11n
99
(K11n
99
)
1
Arc Sequences
7 1 8 7 10 2 4 11 1 5 9
Solving Sequence
8,11
9
1,4
3 2 7 5 10 6
c
8
c
11
c
3
c
2
c
7
c
4
c
10
c
5
c
1
, c
6
, c
9
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu + 1, a + 1, 2b 1i
I
u
2
= ha
4
+ 3a
2
+ 1, a
3
+ 2a + u, a
3
+ a
2
+ b + 3a + 2i
I
u
3
= hu
12
u
11
+ 8u
10
11u
9
+ 26u
8
40u
7
+ 45u
6
58u
5
+ 38u
4
27u
3
+ 3u
2
+ u 1,
7u
11
6u
10
+ 62u
9
75u
8
+ 201u
7
297u
6
+ 316u
5
434u
4
+ 236u
3
169u
2
+ 32b + 6u + 1,
9u
11
+ 10u
10
74u
9
+ 109u
8
247u
7
+ 399u
6
444u
5
+ 582u
4
412u
3
+ 271u
2
+ 16a 82u + 1i
I
u
4
= hu
12
+ 3u
11
+ 10u
10
+ 22u
9
+ 41u
8
+ 63u
7
+ 81u
6
+ 85u
5
+ 75u
4
+ 53u
3
+ 32u
2
+ 12u + 9,
5444u
11
27924u
10
+ ··· + 35601b 15318, 8542u
11
+ 46692u
10
+ ··· + 106803a + 142269i
There are 4 irreducible components with 29 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, a + 1, 2b 1i
(i) Arc colorings
a
8
=
0
1
a
11
=
1
1
2
a
9
=
1
1
2
a
1
=
0
1
a
4
=
1
0
a
3
=
1
1
a
2
=
1
0
a
7
=
1
1
a
5
=
2
1
a
10
=
1
1
2
a
6
=
2
1
a
6
=
2
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 = 1.00000
a = 1.00000
b = 0.500000
3.28987 14.2500
3
II. I
u
2
= ha
4
+ 3a
2
+ 1, a
3
+ 2a + u, a
3
+ a
2
+ b + 3a + 2i
(i) Arc colorings
a
8
=
0
a
3
2a
a
11
=
a
a
3
a
2
3a 2
a
9
=
a
3
+ a
2a
3
4a 1
a
1
=
a
3
+ a
a
3
a 1
a
4
=
1
0
a
3
=
1
1
a
2
=
a
3
+ a + 1
a
3
a 2
a
7
=
a
3
+ 2a
a
3
2a
a
5
=
0
1
a
10
=
a
a
3
a
2
4a 2
a
6
=
a
2
a
3
+ a
2
+ 2a
a
6
=
a
2
a
3
+ a
2
+ 2a
(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 = 1.00000I
a = 0.618034I
b = 1.61803 + 1.61803I
2.30291 4.00000
u = 1.00000I
a = 0.618034I
b = 1.61803 1.61803I
2.30291 4.00000
u = 1.00000I
a = 1.61803I
b = 0.618034 + 0.618034I
5.59278 4.00000
u = 1.00000I
a = 1.61803I
b = 0.618034 0.618034I
5.59278 4.00000
5
III. I
u
3
=
hu
12
u
11
+· · ·+u1, 7u
11
6u
10
+· · ·+32b+1, 9u
11
+10u
10
+· · ·+16a+1i
(i) Arc colorings
a
8
=
0
u
a
11
=
9
16
u
11
5
8
u
10
+ ··· +
41
8
u
1
16
0.218750u
11
+ 0.187500u
10
+ ··· 0.187500u 0.0312500
a
9
=
1
16
u
11
3
8
u
10
+ ··· +
47
8
u
13
16
0.0312500u
11
+ 0.0625000u
10
+ ··· 0.0625000u + 0.0312500
a
1
=
1
4
u
10
+
1
4
u
9
+ ···
1
4
u +
5
4
1
4
u
10
1
4
u
9
+ ··· +
1
4
u
1
4
a
4
=
1
0
a
3
=
1
u
2
a
2
=
1
4
u
10
+
1
4
u
9
+ ···
1
4
u +
5
4
1
4
u
10
1
4
u
9
+ ··· +
1
4
u
1
4
a
7
=
u
u
a
5
=
u
2
+ 1
u
2
a
10
=
5
16
u
11
5
8
u
10
+ ··· +
41
8
u
5
16
0.343750u
11
+ 0.437500u
10
+ ··· + 0.0625000u + 0.0937500
a
6
=
1
4
u
11
+
1
4
u
10
+ ···
1
4
u
2
+
9
4
u
1
4
u
11
1
4
u
10
+ ··· +
1
4
u
2
5
4
u
a
6
=
1
4
u
11
+
1
4
u
10
+ ···
1
4
u
2
+
9
4
u
1
4
u
11
1
4
u
10
+ ··· +
1
4
u
2
5
4
u
(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 = 0.46036 1.59632I
a = 0.617140 0.709434I
b = 1.96352 + 0.61798I
9.7228 + 11.6344I 3.05947 5.63312I
u = 0.46036 + 1.59632I
a = 0.617140 + 0.709434I
b = 1.96352 0.61798I
9.7228 11.6344I 3.05947 + 5.63312I
u = 0.286848
a = 3.85090
b = 0.748460
2.08000 2.69919
u = 0.22221 1.69295I
a = 0.740223 0.119924I
b = 1.93980 0.00808I
13.9790 + 4.8530I 0.50797 2.30086I
u = 0.22221 + 1.69295I
a = 0.740223 + 0.119924I
b = 1.93980 + 0.00808I
13.9790 4.8530I 0.50797 + 2.30086I
u = 0.06595 1.61394I
a = 0.453344 + 0.915914I
b = 1.50627 0.49543I
9.24108 2.07346I 2.05543 + 1.04459I
u = 0.06595 + 1.61394I
a = 0.453344 0.915914I
b = 1.50627 + 0.49543I
9.24108 + 2.07346I 2.05543 1.04459I
u = 0.257303 0.306472I
a = 1.046990 + 0.141189I
b = 0.098944 0.484685I
0.367468 0.926038I 6.49064 + 7.55473I
u = 0.257303 + 0.306472I
a = 1.046990 0.141189I
b = 0.098944 + 0.484685I
0.367468 + 0.926038I 6.49064 7.55473I
u = 0.376488 0.828316I
a = 0.62795 1.54039I
b = 0.204419 + 0.641684I
6.26923 1.30619I 9.66269 + 5.18573I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.376488 + 0.828316I
a = 0.62795 + 1.54039I
b = 0.204419 0.641684I
6.26923 + 1.30619I 9.66269 5.18573I
u = 1.25251
a = 0.647705
b = 0.600571
2.82934 4.50161
8
IV. I
u
4
= hu
12
+ 3u
11
+ · · · + 12u + 9, 5444u
11
27924u
10
+ · · · + 35601b
15318, 8542u
11
+ 46692u
10
+ · · · + 106803a + 142269i
(i) Arc colorings
a
8
=
0
u
a
11
=
0.0799790u
11
0.437179u
10
+ ··· 3.87384u 1.33207
0.152917u
11
+ 0.784360u
10
+ ··· + 1.96624u + 0.430269
a
9
=
0.133517u
11
0.508019u
10
+ ··· 3.93721u 1.26106
0.170388u
11
+ 0.741721u
10
+ ··· + 2.13137u + 0.590377
a
1
=
0.0356638u
11
+ 0.0533693u
10
+ ··· 0.249731u 0.511334
1
a
4
=
1
0
a
3
=
1
u
2
a
2
=
0.144116u
11
+ 0.338530u
10
+ ··· + 0.0727601u 0.0287351
0.108452u
11
0.285161u
10
+ ··· 0.322491u 1.48260
a
7
=
u
u
a
5
=
u
2
+ 1
u
2
a
10
=
0.152056u
11
0.454201u
10
+ ··· 4.64191u 1.49740
0.367630u
11
+ 1.16710u
10
+ ··· + 2.50369u 0.129435
a
6
=
0.151307u
11
+ 0.543917u
10
+ ··· + 3.37438u + 2.30940
0.0134266u
11
0.158170u
10
+ ··· + 1.12047u 0.655094
a
6
=
0.151307u
11
+ 0.543917u
10
+ ··· + 3.37438u + 2.30940
0.0134266u
11
0.158170u
10
+ ··· + 1.12047u 0.655094
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.172057 0.407463I
a = 0.965356 + 0.141124I
b = 0.891259 + 0.143836I
3.28987 + 5.69302I 4.00000 5.51057I
u = 1.172057 + 0.407463I
a = 0.965356 0.141124I
b = 0.891259 0.143836I
3.28987 5.69302I 4.00000 + 5.51057I
u = 0.793458 0.920250I
a = 0.184052 0.624049I
b = 0.0718625 0.0733467I
5.18047 + 0.92430I 0.283283 0.794226I
u = 0.793458 + 0.920250I
a = 0.184052 + 0.624049I
b = 0.0718625 + 0.0733467I
5.18047 0.92430I 0.283283 + 0.794226I
u = 0.12359 1.51263I
a = 0.459386 0.245578I
b = 1.50589 0.10109I
5.18047 0.92430I 0.283283 + 0.794226I
u = 0.12359 + 1.51263I
a = 0.459386 + 0.245578I
b = 1.50589 + 0.10109I
5.18047 + 0.92430I 0.283283 0.794226I
u = 0.007520 1.191128I
a = 0.242797 + 0.842914I
b = 1.82424 1.33658I
1.39926 + 0.92430I 7.71672 0.79423I
u = 0.007520 + 1.191128I
a = 0.242797 0.842914I
b = 1.82424 + 1.33658I
1.39926 0.92430I 7.71672 + 0.79423I
u = 0.250689 0.621966I
a = 0.96746 + 1.22138I
b = 1.37089 + 1.38232I
1.39926 0.92430I 7.71672 + 0.79423I
u = 0.250689 + 0.621966I
a = 0.96746 1.22138I
b = 1.37089 1.38232I
1.39926 + 0.92430I 7.71672 0.79423I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.33089 1.60761I
a = 0.465356 0.572259I
b = 1.51567 + 0.58147I
3.28987 5.69302I 4.00000 + 5.51057I
u = 0.33089 + 1.60761I
a = 0.465356 + 0.572259I
b = 1.51567 0.58147I
3.28987 + 5.69302I 4.00000 5.51057I
11
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
(u + 1)(u
2
+ 1)
2
(u
12
u
11
+ ··· + u 1)(u
12
+ 3u
11
+ ··· + 12u + 9)
c
2
(u 1)(u + 1)
4
(u
12
+ 11u
11
+ ··· + 432u + 81)
(u
12
+ 15u
11
+ ··· 7u + 1)
c
5
, c
10
(u)(u
4
+ 3u
2
+ 1)(1 u + 2u
3
u
4
u
5
+ u
6
)
2
(u
12
+ 3u
11
+ ··· + 30u + 8)
c
6
, c
7
(u 1)(u
2
+ 1)
2
(u
12
u
11
+ ··· + u 1)(u
12
+ 3u
11
+ ··· + 12u + 9)
c
8
, c
9
(u + 1)(u
2
u 1)
2
(u
6
+ u
5
u
4
2u
3
+ u + 1)
2
(u
12
+ 2u
11
+ ··· + 7u 4)
c
11
(u 1)(u
2
+ u 1)
2
(u
6
+ u
5
u
4
2u
3
+ u + 1)
2
(u
12
+ 2u
11
+ ··· + 7u 4)
12
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
6
, c
7
(y 1)(y + 1)
4
(y
12
+ 11y
11
+ ··· + 432y + 81)
(y
12
+ 15y
11
+ ··· 7y + 1)
c
2
(y 1)
5
(y
12
37y
11
+ ··· 75y + 1)
(y
12
21y
11
+ ··· 8100y + 6561)
c
5
, c
10
y(y
2
+ 3y + 1)
2
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
2
(y
12
+ 3y
11
+ ··· 180y + 64)
c
8
, c
9
, c
11
(y 1)(y
2
3y + 1)
2
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
2
(y
12
10y
11
+ ··· 65y + 16)
13