11a
71
(K11a
71
)
1
Arc Sequences
5 1 9 10 2 3 4 11 7 8 6
Solving Sequence
2,6
5 1
3,9
4 7 11 8 10
c
5
c
1
c
2
c
3
c
6
c
11
c
8
c
10
c
4
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu 1, a + 1, b + 2i
I
u
2
= hu
80
+ 2u
79
+ ··· + 5u + 1, 2.93692 × 10
25
u
79
+ 1.32700 × 10
26
u
78
+ ··· + 1.29163 × 10
26
a 1.04145 × 10
26
,
3.13922 × 10
25
u
79
+ 5.43494 × 10
25
u
78
+ ··· + 1.43514 × 10
25
b + 5.43494 × 10
25
i
There are 2 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
= hu 1, a + 1, b + 2i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
5
=
1
1
a
1
=
0
1
a
3
=
1
1
a
9
=
1
2
a
4
=
2
3
a
7
=
1
2
a
11
=
0
1
a
8
=
1
1
a
10
=
1
2
a
10
=
1
2
(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 = 2.00000
3.28987 12.0000
3
II. I
u
2
=
hu
80
+2u
79
+· · · +5u + 1, 2.94 × 10
25
u
79
+1.33 × 10
26
u
78
+· · · +1.29 × 10
26
a
1.04 × 10
26
, 3.14 × 10
25
u
79
+ 5.43 × 10
25
u
78
+ · · · + 1.44 × 10
25
b + 5.43 × 10
25
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
9
=
0.227381u
79
1.02738u
78
+ ··· 0.813586u + 0.806310
2.18739u
79
3.78703u
78
+ ··· 10.3818u 3.78703
a
4
=
2.35995u
79
+ 3.16571u
78
+ ··· + 9.85817u + 2.44324
7.31499u
79
+ 8.92076u
78
+ ··· + 30.7605u + 8.92076
a
7
=
u
9
2u
7
+ 3u
5
2u
3
+ u
u
9
u
7
+ u
5
+ u
a
11
=
u
2
+ 1
u
4
a
8
=
0.193492u
79
0.993492u
78
+ ··· + 0.0597437u + 0.0232540
1.95349u
79
3.55342u
78
+ ··· 9.43041u 3.55341
a
10
=
0.259921u
79
1.05992u
78
+ ··· 1.11230u + 0.690039
2.41992u
79
4.01996u
78
+ ··· 11.2298u 4.01996
a
10
=
0.259921u
79
1.05992u
78
+ ··· 1.11230u + 0.690039
2.41992u
79
4.01996u
78
+ ··· 11.2298u 4.01996
(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 = 1.191153 0.227672I
a = 0.634306 0.761625I
b = 0.88100 1.49748I
3.84602 + 1.16588I 5.92246 9.99851I
u = 1.191153 + 0.227672I
a = 0.634306 + 0.761625I
b = 0.88100 + 1.49748I
3.84602 1.16588I 5.92246 + 9 .99851I
u = 1.17701
a = 0.632361
b = 0.893580
2.75709 8.86822
u = 1.169165 0.443646I
a = 0.023634 0.845694I
b = 0.192160 0.753579I
7.65032 + 0.52064I 10.91177 2.11369I
u = 1.169165 + 0.443646I
a = 0.023634 + 0.845694I
b = 0.192160 + 0.753579I
7.65032 0.52064I 10.91177 + 2.11369I
u = 1.130221 0.587955I
a = 2.01348 0.72410I
b = 3.78872 1.49958I
2.6682 17.3873I 5.42575 + 10.65398I
u = 1.130221 + 0.587955I
a = 2.01348 + 0.72410I
b = 3.78872 + 1.49958I
2.6682 + 17 .3873I 5.42575 10.65398I
u = 1.107908 0.578539I
a = 1.010647 + 0.936041I
b = 2.31486 + 2.22056I
1.28682 11.11430I 2.75291 + 9.85263I
u = 1.107908 + 0.578539I
a = 1.010647 0.936041I
b = 2.31486 2.22056I
1.28682 + 11.11430I 2.75291 9.85263I
u = 1.103748 0.554364I
a = 1.95098 + 0.64959I
b = 3.90757 + 0.89925I
3.76245 8.78286I 9.29177 + 9 .93226I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.103748 + 0.554364I
a = 1.95098 0.64959I
b = 3.90757 0.89925I
3.76245 + 8.78286I 9.29177 9.93226I
u = 1.092077 0.530795I
a = 0.653426 + 0.286604I
b = 0.809905 0.623961I
4.75974 4.95618I 11.44465 + 4.26355I
u = 1.092077 + 0.530795I
a = 0.653426 0.286604I
b = 0.809905 + 0.623961I
4.75974 + 4.95618I 11.44465 4.26355I
u = 1.045003 0.507333I
a = 0.438277 0.940664I
b = 1.44240 2.26009I
2.30319 1.88747I 6.90646 0.67672I
u = 1.045003 + 0.507333I
a = 0.438277 + 0.940664I
b = 1.44240 + 2.26009I
2.30319 + 1.88747I 6.90646 + 0.67672I
u = 1.031484 0.286770I
a = 2.22695 + 2.33502I
b = 1.88467 + 5.18266I
3.78449 0.73170I 12.8102 + 21.5381I
u = 1.031484 + 0.286770I
a = 2.22695 2.33502I
b = 1.88467 5.18266I
3.78449 + 0.73170I 12.8102 21.5381I
u = 1.017555 0.366491I
a = 0.555590 0.608325I
b = 0.56573 2.02589I
2.69872 1.44530I 8.31672 + 0.31990I
u = 1.017555 + 0.366491I
a = 0.555590 + 0.608325I
b = 0.56573 + 2.02589I
2.69872 + 1.44530I 8.31672 0.31990I
u = 1.001244 0.581180I
a = 1.25904 0.79577I
b = 2.23468 1.14533I
3.04528 1.12254I 0.397140 + 0 .783025I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.001244 + 0.581180I
a = 1.25904 + 0.79577I
b = 2.23468 + 1.14533I
3.04528 + 1.12254I 0.397140 0.783025I
u = 0.993582 0.219652I
a = 0.084489 0.170421I
b = 0.804974 0.110018I
1.76831 0.41384I 5.89768 + 1.36191I
u = 0.993582 + 0.219652I
a = 0.084489 + 0.170421I
b = 0.804974 + 0.110018I
1.76831 + 0.41384I 5.89768 1.36191I
u = 0.950417 0.625173I
a = 0.627140 + 0.807365I
b = 1.14563 + 1.97930I
0.16239 + 4.42235I 2.61284 2.30309I
u = 0.950417 + 0.625173I
a = 0.627140 0.807365I
b = 1.14563 1.97930I
0.16239 4.42235I 2.61284 + 2.30309I
u = 0.633725 0.710008I
a = 1.45155 0.75488I
b = 0.256439 0.056529I
1.10017 9.52896I 1.23112 + 7.63149I
u = 0.633725 + 0.710008I
a = 1.45155 + 0.75488I
b = 0.256439 + 0.056529I
1.10017 + 9.52896I 1.23112 7.63149I
u = 0.562648 0.677266I
a = 1.15591 + 1.53865I
b = 0.303931 + 0.253512I
4.34092 3.75201I 2.40820 + 5.29181I
u = 0.562648 + 0.677266I
a = 1.15591 1.53865I
b = 0.303931 0.253512I
4.34092 + 3.75201I 2.40820 5.29181I
u = 0.527849 0.562689I
a = 1.77862 + 0.62507I
b = 0.143641 0.453635I
0.74428 2.39884I 3.80212 + 6.62818I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.527849 + 0.562689I
a = 1.77862 0.62507I
b = 0.143641 + 0.453635I
0.74428 + 2.39884I 3.80212 6.62818I
u = 0.381093 0.751489I
a = 1.56876 1.20506I
b = 0.224685 + 0.249500I
3.42648 + 6.07271I 0.67730 5.78894I
u = 0.381093 + 0.751489I
a = 1.56876 + 1.20506I
b = 0.224685 0.249500I
3.42648 6.07271I 0.67730 + 5 .78894I
u = 0.363856
a = 0.858090
b = 1.39437
2.38530 2.53006
u = 0.358068 0.799824I
a = 1.24458 + 2.39226I
b = 0.060260 + 0.174624I
0.37727 + 12.19403I 2.40584 6.91147I
u = 0.358068 + 0.799824I
a = 1.24458 2.39226I
b = 0.060260 0.174624I
0.37727 12.19403I 2.40584 + 6 .91147I
u = 0.349116 0.697052I
a = 1.01180 2.57817I
b = 0.311489 0.024997I
1.57621 + 3.96915I 5.87435 6.27693I
u = 0.349116 + 0.697052I
a = 1.01180 + 2.57817I
b = 0.311489 + 0.024997I
1.57621 3.96915I 5.87435 + 6 .27693I
u = 0.325329 0.604214I
a = 0.043257 0.905183I
b = 0.883800 0.481699I
2.59935 + 0.42981I 7.94943 0.49689I
u = 0.325329 + 0.604214I
a = 0.043257 + 0.905183I
b = 0.883800 + 0.481699I
2.59935 0.42981I 7.94943 + 0 .49689I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.042883 0.748508I
a = 0.807515 0.085962I
b = 0.620447 + 0.295702I
4.35199 4.81540I 6.00555 + 5 .45939I
u = 0.042883 + 0.748508I
a = 0.807515 + 0.085962I
b = 0.620447 0.295702I
4.35199 + 4 .81540I 6.00555 5.45939I
u = 0.022176 0.470783I
a = 1.223715 + 0.049549I
b = 0.228724 0.451028I
0.24724 1.52128I 2.36985 + 4 .33953I
u = 0.022176 + 0.470783I
a = 1.223715 0.049549I
b = 0.228724 + 0.451028I
0.24724 + 1 .52128I 2.36985 4.33953I
u = 0.322943 0.804382I
a = 1.18389 1.24966I
b = 0.081562 0.237623I
0.97731 4.20187I 0.27939 + 7 .34151I
u = 0.322943 + 0.804382I
a = 1.18389 + 1.24966I
b = 0.081562 + 0.237623I
0.97731 + 4 .20187I 0.27939 7.34151I
u = 0.393973 0.671200I
a = 2.83796 + 3.66333I
b = 0.129044 + 1.075226I
0.103983 1.397742I 2.4292 14.3351I
u = 0.393973 + 0.671200I
a = 2.83796 3.66333I
b = 0.129044 1.075226I
0.103983 + 1.397742I 2.4292 + 14.3351I
u = 0.435068 0.737884I
a = 0.075626 + 0.399338I
b = 0.232160 0.204773I
2.69064 1.31991I 2.13735 0.80724I
u = 0.435068 + 0.737884I
a = 0.075626 0.399338I
b = 0.232160 + 0.204773I
2.69064 + 1 .31991I 2.13735 + 0.80724I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.457682 0.625241I
a = 2.19567 2.35041I
b = 0.829055 0.454704I
0.412068 0.321069I 4.01206 + 8 .91676I
u = 0.457682 + 0.625241I
a = 2.19567 + 2.35041I
b = 0.829055 + 0.454704I
0.412068 + 0.321069I 4.01206 8.91676I
u = 0.529228 0.750028I
a = 0.099506 0.743958I
b = 0.170641 0.235530I
3.02838 1.37936I 3.64616 + 3 .11353I
u = 0.529228 + 0.750028I
a = 0.099506 + 0.743958I
b = 0.170641 + 0.235530I
3.02838 + 1 .37936I 3.64616 3.11353I
u = 0.666948 0.650580I
a = 0.265794 + 0.880885I
b = 0.417972 0.103624I
2.84471 + 1 .91218I 3.52882 5.17190I
u = 0.666948 + 0.650580I
a = 0.265794 0.880885I
b = 0.417972 + 0.103624I
2.84471 1.91218I 3.52882 + 5 .17190I
u = 0.924365 0.563218I
a = 0.808180 + 0.184164I
b = 1.61913 0.24777I
2.08735 + 2 .83654I 2.00024 2.68276I
u = 0.924365 + 0.563218I
a = 0.808180 0.184164I
b = 1.61913 + 0.24777I
2.08735 2.83654I 2.00024 + 2 .68276I
u = 1.027725 0.623923I
a = 0.763882 + 0.035932I
b = 1.229081 0.172448I
1.55978 + 6 .58436I 0.38599 8.94891I
u = 1.027725 + 0.623923I
a = 0.763882 0.035932I
b = 1.229081 + 0.172448I
1.55978 6.58436I 0.38599 + 8 .94891I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.059009 0.544082I
a = 1.81953 + 1.04085I
b = 3.02873 + 2.97647I
1.35793 + 4.93977I 6.22356 11.09851I
u = 1.059009 + 0.544082I
a = 1.81953 1.04085I
b = 3.02873 2.97647I
1.35793 4.93977I 6.22356 + 11 .09851I
u = 1.067679 0.392778I
a = 0.111831 + 0.610879I
b = 0.59539 + 1.61424I
2.94524 + 4.81984I 7.83012 8.40965I
u = 1.067679 + 0.392778I
a = 0.111831 0.610879I
b = 0.59539 1.61424I
2.94524 4.81984I 7.83012 + 8.40965I
u = 1.075993 0.321255I
a = 0.969673 0.596409I
b = 0.935782 + 0.089198I
6.18593 + 2.21505I 14.4791 3.5229I
u = 1.075993 + 0.321255I
a = 0.969673 + 0.596409I
b = 0.935782 0.089198I
6.18593 2.21505I 14.4791 + 3.5229I
u = 1.078906 0.266597I
a = 1.36745 1.21556I
b = 2.59743 1.94477I
5.69690 1.47728I 13.26865 + 3 .59149I
u = 1.078906 + 0.266597I
a = 1.36745 + 1.21556I
b = 2.59743 + 1.94477I
5.69690 + 1.47728I 13.26865 3.59149I
u = 1.085722 0.197443I
a = 0.023020 1.044414I
b = 0.63204 2.26814I
1.19443 3.73477I 5.95301 + 4.95422I
u = 1.085722 + 0.197443I
a = 0.023020 + 1.044414I
b = 0.63204 + 2.26814I
1.19443 + 3.73477I 5.95301 4.95422I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.086100 0.555013I
a = 3.04220 2.11141I
b = 4.63858 4.07536I
1.91081 + 6.16425I 2.38713 + 7.83371I
u = 1.086100 + 0.555013I
a = 3.04220 + 2.11141I
b = 4.63858 + 4.07536I
1.91081 6.16425I 2.38713 7.83371I
u = 1.087297 0.584528I
a = 0.429157 + 0.100796I
b = 1.096520 0.160168I
0.75605 + 6.35925I 1.42144 3.55958I
u = 1.087297 + 0.584528I
a = 0.429157 0.100796I
b = 1.096520 + 0.160168I
0.75605 6.35925I 1.42144 + 3.55958I
u = 1.142032 0.580131I
a = 1.21044 + 0.78836I
b = 2.13356 + 1.43409I
1.44455 + 9.36809I 4.50729 10.69470I
u = 1.142032 + 0.580131I
a = 1.21044 0.78836I
b = 2.13356 1.43409I
1.44455 9.36809I 4.50729 + 10 .69470I
u = 1.155907 0.191256I
a = 1.18018 + 1.12873I
b = 2.14083 + 2.45001I
5.29538 9.47417I 8.59979 + 5.58527I
u = 1.155907 + 0.191256I
a = 1.18018 1.12873I
b = 2.14083 2.45001I
5.29538 + 9.47417I 8.59979 5.58527I
u = 1.165949 0.406590I
a = 0.106217 + 0.764241I
b = 0.424755 + 0.599433I
7.89762 + 8.83460I 10.28104 8.00686I
u = 1.165949 + 0.406590I
a = 0.106217 0.764241I
b = 0.424755 0.599433I
7.89762 8.83460I 10.28104 + 8 .00686I
12
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)(u
80
+ 2u
79
+ ··· + 5u + 1)
c
2
(u + 1)(u
80
+ 38u
79
+ ··· + 5u + 1)
c
3
(u + 1)(u
80
+ 35u
78
+ ··· + 27u 1)
c
4
(u + 1)(u
80
+ 2u
79
+ ··· + 89u + 19)
c
5
(u + 1)(u
80
+ 2u
79
+ ··· + 5u + 1)
c
6
(u + 1)(u
80
15u
78
+ ··· + 56453u + 8017)
c
7
(u + 1)(u
80
+ 4u
79
+ ··· u 1)
c
8
(u 1)(u
80
+ 2u
79
+ ··· 5u 1)
c
9
(u)(u
80
+ 13u
79
+ ··· + 6u + 2)
c
10
(u + 1)(u
80
+ 2u
79
+ ··· 5u 1)
c
11
(u)(u
80
+ 3u
79
+ ··· 1824u 288)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)(y
80
38y
79
+ ··· 5y + 1)
c
2
(y 1)(y
80
+ 10y
79
+ ··· 73y + 1)
c
3
(y 1)(y
80
+ 70y
79
+ ··· 129y + 1)
c
4
(y 1)(y
80
+ 78y
79
+ ··· + 10547y + 361)
c
6
(y 1)(y
80
30y
79
+ ··· 2.16019 × 10
9
y + 6.42723 × 10
7
)
c
7
(y 1)(y
80
14y
79
+ ··· 5y + 1)
c
8
(y 1)(y
80
50y
79
+ ··· + 55y + 1)
c
9
(y)(y
80
9y
79
+ ··· 64y + 4)
c
10
(y 1)(y
80
50y
79
+ ··· + 55y + 1)
c
11
(y)(y
80
+ 15y
79
+ ··· + 2084544y + 82944)
14