Sequences
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
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
6
5a
5
+ 11a
4
12a
3
+ 6a
2
a + 1, a
4
3a
3
+ 4a
2
2a + u + 1, a
5
4a
4
+ 7a
3
6a
2
+ b + 2ai
I
u
2
= hu
32
4u
31
+ ··· + 4u + 1,
17327884311u
31
51370177786u
30
+ ··· + 50776700428b + 45898407811,
65060365722u
31
300114124597u
30
+ ··· + 50776700428a + 117492282989i
There are 2 irreducible components with 38 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= ha
6
5a
5
+ 11a
4
12a
3
+ 6a
2
a + 1, a
4
3a
3
+ 4a
2
2a + u +
1, a
5
4a
4
+ 7a
3
6a
2
+ b + 2ai
(i) Arc colorings
a
1
=
1
0
a
5
=
0
a
4
+ 3a
3
4a
2
+ 2a 1
a
2
=
1
a
4
+ 3a
3
4a
2
+ 2a
a
3
=
a
4
3a
3
+ 4a
2
2a + 1
a
4
+ 3a
3
4a
2
+ 2a
a
6
=
1
0
a
4
=
a
4
3a
3
+ 4a
2
2a + 1
a
4
+ 3a
3
4a
2
+ 2a
a
9
=
a
a
5
+ 4a
4
7a
3
+ 6a
2
2a
a
11
=
a
5
+ 4a
4
6a
3
+ 4a
2
a + 2
a
3
3a
2
+ 3a
a
8
=
a
5
5a
4
+ 10a
3
9a
2
+ 3a
a
5
+ 4a
4
6a
3
+ 3a
2
+ a 1
a
7
=
a
5
5a
4
+ 10a
3
9a
2
+ 3a
a
5
+ 4a
4
6a
3
+ 3a
2
+ a 1
a
10
=
a
5
+ 4a
4
7a
3
+ 7a
2
4a + 2
a
3
3a
2
+ 3a
a
10
=
a
5
+ 4a
4
7a
3
+ 7a
2
4a + 2
a
3
3a
2
+ 3a
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
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 = 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 = 1.20635 + 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 = 1.37744 + 0.65374I
b = 0.754878
1.11345 + 2.02988I 15.8142 + 4.6579I
3
II. I
u
2
=
hu
32
4u
31
+ · · · +4u + 1, 1.73 × 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
i
(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 =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.14802
a = 1.35165
b = 1.22763
5.55891 17.6890
u = 0.988672 0.933470I
a = 1.47255 + 0.51973I
b = 1.198748 + 0.108112I
4.49287 + 3.58059I 15.1628 6.1458I
u = 0.988672 + 0.933470I
a = 1.47255 0.51973I
b = 1.198748 0.108112I
4.49287 3.58059I 15.1628 + 6.1458I
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 = 0.512800 0.618429I
a = 2.43693 + 0.85304I
b = 1.158389 0.012405I
2.28712 + 1.38183I 3.82254 3.38886I
u = 0.512800 + 0.618429I
a = 2.43693 0.85304I
b = 1.158389 + 0.012405I
2.28712 1.38183I 3.82254 + 3.38886I
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 = 0.407352 1.294032I
a = 0.463231 + 0.830063I
b = 1.120699 + 0.274244I
1.05045 + 5.46747I 8.44967 8.57452I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.407352 + 1.294032I
a = 0.463231 0.830063I
b = 1.120699 0.274244I
1.05045 5.46747I 8.44967 + 8.57452I
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 = 0.222533
a = 3.08647
b = 0.525361
0.954521 10.1034
u = 0.157159 0.395434I
a = 0.59313 + 2.58536I
b = 0.687544 + 0.278246I
0.876896 0.039424I 8.12095 + 0.03456I
u = 0.157159 + 0.395434I
a = 0.59313 2.58536I
b = 0.687544 0.278246I
0.876896 + 0.039424I 8.12095 0.03456I
u = 0.237631 0.764192I
a = 0.549640 0.448330I
b = 0.754657 + 0.723151I
3.57711 + 1.46097I 0.39348 5.16672I
u = 0.237631 + 0.764192I
a = 0.549640 + 0.448330I
b = 0.754657 0.723151I
3.57711 1.46097I 0.39348 + 5.16672I
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
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.829030 0.999225I
a = 0.630834 0.065885I
b = 0.039106 + 1.099538I
5.75444 7.24046I 7.29722 + 4.74884I
u = 0.829030 + 0.999225I
a = 0.630834 + 0.065885I
b = 0.039106 1.099538I
5.75444 + 7.24046I 7.29722 4.74884I
u = 0.842421 1.093728I
a = 1.53412 1.22066I
b = 1.35984 0.55004I
9.8984 13.0980I 9.25603 + 7.22038I
u = 0.842421 + 1.093728I
a = 1.53412 + 1.22066I
b = 1.35984 + 0.55004I
9.8984 + 13.0980I 9.25603 7.22038I
u = 0.883634 0.956116I
a = 1.60874 + 1.14616I
b = 1.34427 + 0.57687I
10.18448 5.08130I 10.26358 + 3.28255I
u = 0.883634 + 0.956116I
a = 1.60874 1.14616I
b = 1.34427 0.57687I
10.18448 + 5.08130I 10.26358 3.28255I
u = 0.901533 0.826115I
a = 0.534327 + 0.197105I
b = 0.091685 1.072877I
6.30379 + 0.82960I 8.30496 0.12180I
u = 0.901533 + 0.826115I
a = 0.534327 0.197105I
b = 0.091685 + 1.072877I
6.30379 0.82960I 8.30496 + 0.12180I
u = 0.907454 0.905430I
a = 1.190936 + 0.400729I
b = 1.40337 0.49810I
10.34722 1.52704I 10.46079 + 1.65098I
u = 0.907454 + 0.905430I
a = 1.190936 0.400729I
b = 1.40337 + 0.49810I
10.34722 + 1.52704I 10.46079 1.65098I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.030376 0.755077I
a = 1.378019 0.265871I
b = 1.38664 + 0.47203I
11.00118 + 6.27983I 10.83996 2.99292I
u = 1.030376 + 0.755077I
a = 1.378019 + 0.265871I
b = 1.38664 0.47203I
11.00118 6.27983I 10.83996 + 2.99292I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
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)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
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)
9