11n
86
(K11n
86
)
1
Arc Sequences
6 1 8 9 2 3 10 11 1 3 4
Solving Sequence
4,8 1,3
2 11 10 7 6 9 5
c
3
c
2
c
11
c
10
c
7
c
6
c
9
c
4
c
1
, c
5
, c
8
Representation Ideals
I =
5
\
i=1
I
u
i
I
u
1
= hu + 1, b + 1, a 1i
I
u
2
= hu
4
u
3
u
2
+ u + 1, b + 1, u
3
+ 2u
2
+ a 2i
I
u
3
= hu
5
u
3
+ u
2
+ u 1, a + 1, u
4
u
3
+ b 3i
I
u
4
= hu
10
u
9
u
8
+ u
7
+ 5u
6
4u
5
3u
4
+ u
3
+ u
2
+ 2u 1, a 1,
107u
9
+ 64u
8
+ 132u
7
41u
6
545u
5
+ 173u
4
+ 397u
3
+ 172u
2
+ 77b 35u 200i
I
u
5
= hu
14
+ 6u
10
u
9
u
8
4u
7
+ 12u
6
4u
5
+ 5u
4
10u
3
+ 11u
2
5u + 1, b 1,
303u
13
+ 398u
12
+ ··· + 481a + 3023i
There are 5 irreducible components with 34 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
= hu + 1, b + 1, a 1i
(i) Arc colorings
a
4
=
0
1
a
8
=
1
1
a
1
=
1
0
a
3
=
1
0
a
2
=
1
0
a
11
=
1
1
a
10
=
2
1
a
7
=
1
0
a
6
=
1
0
a
9
=
1
1
a
5
=
1
0
a
5
=
1
0
(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 = 1.00000
a = 1.00000
b = 1.00000
1.64493 6.00000
3
II. I
u
2
= hu
4
u
3
u
2
+ u + 1, b + 1, u
3
+ 2u
2
+ a 2i
(i) Arc colorings
a
4
=
0
u
a
8
=
u
3
2u
2
+ 2
1
a
1
=
1
0
a
3
=
2u
3
+ 3u
2
2
u
3
u
2
+ 1
a
2
=
u
3
+ 2u
2
1
u
3
u
2
+ 1
a
11
=
1
u
2
a
10
=
u
3
2u
2
+ 3
1
a
7
=
1
0
a
6
=
u
3
+ 2u
2
u 1
u
3
u
2
a
9
=
u
3
2u
2
+ 2
1
a
5
=
2u
3
+ 3u
2
2
u
3
u
2
+ 1
a
5
=
2u
3
+ 3u
2
2
u
3
u
2
+ 1
(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 = 0.692440 0.318148I
a = 1.12174 1.30662I
b = 1.00000
1.64493 + 2.02988I 3.50000 0.86603I
u = 0.692440 + 0.318148I
a = 1.12174 + 1.30662I
b = 1.00000
1.64493 2.02988I 3.50000 + 0.86603I
u = 1.192440 0.547877I
a = 0.378256 + 0.440597I
b = 1.00000
1.64493 + 2.02988I 3.50000 0.86603I
u = 1.192440 + 0.547877I
a = 0.378256 0.440597I
b = 1.00000
1.64493 2.02988I 3.50000 + 0.86603I
5
III. I
u
3
= hu
5
u
3
+ u
2
+ u 1, a + 1, u
4
u
3
+ b 3i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
u
4
+ u
3
+ 3
a
1
=
1
0
a
3
=
u
u
4
u
3
+ u
2
u 1
a
2
=
u
4
u
3
+ u
2
1
u
4
u
3
+ u
2
u 1
a
11
=
1
u
2
a
10
=
0
u
4
+ u
3
u
2
+ 2
a
7
=
1
3u
4
+ 3u
3
u
2
+ u + 6
a
6
=
u
4
+ u
3
+ 1
u
3
+ 1
a
9
=
u
4
+ u
3
u
2
+ 2
u
4
+ u
3
u
2
+ 2
a
5
=
2u
4
+ u
3
u
2
+ u + 2
2u
4
+ u
3
u
2
+ 2u + 2
a
5
=
2u
4
+ u
3
u
2
+ u + 2
2u
4
+ u
3
u
2
+ 2u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.045747 0.405588I
a = 1.00000
b = 2.51608 + 0.31234I
3.01018 5.17259I 5.18262 + 7.13326I
u = 1.045747 + 0.405588I
a = 1.00000
b = 2.51608 0.31234I
3.01018 + 5.17259I 5.18262 7.13326I
u = 0.692872
a = 1.00000
b = 3.56310
2.14584 10.4210
u = 0.699311 0.811268I
a = 1.00000
b = 0.702373 0.272489I
0.29233 + 3.70382I 1.60688 5.64419I
u = 0.699311 + 0.811268I
a = 1.00000
b = 0.702373 + 0.272489I
0.29233 3.70382I 1.60688 + 5.64419I
7
IV. I
u
4
= hu
10
u
9
+ · · · + 2u 1, a 1, 107u
9
+ 64u
8
+ · · · + 77b 200i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
1.38961u
9
0.831169u
8
+ ··· + 0.454545u + 2.59740
a
1
=
1
0
a
3
=
u
0.558442u
9
0.324675u
8
+ ··· + 0.818182u + 1.38961
a
2
=
0.558442u
9
0.324675u
8
+ ··· + 1.81818u + 1.38961
0.558442u
9
0.324675u
8
+ ··· + 0.818182u + 1.38961
a
11
=
1
u
2
a
10
=
0.233766u
9
0.298701u
8
+ ··· + 0.272727u + 1.55844
1.15584u
9
0.532468u
8
+ ··· + 0.181818u + 2.03896
a
7
=
1.48052u
9
0.558442u
8
+ ··· + 0.727273u + 3.87013
2.11688u
9
1.64935u
8
+ ··· + 1.63636u + 4.77922
a
6
=
1.15584u
9
0.532468u
8
+ ··· + 0.181818u + 3.03896
0.662338u
9
0.0129870u
8
+ ··· + 0.272727u + 1.41558
a
9
=
1.38961u
9
0.831169u
8
+ ··· + 0.454545u + 3.59740
1.15584u
9
0.532468u
8
+ ··· + 0.181818u + 2.03896
a
5
=
1.97403u
9
1.07792u
8
+ ··· + 1.63636u + 4.49351
0.831169u
9
0.506494u
8
+ ··· + 1.63636u + 2.20779
a
5
=
1.97403u
9
1.07792u
8
+ ··· + 1.63636u + 4.49351
0.831169u
9
0.506494u
8
+ ··· + 1.63636u + 2.20779
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.07634 0.95572I
a = 1.00000
b = 2.26100 1.21483I
12.08105 3.47973I 0.63239 + 2.31358I
u = 1.07634 + 0.95572I
a = 1.00000
b = 2.26100 + 1.21483I
12.08105 + 3.47973I 0.63239 2.31358I
u = 0.891654
a = 1.00000
b = 1.12961
1.69527 5.20415
u = 0.291247 0.679656I
a = 1.00000
b = 0.101239 0.236355I
0.05612 1.78093I 0.00118 + 2.91964I
u = 0.291247 + 0.679656I
a = 1.00000
b = 0.101239 + 0.236355I
0.05612 + 1.78093I 0.00118 2.91964I
u = 0.431833
a = 1.00000
b = 1.98848
2.62770 7.06151
u = 0.959690 0.284587I
a = 1.00000
b = 2.16850 + 0.95364I
3.79026 + 3.69224I 7.80243 4.12303I
u = 0.959690 + 0.284587I
a = 1.00000
b = 2.16850 0.95364I
3.79026 3.69224I 7.80243 + 4.12303I
u = 1.13781 0.99669I
a = 1.00000
b = 2.10117 + 1.18896I
11.8608 + 11.7195I 0.24253 5.99452I
u = 1.13781 + 0.99669I
a = 1.00000
b = 2.10117 1.18896I
11.8608 11.7195I 0.24253 + 5.99452I
9
V.
I
u
5
= hu
14
+ 6u
10
+ · · · 5u + 1, b 1, 303u
13
+ 398u
12
+ · · · + 481a + 3023i
(i) Arc colorings
a
4
=
0
u
a
8
=
0.629938u
13
0.827443u
12
+ ··· + 10.7069u 6.28482
1
a
1
=
1
0
a
3
=
3.18711u
13
+ 2.55925u
12
+ ··· + 14.2640u + 1.16216
0.827443u
13
0.844075u
12
+ ··· 2.13514u 0.629938
a
2
=
2.35967u
13
+ 1.71518u
12
+ ··· + 12.1289u + 0.532225
0.827443u
13
0.844075u
12
+ ··· 2.13514u 0.629938
a
11
=
1
u
2
a
10
=
0.334719u
13
1.19958u
12
+ ··· + 4.43451u 4.98753
0.120582u
13
0.0977131u
12
+ ··· + 1.50520u + 0.530146
a
7
=
4.03950u
13
+ 0.765073u
12
+ ··· + 33.7318u 10.1247
1.28690u
13
1.23701u
12
+ ··· 3.44075u + 0.812890
a
6
=
5.81289u
13
+ 1.44075u
12
+ ··· + 48.7360u 13.1622
1.48233u
13
0.762994u
12
+ ··· 9.25156u + 2.41788
a
9
=
0.214137u
13
1.29730u
12
+ ··· + 5.93971u 4.45738
0.120582u
13
0.0977131u
12
+ ··· + 1.50520u + 0.530146
a
5
=
2.78794u
13
+ 0.925156u
12
+ ··· + 21.1726u 3.70686
0.787942u
13
1.07900u
12
+ ··· 0.403326u 0.754678
a
5
=
2.78794u
13
+ 0.925156u
12
+ ··· + 21.1726u 3.70686
0.787942u
13
1.07900u
12
+ ··· 0.403326u 0.754678
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.185389 0.692372I
a = 0.319038 0.026392I
b = 1.00000
1.50295 3.09849I 4.37162 + 6.44758I
u = 1.185389 + 0.692372I
a = 0.319038 + 0.026392I
b = 1.00000
1.50295 + 3.09849I 4.37162 6.44758I
u = 0.933345 1.046642I
a = 0.171632 + 1.057991I
b = 1.00000
12.56521 3.87242I 1.20776 + 2.37795I
u = 0.933345 + 1.046642I
a = 0.171632 1.057991I
b = 1.00000
12.56521 + 3.87242I 1.20776 2.37795I
u = 0.515925 0.958517I
a = 0.864419 + 0.444469I
b = 1.00000
1.69011 4.26740I 3.53857 + 7.16930I
u = 0.515925 + 0.958517I
a = 0.864419 0.444469I
b = 1.00000
1.69011 + 4.26740I 3.53857 7.16930I
u = 0.359911 0.252178I
a = 3.11312 0.25753I
b = 1.00000
1.50295 + 3.09849I 4.37162 6.44758I
u = 0.359911 + 0.252178I
a = 3.11312 + 0.25753I
b = 1.00000
1.50295 3.09849I 4.37162 + 6.44758I
u = 0.455596 0.508546I
a = 0.109510 + 0.993986I
b = 1.00000
2.45915 4.25058
u = 0.455596 + 0.508546I
a = 0.109510 0.993986I
b = 1.00000
2.45915 4.25058
11
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.872006 0.599247I
a = 0.914949 + 0.470451I
b = 1.00000
1.69011 + 4.26740I 3.53857 7.16930I
u = 0.872006 + 0.599247I
a = 0.914949 0.470451I
b = 1.00000
1.69011 4.26740I 3.53857 + 7.16930I
u = 0.94715 1.16711I
a = 0.149400 0.920951I
b = 1.00000
12.56521 3.87242I 1.20776 + 2.37795I
u = 0.94715 + 1.16711I
a = 0.149400 + 0.920951I
b = 1.00000
12.56521 + 3.87242I 1.20776 2.37795I
12
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
u(u
2
+ u + 1)
2
(u
5
2u
4
+ 3u
3
3u
2
+ u 1)
(u
7
2u
6
+ 2u
5
+ u
2
2u + 1)
2
(u
10
+ 5u
9
+ 13u
8
+ 21u
7
+ 27u
6
+ 32u
5
+ 35u
4
+ 27u
3
+ 11u
2
3)
c
2
u(u
2
+ u + 1)
2
(u
5
+ 2u
4
u
3
7u
2
5u 1)
(1 + 2u u
2
4u
3
+ 4u
5
+ u
7
)
2
(u
10
+ u
9
+ ··· 66u + 9)
c
3
, c
11
(u 1)(u
4
u
3
u
2
+ u + 1)(u
5
u
3
+ u
2
+ u 1)
(u
10
+ u
9
u
8
u
7
+ 5u
6
+ 4u
5
3u
4
u
3
+ u
2
2u 1)
(u
14
+ 6u
10
+ u
9
u
8
+ 4u
7
+ 12u
6
+ 4u
5
+ 5u
4
+ 10u
3
+ 11u
2
+ 5u + 1)
c
4
, c
10
(u + 1)(u
4
u
3
u
2
+ u + 1)(u
5
+ u
4
u
3
u
2
+ 1)
(u
10
8u
8
u
7
+ 19u
6
+ 4u
5
4u
4
10u
3
8u
2
3u 1)
(u
14
10u
12
+ ··· + 143u + 43)
c
5
u(u
2
u + 1)
2
(u
5
+ 2u
4
+ 3u
3
+ 3u
2
+ u + 1)
(u
7
2u
6
+ 2u
5
+ u
2
2u + 1)
2
(u
10
+ 5u
9
+ 13u
8
+ 21u
7
+ 27u
6
+ 32u
5
+ 35u
4
+ 27u
3
+ 11u
2
3)
c
6
u(u
2
+ u + 1)
2
(u
5
+ u
4
8u
3
+ 7u
2
u + 1)
(u
7
+ 2u
6
+ 10u
5
8u
4
18u
3
+ 39u
2
22u + 5)
2
(u
10
2u
9
+ ··· + 1236u 471)
c
7
, c
9
(u 1)
5
(u
5
+ 3u
4
+ 3u
3
+ 3u
2
+ 2u + 1)
(u
10
+ 10u
8
11u
7
+ 13u
6
54u
5
66u
4
+ 220u
3
96u
2
13u 1)
(u
14
+ 5u
13
+ ··· + 198u + 121)
c
8
u
5
(u
5
+ 8u
4
+ 25u
3
+ 40u
2
+ 34u + 13)
(u
7
3u
6
+ 2u
5
+ 5u
4
9u
3
+ u
2
+ 6u 4)
2
(u
10
+ 9u
9
+ ··· 33u 3)
13
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
y(y
2
+ y + 1)
2
(y
5
+ 2y
4
y
3
7y
2
5y 1)
(1 + 2y y
2
4y
3
+ 4y
5
+ y
7
)
2
(y
10
+ y
9
+ ··· 66y + 9)
c
2
y(y
2
+ y + 1)
2
(y
5
6y
4
+ 19y
3
35y
2
+ 11y 1)
(y
7
+ 8y
6
+ 8y
5
28y
4
+ 32y
3
17y
2
+ 2y 1)
2
(y
10
+ 25y
9
+ ··· 5958y + 81)
c
3
, c
11
(y 1)(y
4
3y
3
+ 5y
2
3y + 1)(y
5
2y
4
+ 3y
3
3y
2
+ 3y 1)
(y
10
3y
9
+ 13y
8
25y
7
+ 43y
6
48y
5
+ 25y
4
y
3
+ 3y
2
6y + 1)
(y
14
+ 12y
12
+ ··· 3y + 1)
c
4
, c
10
(y 1)(y
4
3y
3
+ 5y
2
3y + 1)(y
5
3y
4
+ 3y
3
3y
2
+ 2y 1)
(y
10
16y
9
+ ··· + 7y + 1)(y
14
20y
13
+ ··· 12623y + 1849)
c
6
y(y
2
+ y + 1)
2
(y
5
17y
4
+ 48y
3
35y
2
13y 1)
(y
7
+ 16y
6
+ 96y
5
624y
4
+ 488y
3
649y
2
+ 94y 25)
2
(y
10
+ 46y
9
+ ··· 1142418y + 221841)
c
7
, c
9
(y 1)
5
(y
5
3y
4
+ ··· 2y 1)(y
10
+ 20y
9
+ ··· + 23y + 1)
(y
14
+ 23y
13
+ ··· 22264y + 14641)
c
8
y
5
(y
5
14y
4
+ 53y
3
108y
2
+ 116y 169)
(y
7
5y
6
+ 16y
5
43y
4
+ 71y
3
69y
2
+ 44y 16)
2
(y
10
19y
9
+ ··· 183y + 9)
14