11n
176
(K11n
176
)
1
Arc Sequences
6 9 1 7 2 10 11 2 4 5 9
Solving Sequence
1,6 2,9
3 4 5 8 11 7 10
c
1
c
2
c
3
c
5
c
8
c
11
c
7
c
10
c
4
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
16
+ 4u
14
+ u
13
+ 3u
12
10u
10
9u
9
30u
8
19u
7
33u
6
18u
5
21u
4
10u
3
7u
2
2u 1,
411u
15
+ 80u
14
+ ··· + 327b 544, 1184u
15
+ 48u
14
+ ··· + 327a + 175i
I
u
2
= hu
47
u
46
+ ··· 21u + 6, 9.28358 × 10
73
u
46
+ 4.54756 × 10
74
u
45
+ ··· + 1.24850 × 10
77
b + 1.15944 × 10
77
,
2.16945 × 10
75
u
46
2.78753 × 10
76
u
45
+ ··· + 3.74550 × 10
77
a 4.53039 × 10
78
i
There are 2 irreducible components with 63 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
16
+ 4u
14
+ · · · 2u 1, 411u
15
+ 80u
14
+ · · · + 327b
544, 1184u
15
+ 48u
14
+ · · · + 327a + 175i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
9
=
3.62080u
15
0.146789u
14
+ ··· + 7.28440u 0.535168
1.25688u
15
0.244648u
14
+ ··· + 3.14067u + 1.66361
a
3
=
3.38838u
15
+ 0.290520u
14
+ ··· 6.79205u + 2.69113
0.244648u
15
+ 0.782875u
14
+ ··· 0.850153u 0.256881
a
4
=
3.63303u
15
0.492355u
14
+ ··· 5.94190u + 2.94801
0.244648u
15
+ 0.782875u
14
+ ··· 0.850153u 0.256881
a
5
=
u
u
3
+ u
a
8
=
4.36697u
15
+ 0.507645u
14
+ ··· + 8.05810u 2.05199
1.22936u
15
0.266055u
14
+ ··· + 2.57798u + 1.00917
a
11
=
0.996942u
15
+ 0.743119u
14
+ ··· 5.75229u 0.519878
0.155963u
15
+ 0.100917u
14
+ ··· 1.63303u 0.486239
a
7
=
0.345566u
15
+ 0.972477u
14
+ ··· 6.00917u 4.74618
0.207951u
15
0.865443u
14
+ ··· 1.17737u + 0.0183486
a
10
=
1.65749u
15
0.103976u
14
+ ··· 4.59021u 0.559633
0.0152905u
15
0.284404u
14
+ ··· 1.76147u + 0.400612
a
10
=
1.65749u
15
0.103976u
14
+ ··· 4.59021u 0.559633
0.0152905u
15
0.284404u
14
+ ··· 1.76147u + 0.400612
(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 = 1.33401
a = 1.14234
b = 1.45019
4.66021 14.5887
u = 0.426918 0.416046I
a = 0.794820 + 0.479672I
b = 0.231103 + 1.368398I
2.10953 + 4.31438I 3.92733 9.24718I
u = 0.426918 + 0.416046I
a = 0.794820 0.479672I
b = 0.231103 1.368398I
2.10953 4.31438I 3.92733 + 9.24718I
u = 0.426826 0.970389I
a = 0.491146 + 0.758867I
b = 0.744279 + 0.333017I
0.63338 + 3.05324I 12.18746 3.83561I
u = 0.426826 + 0.970389I
a = 0.491146 0.758867I
b = 0.744279 0.333017I
0.63338 3.05324I 12.18746 + 3.83561I
u = 0.351578 0.904816I
a = 1.052934 0.365066I
b = 0.941434 0.633888I
1.037664 + 0.078170I 8.99757 + 0.56181I
u = 0.351578 + 0.904816I
a = 1.052934 + 0.365066I
b = 0.941434 + 0.633888I
1.037664 0.078170I 8.99757 0.56181I
u = 0.045118 0.600191I
a = 4.18562 + 0.13881I
b = 0.457773 0.219345I
4.54626 + 3.92164I 0.825014 0.819860I
u = 0.045118 + 0.600191I
a = 4.18562 0.13881I
b = 0.457773 + 0.219345I
4.54626 3.92164I 0.825014 + 0.819860I
u = 0.15358 1.53821I
a = 0.13199 1.70834I
b = 0.13262 1.73681I
8.74229 3.03673I 2.57896 2.26924I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.15358 + 1.53821I
a = 0.13199 + 1.70834I
b = 0.13262 + 1.73681I
8.74229 + 3.03673I 2.57896 + 2.26924I
u = 0.437533 0.756284I
a = 0.346240 0.869346I
b = 0.380134 + 0.976408I
4.94918 0.09603I 3.84796 1.04513I
u = 0.437533 + 0.756284I
a = 0.346240 + 0.869346I
b = 0.380134 0.976408I
4.94918 + 0.09603I 3.84796 + 1.04513I
u = 0.473070 1.323391I
a = 0.394143 0.116229I
b = 0.686442 0.076962I
0.84655 6.12118I 6.68270 + 6.03281I
u = 0.473070 + 1.323391I
a = 0.394143 + 0.116229I
b = 0.686442 + 0.076962I
0.84655 + 6.12118I 6.68270 6.03281I
u = 1.52605
a = 1.18323
b = 1.02090
6.23435 3.38258
4
II. I
u
2
=
hu
47
u
46
+ · · · 21u + 6, 9.28 × 10
73
u
46
+ 4.55 × 10
74
u
45
+ · · · + 1.25 × 10
77
b +
1.16 × 10
77
, 2.17 × 10
75
u
46
2.79 × 10
76
u
45
+ · · · + 3.75 × 10
77
a 4.53 × 10
78
i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
9
=
0.00579214u
46
+ 0.0744235u
45
+ ··· + 7.90890u + 12.0956
0.000743579u
46
0.00364242u
45
+ ··· + 0.922897u 0.928667
a
3
=
0.0143049u
46
+ 0.0978976u
45
+ ··· + 4.71215u + 13.6102
0.0228505u
46
0.0369206u
45
+ ··· + 1.75270u 0.870753
a
4
=
0.0371555u
46
+ 0.134818u
45
+ ··· + 2.95945u + 14.4810
0.0228505u
46
0.0369206u
45
+ ··· + 1.75270u 0.870753
a
5
=
u
u
3
+ u
a
8
=
0.0170297u
46
+ 0.0889078u
45
+ ··· + 5.50999u + 13.4360
0.00951166u
46
+ 0.00323251u
45
+ ··· + 0.787290u 0.909186
a
11
=
0.0661464u
46
+ 0.0255355u
45
+ ··· 15.5094u 5.85559
0.00201782u
46
+ 0.0234904u
45
+ ··· 1.29675u + 0.627088
a
7
=
0.235637u
46
+ 0.255083u
45
+ ··· 29.2766u + 8.54904
0.0237208u
46
+ 0.0372482u
45
+ ··· + 0.0651824u 0.264345
a
10
=
0.0864183u
46
+ 0.0400257u
45
+ ··· 15.5328u 5.69635
0.00595648u
46
+ 0.0186824u
45
+ ··· 1.27359u + 0.433152
a
10
=
0.0864183u
46
+ 0.0400257u
45
+ ··· 15.5328u 5.69635
0.00595648u
46
+ 0.0186824u
45
+ ··· 1.27359u + 0.433152
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.36636
a = 1.35558
b = 1.02511
6.55372 22.3581
u = 1.089521 0.393683I
a = 0.293954 0.011791I
b = 0.37684 + 1.53640I
6.00687 + 1.21428I 0.38857 1.82170I
u = 1.089521 + 0.393683I
a = 0.293954 + 0.011791I
b = 0.37684 1.53640I
6.00687 1.21428I 0.38857 + 1.82170I
u = 0.773512 0.492443I
a = 0.425302 + 0.252935I
b = 0.018902 0.913672I
1.18695 2.20702I 6.40904 + 2.67371I
u = 0.773512 + 0.492443I
a = 0.425302 0.252935I
b = 0.018902 + 0.913672I
1.18695 + 2.20702I 6.40904 2.67371I
u = 0.76218 1.43981I
a = 0.91525 1.17056I
b = 0.76117 1.69718I
9.04480 + 6.06544I 2.29283 5.25904I
u = 0.76218 + 1.43981I
a = 0.91525 + 1.17056I
b = 0.76117 + 1.69718I
9.04480 6.06544I 2.29283 + 5.25904I
u = 0.559473 1.085529I
a = 0.723467 + 0.239741I
b = 0.109730 + 0.820940I
0.67448 + 7.25675I 1.90652 8.76697I
u = 0.559473 + 1.085529I
a = 0.723467 0.239741I
b = 0.109730 0.820940I
0.67448 7.25675I 1.90652 + 8.76697I
u = 0.43399 1.44976I
a = 0.68286 + 1.56290I
b = 0.05165 + 1.95982I
11.62383 + 6.41934I 0.63591 5.02747I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.43399 + 1.44976I
a = 0.68286 1.56290I
b = 0.05165 1.95982I
11.62383 6.41934I 0.63591 + 5.02747I
u = 0.225548 0.638119I
a = 1.033439 0.803339I
b = 0.046181 + 0.992925I
5.18251 + 1.11302I 0.53819 5.95442I
u = 0.225548 + 0.638119I
a = 1.033439 + 0.803339I
b = 0.046181 0.992925I
5.18251 1.11302I 0.53819 + 5.95442I
u = 0.222144 1.218962I
a = 0.233025 0.690594I
b = 0.662108 0.421221I
3.99416 0.19865I 0.603828 + 0.009130I
u = 0.222144 + 1.218962I
a = 0.233025 + 0.690594I
b = 0.662108 + 0.421221I
3.99416 + 0.19865I 0.603828 0.009130I
u = 0.190847 0.310941I
a = 1.112969 0.105678I
b = 0.339070 1.341847I
2.24296 3.68122I 1.56886 2.39852I
u = 0.190847 + 0.310941I
a = 1.112969 + 0.105678I
b = 0.339070 + 1.341847I
2.24296 + 3.68122I 1.56886 + 2.39852I
u = 0.142302 1.399618I
a = 0.437418 0.681680I
b = 0.707328 0.658989I
0.04235 + 4.49004I 3.07286 2.52510I
u = 0.142302 + 1.399618I
a = 0.437418 + 0.681680I
b = 0.707328 + 0.658989I
0.04235 4.49004I 3.07286 + 2.52510I
u = 0.135940 1.249239I
a = 0.24140 + 2.25011I
b = 0.46236 + 1.85158I
5.37560 + 5.09376I 4.17663 5.21914I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.135940 + 1.249239I
a = 0.24140 2.25011I
b = 0.46236 1.85158I
5.37560 5.09376I 4.17663 + 5.21914I
u = 0.008414 1.269400I
a = 0.17728 2.28677I
b = 0.34057 1.47157I
7.89928 0.59442I 0.111195 0.634931I
u = 0.008414 + 1.269400I
a = 0.17728 + 2.28677I
b = 0.34057 + 1.47157I
7.89928 + 0.59442I 0.111195 + 0.634931I
u = 0.038224 0.668626I
a = 1.304073 + 0.196382I
b = 0.813691 0.238509I
0.890036 1.096420I 7.02928 + 5.72666I
u = 0.038224 + 0.668626I
a = 1.304073 0.196382I
b = 0.813691 + 0.238509I
0.890036 + 1.096420I 7.02928 5.72666I
u = 0.04978 1.52738I
a = 0.10416 1.74062I
b = 0.01532 1.71376I
8.82034 3.57353I 0.18580 + 11.85643I
u = 0.04978 + 1.52738I
a = 0.10416 + 1.74062I
b = 0.01532 + 1.71376I
8.82034 + 3.57353I 0.18580 11.85643I
u = 0.078087 0.175150I
a = 10.95150 4.13339I
b = 0.620164 + 0.192865I
4.93463 4.05841I 18.1534 + 7.4874I
u = 0.078087 + 0.175150I
a = 10.95150 + 4.13339I
b = 0.620164 0.192865I
4.93463 + 4.05841I 18.1534 7.4874I
u = 0.120942 1.100587I
a = 0.380545 + 1.251692I
b = 1.012371 + 0.893281I
0.026486 + 0.766540I 4.20873 2.92644I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.120942 + 1.100587I
a = 0.380545 1.251692I
b = 1.012371 0.893281I
0.026486 0.766540I 4.20873 + 2.92644I
u = 0.32663 1.48436I
a = 0.179586 + 0.501442I
b = 1.55279 + 0.21910I
1.89529 6.16204I 2.32673 + 5.42090I
u = 0.32663 + 1.48436I
a = 0.179586 0.501442I
b = 1.55279 0.21910I
1.89529 + 6.16204I 2.32673 5.42090I
u = 0.362669
a = 0.988687
b = 0.580670
1.51314 7.48836
u = 0.397415 0.585722I
a = 0.858641 + 0.085959I
b = 0.442468 + 0.043886I
0.847784 0.991810I 7.10163 + 6.37542I
u = 0.397415 + 0.585722I
a = 0.858641 0.085959I
b = 0.442468 0.043886I
0.847784 + 0.991810I 7.10163 6.37542I
u = 0.47098 1.75190I
a = 0.432971 1.203297I
b = 0.09716 2.03256I
9.95883 0.76536I 1.86673 0.76605I
u = 0.47098 + 1.75190I
a = 0.432971 + 1.203297I
b = 0.09716 + 2.03256I
9.95883 + 0.76536I 1.86673 + 0.76605I
u = 0.532721 0.951033I
a = 0.114543 0.789067I
b = 0.205803 0.637663I
0.32563 3.32812I 4.47407 + 5.56859I
u = 0.532721 + 0.951033I
a = 0.114543 + 0.789067I
b = 0.205803 + 0.637663I
0.32563 + 3.32812I 4.47407 5.56859I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.566468 0.943415I
a = 0.177396 0.352172I
b = 0.162886 0.570692I
0.16355 3.21246I 2.46572 + 3.89028I
u = 0.566468 + 0.943415I
a = 0.177396 + 0.352172I
b = 0.162886 + 0.570692I
0.16355 + 3.21246I 2.46572 3.89028I
u = 0.65497 1.51204I
a = 0.77642 + 1.33598I
b = 0.62901 + 1.84217I
8.5771 14.5345I 3.00788 + 7.06087I
u = 0.65497 + 1.51204I
a = 0.77642 1.33598I
b = 0.62901 1.84217I
8.5771 + 14.5345I 3.00788 7.06087I
u = 1.52172 0.04362I
a = 0.1136505 0.0768899I
b = 0.26031 1.86850I
3.81730 + 7.05587I 3.23734 5.50108I
u = 1.52172 + 0.04362I
a = 0.1136505 + 0.0768899I
b = 0.26031 + 1.86850I
3.81730 7.05587I 3.23734 + 5.50108I
u = 1.54191
a = 1.02710
b = 1.53198
4.27834 6.95466
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
16
+ 4u
14
+ ··· 2u 1)(u
47
+ u
46
+ ··· 21u 6)
c
2
(u
16
+ 4u
14
+ ··· 3u 1)(u
47
+ u
46
+ ··· 17271u 4993)
c
3
(u
16
+ 7u
15
+ ··· + 27u + 9)(u
47
+ 4u
46
+ ··· + 503u + 103)
c
4
(u
16
+ 4u
15
+ ··· + u + 1)(u
47
+ 3u
46
+ ··· + 3u + 1)
c
5
(u
16
+ 4u
14
+ ··· + 2u 1)(u
47
+ u
46
+ ··· 21u 6)
c
6
(u
16
+ 2u
14
+ ··· + 3u + 1)(u
47
+ 5u
46
+ ··· + 25u 25)
c
7
(u
16
2u
15
+ ··· + 18u 1)(u
47
+ u
46
+ ··· 9496u + 1136)
c
8
(u
16
+ 4u
14
+ ··· + 3u 1)(u
47
+ u
46
+ ··· 17271u 4993)
c
9
(u
16
u
15
+ ··· 21u
2
+ 5)(u
47
+ 2u
46
+ ··· 115u 38)
c
10
(u
16
u
15
+ ··· u 1)(u
47
9u
45
+ ··· + 1771u 137)
c
11
(u
16
3u
15
+ ··· 2u 1)(u
47
+ 26u
45
+ ··· 7818u + 1097)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
16
+ 8y
15
+ ··· + 10y + 1)(y
47
+ 35y
46
+ ··· 1923y 36)
c
2
, c
8
(y
16
+ 8y
15
+ ··· 15y + 1)
(y
47
+ 67y
46
+ ··· 225548161y 24930049)
c
3
(y
16
17y
15
+ ··· 225y + 81)
(y
47
58y
46
+ ··· + 257953y 10609)
c
4
(y
16
6y
15
+ ··· 9y + 1)(y
47
7y
46
+ ··· + 9y 1)
c
6
(y
16
+ 4y
15
+ ··· 15y + 1)(y
47
y
46
+ ··· + 17375y 625)
c
7
(y
16
+ 6y
15
+ ··· 364y + 1)
(y
47
+ 33y
46
+ ··· + 121024y 1290496)
c
9
(y
16
13y
15
+ ··· 210y + 25)(y
47
22y
46
+ ··· + 9957y 1444)
c
10
(y
16
y
15
+ ··· 13y + 1)(y
47
18y
46
+ ··· + 2612553y 18769)
c
11
(y
16
+ 5y
15
+ ··· 10y + 1)
(y
47
+ 52y
46
+ ··· + 1652754y 1203409)
12