11n
155
(K11n
155
)
1
Arc Sequences
10 8 1 2 9 3 1 6 5 4 8
Solving Sequence
2,10 1,5
4 3 9 6 7 8 11
c
1
c
4
c
3
c
9
c
5
c
6
c
8
c
11
c
2
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
9
+ u
8
3u
6
+ 2u
4
+ 3u
3
u
2
u 1, b + u, u
8
+ u
6
+ 3u
5
3u
4
2u
3
u
2
+ a + 3ui
I
u
2
= hu
16
7u
15
+ ··· 26u + 7,
5989474225u
15
+ 35317058852u
14
+ ··· + 7635864529b 29727831515,
34290455046u
15
193768360261u
14
+ ··· + 53451051703a + 406627298359i
I
u
3
= hu
18
u
17
+ ··· + 6u + 1, b + u, 95742u
17
272489u
16
+ ··· + 148175a + 581762i
There are 3 irreducible components with 43 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
9
+ u
8
3u
6
+ 2u
4
+ 3u
3
u
2
u 1, b + u, u
8
+ u
6
+ 3u
5
3u
4
2u
3
u
2
+ a + 3ui
(i) Arc colorings
a
2
=
1
0
a
10
=
u
8
u
6
3u
5
+ 3u
4
+ 2u
3
+ u
2
3u
u
a
1
=
u
8
+ u
7
3u
5
+ 2u
3
+ 2u
2
u
u
2
a
5
=
0
u
a
4
=
u
u
a
3
=
u
8
+ u
7
3u
5
+ u
3
+ 2u
2
u + 1
u
3
+ u
2
+ u
a
9
=
u
8
u
6
3u
5
+ 3u
4
+ 2u
3
+ u
2
3u
u
3
u 1
a
6
=
u
7
+ u
5
+ 3u
4
2u
3
2u
2
u + 2
u
8
+ u
7
2u
5
+ u
3
+ u
2
a
7
=
u
8
2u
7
+ 3u
5
+ u
4
6u
3
4u
2
+ u + 3
u
7
2u
6
u
5
+ 2u
4
+ 2u
3
u
2
2u 1
a
8
=
u
7
+ u
5
+ 2u
4
3u
3
2u
2
+ 2
u
6
u
5
+ 2u
3
u 1
a
11
=
u
8
u
6
3u
5
+ 3u
4
+ 2u
3
+ u
2
3u + 1
u 1
a
11
=
u
8
u
6
3u
5
+ 3u
4
+ 2u
3
+ u
2
3u + 1
u 1
(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 = 0.93778 1.07792I
a = 0.869995 0.592754I
b = 0.93778 + 1.07792I
4.99677 + 5.78819I 2.01216 5.60852I
u = 0.93778 + 1.07792I
a = 0.869995 + 0.592754I
b = 0.93778 1.07792I
4.99677 5.78819I 2.01216 + 5.60852I
u = 0.704599 0.747798I
a = 0.648299 + 0.197800I
b = 0.704599 + 0.747798I
1.13540 + 3.06246I 3.40537 6.53342I
u = 0.704599 + 0.747798I
a = 0.648299 0.197800I
b = 0.704599 0.747798I
1.13540 3.06246I 3.40537 + 6.53342I
u = 0.291926 0.569978I
a = 1.13307 + 1.40877I
b = 0.291926 + 0.569978I
0.857058 0.898737I 7.48049 2.86554I
u = 0.291926 + 0.569978I
a = 1.13307 1.40877I
b = 0.291926 0.569978I
0.857058 + 0.898737I 7.48049 + 2.86554I
u = 0.898266
a = 0.341074
b = 0.898266
6.41317 5.42199
u = 0.985174 0.537720I
a = 0.555759 1.205383I
b = 0.985174 + 0.537720I
11.81419 + 1.53593I 5.20172 0.08744I
u = 0.985174 + 0.537720I
a = 0.555759 + 1.205383I
b = 0.985174 0.537720I
11.81419 1.53593I 5.20172 + 0.08744I
3
II.
I
u
2
= hu
16
7u
15
+ · · · 26u + 7, 5.99 × 10
9
u
15
+ 3.53 × 10
10
u
14
+ · · · + 7.64 ×
10
9
b2.97×10
10
, 3.43×10
10
u
15
1.94×10
11
u
14
+· · ·+5.35×10
10
a+4.07×10
11
i
(i) Arc colorings
a
2
=
1
0
a
10
=
0.641530u
15
+ 3.62516u
14
+ ··· + 13.2090u 7.60747
0.784387u
15
4.62516u
14
+ ··· 1.49472u + 3.89318
a
1
=
0.471721u
15
+ 3.74197u
14
+ ··· 39.9962u + 11.2928
1.02789u
15
8.41954u
14
+ ··· + 77.8512u 23.2585
a
5
=
0
u
a
4
=
u
u
a
3
=
0.403962u
15
+ 3.65834u
14
+ ··· 57.2020u + 14.7733
2.15783u
15
16.6386u
14
+ ··· + 119.703u 31.0972
a
9
=
0.641530u
15
+ 3.62516u
14
+ ··· + 13.2090u 7.60747
0.615453u
15
3.15412u
14
+ ··· 19.5084u + 9.95207
a
6
=
0.201611u
15
0.883525u
14
+ ··· 11.2371u + 4.30542
0.452245u
15
3.25232u
14
+ ··· + 11.3384u 0.392224
a
7
=
0.296286u
15
2.70439u
14
+ ··· + 52.5692u 12.1830
0.151449u
15
+ 1.45205u
14
+ ··· 37.5254u + 1.92491
a
8
=
0.501789u
15
+ 3.29201u
14
+ ··· 13.2595u + 0.897630
0.335453u
15
2.17614u
14
+ ··· + 15.9763u 1.31484
a
11
=
0.641530u
15
+ 3.62516u
14
+ ··· + 14.2090u 7.60747
0.784387u
15
4.62516u
14
+ ··· 2.49472u + 3.89318
a
11
=
0.641530u
15
+ 3.62516u
14
+ ··· + 14.2090u 7.60747
0.784387u
15
4.62516u
14
+ ··· 2.49472u + 3.89318
(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 = 0.81087 1.46236I
a = 0.849619 + 0.334480I
b = 0.559608 0.857499I
6.57974 + 5.23868I 8.00000 3.04258I
u = 0.81087 + 1.46236I
a = 0.849619 0.334480I
b = 0.559608 + 0.857499I
6.57974 5.23868I 8.00000 + 3.04258I
u = 0.530385 0.793677I
a = 0.814519 0.270987I
b = 0.232467 0.600007I
0.02985 + 2.18536I 4.41681 3.14055I
u = 0.530385 + 0.793677I
a = 0.814519 + 0.270987I
b = 0.232467 + 0.600007I
0.02985 2.18536I 4.41681 + 3.14055I
u = 0.028221 0.727930I
a = 2.08457 + 0.00686I
b = 2.13775 1.37856I
13.18932 2.18536I 11.58319 + 3.14055I
u = 0.028221 + 0.727930I
a = 2.08457 0.00686I
b = 2.13775 + 1.37856I
13.18932 + 2.18536I 11.58319 3.14055I
u = 0.232467 0.600007I
a = 1.091833 0.655444I
b = 0.530385 0.793677I
0.02985 2.18536I 4.41681 + 3.14055I
u = 0.232467 + 0.600007I
a = 1.091833 + 0.655444I
b = 0.530385 + 0.793677I
0.02985 + 2.18536I 4.41681 3.14055I
u = 0.503866 0.651460I
a = 0.636481 0.057780I
b = 1.37934 0.90268I
6.57974 1.04600I 8.00000 + 6.68545I
u = 0.503866 + 0.651460I
a = 0.636481 + 0.057780I
b = 1.37934 + 0.90268I
6.57974 + 1.04600I 8.00000 6.68545I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.559608 0.857499I
a = 1.34461 + 0.64449I
b = 0.81087 1.46236I
6.57974 5.23868I 8.00000 + 3.04258I
u = 0.559608 + 0.857499I
a = 1.34461 0.64449I
b = 0.81087 + 1.46236I
6.57974 + 5.23868I 8.00000 3.04258I
u = 1.37934 0.90268I
a = 0.003729 + 0.319275I
b = 0.503866 0.651460I
6.57974 + 1.04600I 8.00000 6.68545I
u = 1.37934 + 0.90268I
a = 0.003729 0.319275I
b = 0.503866 + 0.651460I
6.57974 1.04600I 8.00000 + 6.68545I
u = 2.13775 1.37856I
a = 0.302168 + 0.514875I
b = 0.028221 0.727930I
13.18932 + 2.18536I 11.58319 3.14055I
u = 2.13775 + 1.37856I
a = 0.302168 0.514875I
b = 0.028221 + 0.727930I
13.18932 2.18536I 11.58319 + 3.14055I
6
III. I
u
3
= hu
18
u
17
+ · · · + 6u + 1, b + u, 9.57 × 10
4
u
17
2.72 × 10
5
u
16
+
· · · + 1.48 × 10
5
a + 5.82 × 10
5
i
(i) Arc colorings
a
2
=
1
0
a
10
=
0.646141u
17
+ 1.83897u
16
+ ··· 18.2110u 3.92618
u
a
1
=
1.19283u
17
+ 2.12046u
16
+ ··· + 0.0493336u + 0.353859
u
2
a
5
=
0
u
a
4
=
u
u
a
3
=
0.863911u
17
0.546617u
16
+ ··· + 6.73865u + 1.95620
0.328915u
17
1.57384u
16
+ ··· 7.78799u 1.31006
a
9
=
0.646141u
17
+ 1.83897u
16
+ ··· 18.2110u 3.92618
0.927633u
17
0.382426u
16
+ ··· 7.51081u 1.19283
a
6
=
2.77863u
17
3.65924u
16
+ ··· + 13.4316u + 6.10807
0.392637u
17
+ 0.644798u
16
+ ··· + 10.0158u + 2.07343
a
7
=
1.22635u
17
2.06353u
16
+ ··· + 18.4709u + 6.70072
0.305969u
17
+ 1.67221u
16
+ ··· + 15.8621u + 3.25843
a
8
=
1.55385u
17
+ 1.56547u
16
+ ··· 16.1116u 4.88383
0.0511355u
17
0.352853u
16
+ ··· 8.53170u 2.08505
a
11
=
0.281491u
17
+ 2.22139u
16
+ ··· 11.7002u 2.73336
0.927633u
17
0.382426u
16
+ ··· 7.51081u 1.19283
a
11
=
0.281491u
17
+ 2.22139u
16
+ ··· 11.7002u 2.73336
0.927633u
17
0.382426u
16
+ ··· 7.51081u 1.19283
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.98678 1.23806I
a = 0.766575 0.547190I
b = 0.98678 + 1.23806I
5.88062 + 4.16437I 5.71584 1.90881I
u = 0.98678 + 1.23806I
a = 0.766575 + 0.547190I
b = 0.98678 1.23806I
5.88062 4.16437I 5.71584 + 1.90881I
u = 0.739762 0.866952I
a = 0.410148 + 0.023946I
b = 0.739762 + 0.866952I
0.25135 + 2.82287I 6.88072 3.05587I
u = 0.739762 + 0.866952I
a = 0.410148 0.023946I
b = 0.739762 0.866952I
0.25135 2.82287I 6.88072 + 3.05587I
u = 0.302221 0.080115I
a = 1.67582 + 2.59626I
b = 0.302221 + 0.080115I
0.45845 + 1.47133I 0.00849 6.64687I
u = 0.302221 + 0.080115I
a = 1.67582 2.59626I
b = 0.302221 0.080115I
0.45845 1.47133I 0.00849 + 6.64687I
u = 0.170673 0.568505I
a = 0.774909 + 0.330314I
b = 0.170673 + 0.568505I
0.543265 + 1.120071I 3.86919 5.32416I
u = 0.170673 + 0.568505I
a = 0.774909 0.330314I
b = 0.170673 0.568505I
0.543265 1.120071I 3.86919 + 5.32416I
u = 0.028471 1.040199I
a = 1.52237 + 0.37502I
b = 0.028471 + 1.040199I
14.5599 + 1.9312I 9.20699 1.03940I
u = 0.028471 + 1.040199I
a = 1.52237 0.37502I
b = 0.028471 1.040199I
14.5599 1.9312I 9.20699 + 1.03940I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.464186 0.991439I
a = 0.047637 + 0.952184I
b = 0.464186 + 0.991439I
7.77265 2.85464I 5.54920 + 2.31741I
u = 0.464186 + 0.991439I
a = 0.047637 0.952184I
b = 0.464186 0.991439I
7.77265 + 2.85464I 5.54920 2.31741I
u = 0.481449 0.953626I
a = 1.250342 0.610225I
b = 0.481449 + 0.953626I
6.93662 + 0.90661I 5.69894 2.68686I
u = 0.481449 + 0.953626I
a = 1.250342 + 0.610225I
b = 0.481449 0.953626I
6.93662 0.90661I 5.69894 + 2.68686I
u = 0.78194 1.18447I
a = 0.693826 + 0.210668I
b = 0.78194 + 1.18447I
8.36952 8.49029I 4.40486 + 6.13776I
u = 0.78194 + 1.18447I
a = 0.693826 0.210668I
b = 0.78194 1.18447I
8.36952 + 8.49029I 4.40486 6.13776I
u = 0.94339 1.44899I
a = 0.851833 0.375849I
b = 0.94339 + 1.44899I
15.7704 12.3848I 6.66577 + 5.56864I
u = 0.94339 + 1.44899I
a = 0.851833 + 0.375849I
b = 0.94339 1.44899I
15.7704 + 12.3848I 6.66577 5.56864I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u
9
u
8
+ ··· u + 1)(u
16
+ 7u
15
+ ··· + 26u + 7)
(u
18
+ u
17
+ ··· 6u + 1)
c
2
(u
9
4u
7
+ ··· + u
2
+ 1)(u
16
+ u
15
+ ··· + 244u + 263)
(u
18
10u
16
+ ··· + u + 23)
c
3
(u
8
+ 7u
7
+ 17u
6
+ 14u
5
u
4
+ 2u
3
+ 6u
2
4u + 1)
2
(u
9
+ 8u
8
+ 28u
7
+ 59u
6
+ 88u
5
+ 99u
4
+ 83u
3
+ 51u
2
+ 21u + 5)
(u
18
11u
17
+ ··· 17u + 24)
c
5
(u
8
u
7
+ 5u
6
4u
5
+ 7u
4
4u
3
+ 2u
2
+ 1)
2
(u
9
+ 2u
8
+ 7u
7
+ 10u
6
+ 16u
5
+ 15u
4
+ 12u
3
+ 5u
2
1)
(u
18
+ 5u
17
+ ··· + 13u + 2)
c
6
, c
11
(u
9
u
8
+ ··· + 3u
2
+ 1)(u
16
+ u
15
+ ··· 54u + 43)
(u
18
u
17
+ ··· u + 1)
c
7
(u
9
+ u
8
+ ··· 3u
2
1)(u
16
+ u
15
+ ··· 54u + 43)
(u
18
u
17
+ ··· u + 1)
c
8
, c
9
(u
8
u
7
+ 5u
6
4u
5
+ 7u
4
4u
3
+ 2u
2
+ 1)
2
(u
9
2u
8
+ 7u
7
10u
6
+ 16u
5
15u
4
+ 12u
3
5u
2
+ 1)
(u
18
+ 5u
17
+ ··· + 13u + 2)
c
10
(u + 1)
16
(u
9
u
8
+ u
7
+ 3u
6
2u
5
+ 3u
3
u + 1)
(u
18
16u
17
+ ··· 1792u + 256)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
9
y
8
+ 6y
7
7y
6
+ 12y
5
8y
4
+ 7y
3
3y
2
y 1)
(y
16
y
15
+ ··· + 472y + 49)(y
18
+ 11y
17
+ ··· 8y + 1)
c
2
(y
9
8y
8
+ 34y
7
78y
6
+ 81y
5
26y
4
7y
3
5y
2
2y 1)
(y
16
17y
15
+ ··· 326744y + 69169)
(y
18
20y
17
+ ··· + 597y + 529)
c
3
(y
8
15y
7
+ 91y
6
246y
5
+ 207y
4
+ 130y
3
+ 50y
2
4y + 1)
2
(y
9
8y
8
+ 16y
7
+ 29y
6
64y
5
115y
4
103y
3
105y
2
69y 25)
(y
18
23y
17
+ ··· 433y + 576)
c
5
, c
8
, c
9
(y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 39y
4
+ 22y
3
+ 18y
2
+ 4y + 1)
2
(y
9
+ 10y
8
+ 41y
7
+ 88y
6
+ 104y
5
+ 63y
4
+ 14y
3
+ 5y
2
+ 10y 1)
(y
18
+ 19y
17
+ ··· + 47y + 4)
c
6
, c
7
, c
11
(y
9
7y
8
+ 13y
7
2y
5
14y
4
16y
3
13y
2
6y 1)
(y
16
21y
15
+ ··· 8076y + 1849)(y
18
27y
17
+ ··· 7y + 1)
c
10
(y 1)
16
(y
9
+ y
8
+ 3y
7
7y
6
+ 8y
5
12y
4
+ 7y
3
6y
2
+ y 1)
(y
18
+ 70y
16
+ ··· + 262144y + 65536)
11