11a
17
(K11a
17
)
1
Arc Sequences
4 1 6 2 8 3 10 5 11 7 9
Solving Sequence
7,10
8
3,11
6 4 5 9 1 2
c
7
c
10
c
6
c
3
c
5
c
9
c
11
c
2
c
1
, c
4
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
\
I
v
1
I
u
1
= ha
6
+ 2a
5
+ 3a
4
a
2
+ a + 1, a
5
+ 2a
4
+ 3a
3
+ b 2a + 1, 6a
5
9a
4
16a
3
+ 3a
2
3a + 5u 7i
I
u
2
= hu
71
3u
70
+ ··· 72u + 16,
3.96590 × 10
111
u
70
+ 1.09500 × 10
112
u
69
+ ··· + 3.39389 × 10
112
a + 2.08337 × 10
113
,
1.55843 × 10
112
u
70
2.60201 × 10
112
u
69
+ ··· + 6.78779 × 10
112
b + 5.34938 × 10
113
i
I
v
1
= hb + v + 2, b
4
+ 3b
3
+ b
2
2b + 1, ai
There are 3 irreducible components with 81 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
+ 2a
5
+ 3a
4
a
2
+ a + 1, a
5
+ 2a
4
+ 3a
3
+ b 2a + 1, 6a
5
9a
4
16a
3
+ 3a
2
3a + 5u 7i
(i) Arc colorings
a
7
=
0
6
5
a
5
+
9
5
a
4
+ ··· +
3
5
a +
7
5
a
10
=
a
a
5
2a
4
3a
3
+ 2a 1
a
8
=
4
5
a
5
6
5
a
4
+ ··· +
3
5
a
3
5
4
5
a
5
6
5
a
4
+ ··· +
3
5
a
3
5
a
3
=
1
0
a
11
=
a
6
5
a
5
9
5
a
4
+ ··· +
2
5
a
7
5
a
6
=
6
5
a
5
+
9
5
a
4
+ ··· +
3
5
a +
7
5
6
5
a
5
+
9
5
a
4
+ ··· +
3
5
a +
7
5
a
4
=
2
5
a
5
+
3
5
a
4
+ ··· +
1
5
a
1
5
2
5
a
5
+
3
5
a
4
+ ··· +
1
5
a
6
5
a
5
=
6
5
a
5
+
9
5
a
4
+ ··· +
3
5
a +
7
5
6
5
a
5
+
9
5
a
4
+ ··· +
3
5
a +
7
5
a
9
=
4
5
a
5
6
5
a
4
+ ··· +
3
5
a
3
5
4
5
a
5
6
5
a
4
+ ··· +
3
5
a
3
5
a
1
=
0
6
5
a
5
9
5
a
4
+ ···
3
5
a
7
5
a
2
=
1
2
5
a
5
3
5
a
4
+ ···
1
5
a +
6
5
a
2
=
1
2
5
a
5
3
5
a
4
+ ···
1
5
a +
6
5
(ii) Obstruction class = 1
(iii) Cusp Shapes =
13
5
a
5
+
7
5
a
4
+
18
5
a
3
19
5
a
2
+
14
5
a +
16
5
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.569840
a = 0.87744 1.51977I
b = 1.16236 1.02627I
1.11345 2.02988I 2.22484 + 11.58609I
u = 0.569840
a = 0.87744 + 1.51977I
b = 1.16236 + 1.02627I
1.11345 + 2.02988I 2.22484 11.58609I
u = 0.215080 1.307141I
a = 0.706350 0.266290I
b = 1.94591 + 0.20102I
3.02413 + 4.85801I 0.92725 3.71146I
u = 0.215080 + 1.307141I
a = 0.706350 + 0.266290I
b = 1.94591 0.20102I
3.02413 4.85801I 0.92725 + 3.71146I
u = 0.215080 1.307141I
a = 0.583789 0.478572I
b = 1.60827 + 0.36126I
3.02413 + 0.79824I 2.65209 0.57512I
u = 0.215080 + 1.307141I
a = 0.583789 + 0.478572I
b = 1.60827 0.36126I
3.02413 0.79824I 2.65209 + 0.57512I
3
II. I
u
2
= hu
71
3u
70
+ · · · 72u + 16, 3.97 × 10
111
u
70
+ 1.09 × 10
112
u
69
+
· · · + 3.39 × 10
112
a + 2.08 × 10
113
, 1.56 × 10
112
u
70
2.60 × 10
112
u
69
+ · · · +
6.79 × 10
112
b + 5.35 × 10
113
i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.116854u
70
0.322638u
69
+ ··· + 27.6437u 6.13860
0.229593u
70
+ 0.383337u
69
+ ··· + 18.6675u 7.88089
a
8
=
0.352237u
70
+ 0.840721u
69
+ ··· 14.6008u + 1.33422
0.332192u
70
+ 1.05815u
69
+ ··· 30.0948u + 5.06594
a
3
=
1
0
a
11
=
0.116854u
70
0.322638u
69
+ ··· + 27.6437u 6.13860
0.199770u
70
+ 0.267189u
69
+ ··· + 18.8084u 8.32769
a
6
=
u
u
a
4
=
u
2
+ 1
u
2
a
5
=
0.217439u
70
0.809829u
69
+ ··· + 36.3452u 10.7861
0.240175u
70
+ 0.419805u
69
+ ··· + 11.4762u 4.98688
a
9
=
0.160473u
70
0.0256137u
69
+ ··· 14.4436u + 7.94918
0.0294757u
70
0.177366u
69
+ ··· + 17.4523u 6.12313
a
1
=
0.457615u
70
+ 1.22963u
69
+ ··· 24.8689u + 5.79918
0.0229582u
70
+ 0.0337975u
69
+ ··· + 8.48701u 2.69554
a
2
=
0.194259u
70
+ 0.601283u
69
+ ··· 17.8491u + 5.87584
0.0383625u
70
0.260260u
69
+ ··· + 24.8249u 7.04452
a
2
=
0.194259u
70
+ 0.601283u
69
+ ··· 17.8491u + 5.87584
0.0383625u
70
0.260260u
69
+ ··· + 24.8249u 7.04452
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1.32757u
70
3.60410u
69
+ ··· + 79.5051u 10.6207
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.108829 0.298809I
a = 0.405496 + 0.963353I
b = 0.464673 0.750133I
4.27963 + 4.12596I 5.64693 2.76613I
u = 1.108829 + 0.298809I
a = 0.405496 0.963353I
b = 0.464673 + 0.750133I
4.27963 4.12596I 5.64693 + 2.76613I
u = 1.044276 0.406192I
a = 0.905243 0.406863I
b = 0.882650 + 0.230624I
4.75730 1.60644I 6.56664 + 2.91512I
u = 1.044276 + 0.406192I
a = 0.905243 + 0.406863I
b = 0.882650 0.230624I
4.75730 + 1.60644I 6.56664 2.91512I
u = 0.869259 0.100924I
a = 0.185713 + 0.970127I
b = 0.481419 0.708766I
0.623561 0.155557I 0.0220175 0.0069970I
u = 0.869259 + 0.100924I
a = 0.185713 0.970127I
b = 0.481419 + 0.708766I
0.623561 + 0.155557I 0.0220175 + 0.0069970I
u = 0.770057 0.536295I
a = 0.051492 1.202999I
b = 1.000923 0.283561I
2.88221 + 3.75765I 5.98986 3.48870I
u = 0.770057 + 0.536295I
a = 0.051492 + 1.202999I
b = 1.000923 + 0.283561I
2.88221 3.75765I 5.98986 + 3.48870I
u = 0.672234 1.164288I
a = 0.123136 + 0.789149I
b = 0.234480 + 0.478546I
2.39090 4.48377I 4.20104 + 2.21421I
u = 0.672234 + 1.164288I
a = 0.123136 0.789149I
b = 0.234480 0.478546I
2.39090 + 4.48377I 4.20104 2.21421I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.653700 1.243696I
a = 0.889734 + 0.117703I
b = 2.76987 0.05666I
1.30783 10.33652I 2.42446 + 6.81123I
u = 0.653700 + 1.243696I
a = 0.889734 0.117703I
b = 2.76987 + 0.05666I
1.30783 + 10.33652I 2.42446 6.81123I
u = 0.625386 1.066573I
a = 0.984933 0.125851I
b = 2.02761 + 0.86595I
1.26633 9.05370I 4.12979 + 7.62616I
u = 0.625386 + 1.066573I
a = 0.984933 + 0.125851I
b = 2.02761 0.86595I
1.26633 + 9.05370I 4.12979 7.62616I
u = 0.617687 0.601618I
a = 1.242853 + 0.140951I
b = 2.13478 + 1.03108I
3.08372 1.52053I 6.23947 + 3.65272I
u = 0.617687 + 0.601618I
a = 1.242853 0.140951I
b = 2.13478 1.03108I
3.08372 + 1.52053I 6.23947 3.65272I
u = 0.572324 0.999549I
a = 0.233123 0.943124I
b = 0.517815 0.001182I
1.88125 3.19322I 5.06706 + 2.97492I
u = 0.572324 + 0.999549I
a = 0.233123 + 0.943124I
b = 0.517815 + 0.001182I
1.88125 + 3.19322I 5.06706 2.97492I
u = 0.518539
a = 0.121980
b = 0.636639
1.19409 8.46118
u = 0.426470 1.144807I
a = 0.885157 + 0.207328I
b = 2.68779 0.48799I
4.31628 4.18567I 2.53385 + 4.06655I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.426470 + 1.144807I
a = 0.885157 0.207328I
b = 2.68779 + 0.48799I
4.31628 + 4.18567I 2.53385 4.06655I
u = 0.419700 1.267451I
a = 0.690447 0.094402I
b = 2.43875 0.49905I
4.39627 4.60339I 3.32762 + 2.73353I
u = 0.419700 + 1.267451I
a = 0.690447 + 0.094402I
b = 2.43875 + 0.49905I
4.39627 + 4.60339I 3.32762 2.73353I
u = 0.406144 0.846985I
a = 0.568192 0.513398I
b = 0.160223 0.333947I
0.11266 1.79745I 3.68305 + 3.00636I
u = 0.406144 + 0.846985I
a = 0.568192 + 0.513398I
b = 0.160223 + 0.333947I
0.11266 + 1.79745I 3.68305 3.00636I
u = 0.21087 1.49010I
a = 0.492915 + 0.502061I
b = 1.74148 0.08557I
2.18680 0.67401I 7.57995 0.70637I
u = 0.21087 + 1.49010I
a = 0.492915 0.502061I
b = 1.74148 + 0.08557I
2.18680 + 0.67401I 7.57995 + 0.70637I
u = 0.176109 1.095201I
a = 0.059666 + 0.439307I
b = 0.036771 0.403240I
2.07880 2.37441I 0.70861 + 4.00251I
u = 0.176109 + 1.095201I
a = 0.059666 0.439307I
b = 0.036771 + 0.403240I
2.07880 + 2.37441I 0.70861 4.00251I
u = 0.139051 0.972541I
a = 0.982199 0.687863I
b = 1.80171 + 0.62956I
1.97142 + 2.68154I 0.29957 2.31966I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.139051 + 0.972541I
a = 0.982199 + 0.687863I
b = 1.80171 0.62956I
1.97142 2.68154I 0.29957 + 2.31966I
u = 0.08760 1.49800I
a = 0.628097 + 0.397155I
b = 2.19274 0.03102I
2.40563 5.71061I 5.14402 + 8.04898I
u = 0.08760 + 1.49800I
a = 0.628097 0.397155I
b = 2.19274 + 0.03102I
2.40563 + 5.71061I 5.14402 8.04898I
u = 0.013074 0.389178I
a = 1.72088 1.06017I
b = 0.011535 0.390861I
0.058062 1.373771I 0.54762 + 4.59641I
u = 0.013074 + 0.389178I
a = 1.72088 + 1.06017I
b = 0.011535 + 0.390861I
0.058062 + 1.373771I 0.54762 4.59641I
u = 0.149859 1.246529I
a = 0.780274 0.040190I
b = 2.55905 0.57530I
5.52736 0.93567I 4.64414 + 2.41235I
u = 0.149859 + 1.246529I
a = 0.780274 + 0.040190I
b = 2.55905 + 0.57530I
5.52736 + 0.93567I 4.64414 2.41235I
u = 0.223038 0.668311I
a = 1.332294 + 0.103105I
b = 3.14217 0.55400I
1.01028 + 1.75385I 0.38756 7.39042I
u = 0.223038 + 0.668311I
a = 1.332294 0.103105I
b = 3.14217 + 0.55400I
1.01028 1.75385I 0.38756 + 7.39042I
u = 0.285900 0.818152I
a = 0.478709 1.027334I
b = 0.323137 0.246398I
0.185937 1.104375I 1.16040 + 2.40265I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.285900 + 0.818152I
a = 0.478709 + 1.027334I
b = 0.323137 + 0.246398I
0.185937 + 1.104375I 1.16040 2.40265I
u = 0.414388 0.586515I
a = 0.40266 + 1.73808I
b = 0.041781 0.513227I
8.63777 2.49143I 6.68851 5.26369I
u = 0.414388 + 0.586515I
a = 0.40266 1.73808I
b = 0.041781 + 0.513227I
8.63777 + 2.49143I 6.68851 + 5.26369I
u = 0.423358 0.977979I
a = 1.092689 0.224936I
b = 2.05311 + 0.81962I
1.01578 + 4.23738I 0.18975 3.40400I
u = 0.423358 + 0.977979I
a = 1.092689 + 0.224936I
b = 2.05311 0.81962I
1.01578 4.23738I 0.18975 + 3.40400I
u = 0.440954 0.146276I
a = 1.57023 + 1.52354I
b = 2.75559 + 1.24867I
1.31688 + 1.88106I 31.3196 4.6871I
u = 0.440954 + 0.146276I
a = 1.57023 1.52354I
b = 2.75559 1.24867I
1.31688 1.88106I 31.3196 + 4.6871I
u = 0.474824 1.073276I
a = 1.022969 + 0.247431I
b = 2.85706 + 0.15996I
6.97127 + 6.31431I 6.68078 5.69078I
u = 0.474824 + 1.073276I
a = 1.022969 0.247431I
b = 2.85706 0.15996I
6.97127 6.31431I 6.68078 + 5.69078I
u = 0.474923 1.017331I
a = 0.730844 0.783505I
b = 1.55954 + 0.72445I
0.54346 + 1.75236I 3.34630 2.70303I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.474923 + 1.017331I
a = 0.730844 + 0.783505I
b = 1.55954 0.72445I
0.54346 1.75236I 3.34630 + 2.70303I
u = 0.503579 0.631395I
a = 1.51985 + 0.19553I
b = 0.569925 0.160071I
8.93127 + 3.93572I 8.19554 8.78480I
u = 0.503579 + 0.631395I
a = 1.51985 0.19553I
b = 0.569925 + 0.160071I
8.93127 3.93572I 8.19554 + 8.78480I
u = 0.555895 1.022587I
a = 0.236549 + 0.930136I
b = 0.553116 + 0.929745I
7.65318 + 0.42476I 8.26553 0.47633I
u = 0.555895 + 1.022587I
a = 0.236549 0.930136I
b = 0.553116 0.929745I
7.65318 0.42476I 8.26553 + 0.47633I
u = 0.613697 1.017340I
a = 0.488779 0.516117I
b = 0.363307 0.244616I
2.42985 + 6.27823I 7.32031 6.32359I
u = 0.613697 + 1.017340I
a = 0.488779 + 0.516117I
b = 0.363307 + 0.244616I
2.42985 6.27823I 7.32031 + 6.32359I
u = 0.628836 1.153550I
a = 0.841830 + 0.293618I
b = 2.63896 0.45159I
2.15092 + 9.48353I 1.21644 8.12337I
u = 0.628836 + 1.153550I
a = 0.841830 0.293618I
b = 2.63896 + 0.45159I
2.15092 9.48353I 1.21644 + 8.12337I
u = 0.712988 0.613513I
a = 0.466264 0.491310I
b = 0.403220 0.683557I
3.67287 1.17347I 10.32734 + 0.68526I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.712988 + 0.613513I
a = 0.466264 + 0.491310I
b = 0.403220 + 0.683557I
3.67287 + 1.17347I 10.32734 0.68526I
u = 0.776807 1.153329I
a = 0.042540 + 0.802326I
b = 0.022765 + 0.518909I
5.27544 + 10.02073I 6.86727 5.48093I
u = 0.776807 + 1.153329I
a = 0.042540 0.802326I
b = 0.022765 0.518909I
5.27544 10.02073I 6.86727 + 5.48093I
u = 0.77993 1.20298I
a = 0.900622 + 0.039066I
b = 2.81120 0.12349I
4.0986 + 16.1013I 5.05780 9.91459I
u = 0.77993 + 1.20298I
a = 0.900622 0.039066I
b = 2.81120 + 0.12349I
4.0986 16.1013I 5.05780 + 9.91459I
u = 0.903463 0.396432I
a = 0.234419 1.036384I
b = 0.912184 + 0.725719I
0.16872 3.86400I 2.80840 + 6.04330I
u = 0.903463 + 0.396432I
a = 0.234419 + 1.036384I
b = 0.912184 0.725719I
0.16872 + 3.86400I 2.80840 6.04330I
u = 1.075773 0.627927I
a = 0.870455 0.165745I
b = 1.003247 + 0.024923I
6.99333 3.33145I 8.40822 + 1.84297I
u = 1.075773 + 0.627927I
a = 0.870455 + 0.165745I
b = 1.003247 0.024923I
6.99333 + 3.33145I 8.40822 1.84297I
u = 1.133831 0.574421I
a = 0.166086 + 0.969653I
b = 0.299220 0.891462I
6.13711 9.23592I 6.69961 + 6.97434I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.133831 + 0.574421I
a = 0.166086 0.969653I
b = 0.299220 + 0.891462I
6.13711 + 9.23592I 6.69961 6.97434I
11
III. I
v
1
= hb + v + 2, b
4
+ 3b
3
+ b
2
2b + 1, ai
(i) Arc colorings
a
7
=
b 2
0
a
10
=
0
b
a
8
=
b 2
b
3
2b
2
a
3
=
1
0
a
11
=
b
3
4b
2
4b
b
a
6
=
b 2
0
a
4
=
1
0
a
5
=
2b
3
7b
2
6b + 1
1
a
9
=
2b
3
6b
2
2b + 5
b
2
+ 2b 1
a
1
=
2b
3
+ 7b
2
+ 6b 1
1
a
2
=
2b
3
+ 7b
2
+ 6b
1
a
2
=
2b
3
+ 7b
2
+ 6b
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2b
3
+ 6b
2
+ 7b 7
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.100768 + 0.400532I
a = 0
b = 1.89923 0.40053I
1.43393 + 1.41510I 11.48794 2.21528I
v = 0.100768 0.400532I
a = 0
b = 1.89923 + 0.40053I
1.43393 1.41510I 11.48794 + 2.21528I
v = 2.39923 + 0.32564I
a = 0
b = 0.399232 0.325640I
8.43568 + 3.16396I 4.01206 4.08190I
v = 2.39923 0.32564I
a = 0
b = 0.399232 + 0.325640I
8.43568 3.16396I 4.01206 + 4.08190I
13
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
3
+ u
2
1)
2
(u
71
+ 7u
70
+ ··· 19u 1)
c
2
(u + 1)
4
(1 + 2u + u
2
+ u
3
)
2
(u
71
+ 35u
70
+ ··· + 91u + 1)
c
3
u
4
(1 + 2u u
2
+ u
3
)
2
(u
71
+ 3u
70
+ ··· 72u 16)
c
4
(u + 1)
4
(u
3
u
2
+ 1)
2
(u
71
+ 7u
70
+ ··· 19u 1)
c
5
u
6
(u
4
+ u
3
+ ··· + 2u + 1)(u
71
+ 2u
70
+ ··· + 224u + 64)
c
6
u
4
(1 + 2u + u
2
+ u
3
)
2
(u
71
+ 3u
70
+ ··· 72u 16)
c
7
(u
2
u + 1)
3
(u
4
+ u
3
+ u
2
+ 1)(u
71
+ 5u
70
+ ··· + 16u + 1)
c
8
u
6
(u
4
u
3
+ ··· 2u + 1)(u
71
+ 2u
70
+ ··· + 224u + 64)
c
9
(u
2
+ u + 1)
3
(u
4
+ u
3
+ ··· + 2u + 1)(u
71
+ 23u
70
+ ··· + 246u 1)
c
10
(u
2
+ u + 1)
3
(u
4
u
3
+ u
2
+ 1)(u
71
+ 5u
70
+ ··· + 16u + 1)
c
11
(u
2
u + 1)
3
(u
4
u
3
+ ··· 2u + 1)(u
71
+ 23u
70
+ ··· + 246u 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(1 + 2y y
2
+ y
3
)
2
(y
71
35y
70
+ ··· + 91y 1)
c
2
(y 1)
4
(1 + 2y + 3y
2
+ y
3
)
2
(y
71
+ 9y
70
+ ··· + 3279y 1)
c
3
, c
6
y
4
(1 + 2y + 3y
2
+ y
3
)
2
(y
71
+ 33y
70
+ ··· 4800y 256)
c
5
, c
8
y
6
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
71
+ 40y
70
+ ··· 39936y 4096)
c
7
, c
10
(y
2
+ y + 1)
3
(y
4
+ y
3
+ ··· + 2y + 1)(y
71
+ 23y
70
+ ··· + 246y 1)
c
9
, c
11
(y
2
+ y + 1)
3
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
71
+ 55y
70
+ ··· + 62490y 1)
15