11a
31
(K11a
31
)
1
Arc Sequences
4 1 7 2 10 11 3 5 6 9 8
Solving Sequence
5,10 2,6
4 1 3 9 11 8 7
c
5
c
4
c
1
c
2
c
9
c
10
c
8
c
7
c
3
, c
6
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
5
+ a
4
+ 2a
3
+ a
2
+ a + 1, u 1, a
4
a
3
a
2
+ b 2ai
I
u
2
= hu
67
6u
66
+ ··· 6u + 1, 16u
66
+ 81u
65
+ ··· + 8a 7, 57u
66
+ 289u
65
+ ··· + 16b + 61i
There are 2 irreducible components with 72 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
5
+ a
4
+ 2a
3
+ a
2
+ a + 1, u 1, a
4
a
3
a
2
+ b 2ai
(i) Arc colorings
a
5
=
0
1
a
10
=
a
a
4
+ a
3
+ a
2
+ 2a
a
2
=
1
0
a
6
=
a
2
a
3
+ a
2
a
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
9
=
a
3
+ a
2a
3
+ 2a
a
11
=
a
4
a
3
a
2
1
a
4
a
3
a
2
2
a
8
=
a
3
+ a
a
3
+ a
a
7
=
a
3
+ a
a
3
+ a
a
7
=
a
3
+ a
a
3
+ a
(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 = 0.766826
b = 1.05077
4.04602 9.32390
u = 1.00000
a = 0.455697 1.200152I
b = 0.05352 3.02220I
7.51750 + 4.40083I 12.31681 3.97407I
u = 1.00000
a = 0.455697 + 1.200152I
b = 0.05352 + 3.02220I
7.51750 4.40083I 12.31681 + 3.97407I
u = 1.00000
a = 0.339110 0.822375I
b = 0.528134 1.303899I
1.97403 1.53058I 12.02124 + 2.62456I
u = 1.00000
a = 0.339110 + 0.822375I
b = 0.528134 + 1.303899I
1.97403 + 1.53058I 12.02124 2.62456I
3
II. I
u
2
= hu
67
6u
66
+ · · · 6u + 1, 16u
66
+ 81u
65
+ · · · + 8a
7, 57u
66
+ 289u
65
+ · · · + 16b + 61i
(i) Arc colorings
a
5
=
0
u
a
10
=
2u
66
81
8
u
65
+ ··· +
29
2
u +
7
8
3.56250u
66
18.0625u
65
+ ··· + 20.8125u 3.81250
a
2
=
1
0
a
6
=
8.68750u
66
+ 36.1875u
65
+ ··· + 6.06250u 2.56250
42.4375u
66
+ 210.188u
65
+ ··· 185.438u + 34.1875
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
9
=
71
4
u
66
317
4
u
65
+ ··· +
111
2
u 8
225
4
u
66
275u
65
+ ··· +
957
4
u 45
a
11
=
0.0625000u
66
0.312500u
65
+ ··· + 1.31250u + 1.93750
0.0625000u
66
0.312500u
65
+ ··· + 1.31250u 0.0625000
a
8
=
71
4
u
66
317
4
u
65
+ ··· +
111
2
u 8
21.2500u
66
103.500u
65
+ ··· + 93.5000u 17.7500
a
7
=
21u
66
381
4
u
65
+ ··· +
265
4
u
19
2
40.5000u
66
194.500u
65
+ ··· + 152.250u 27.2500
a
7
=
21u
66
381
4
u
65
+ ··· +
265
4
u
19
2
40.5000u
66
194.500u
65
+ ··· + 152.250u 27.2500
(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 = 1.286337 0.182808I
a = 0.503749 0.846198I
b = 0.63714 3.43060I
6.72971 + 6.47733I 10.81889 6.36716I
u = 1.286337 + 0.182808I
a = 0.503749 + 0.846198I
b = 0.63714 + 3.43060I
6.72971 6.47733I 10.81889 + 6.36716I
u = 1.254481 0.179113I
a = 0.574319 + 0.170509I
b = 1.144385 0.057300I
3.63648 + 1.86851I 7.74469 3.58479I
u = 1.254481 + 0.179113I
a = 0.574319 0.170509I
b = 1.144385 + 0.057300I
3.63648 1.86851I 7.74469 + 3.58479I
u = 1.251847 0.222499I
a = 0.148651 + 0.974286I
b = 0.40021 + 3.80751I
7.37230 2.08523I 12.61624 + 0.25269I
u = 1.251847 + 0.222499I
a = 0.148651 0.974286I
b = 0.40021 3.80751I
7.37230 + 2.08523I 12.61624 0.25269I
u = 1.225020 0.046584I
a = 0.403607 + 0.626198I
b = 0.496479 + 1.276996I
1.16301 + 1.95072I 3.00568 5.39206I
u = 1.225020 + 0.046584I
a = 0.403607 0.626198I
b = 0.496479 1.276996I
1.16301 1.95072I 3.00568 + 5.39206I
u = 1.058310 0.288428I
a = 0.173496 0.817735I
b = 0.80830 2.45933I
3.49198 1.02780I 15.4724 + 0.3169I
u = 1.058310 + 0.288428I
a = 0.173496 + 0.817735I
b = 0.80830 + 2.45933I
3.49198 + 1.02780I 15.4724 0.3169I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.047785 0.496161I
a = 0.740355 + 0.786160I
b = 0.96285 + 2.08190I
6.70689 0.47788I 13.15939 + 0.62936I
u = 1.047785 + 0.496161I
a = 0.740355 0.786160I
b = 0.96285 2.08190I
6.70689 + 0.47788I 13.15939 0.62936I
u = 1.022760 0.541798I
a = 0.093126 1.107096I
b = 1.26248 3.56745I
5.86773 9.08868I 11.37099 + 7.27344I
u = 1.022760 + 0.541798I
a = 0.093126 + 1.107096I
b = 1.26248 + 3.56745I
5.86773 + 9.08868I 11.37099 7.27344I
u = 1.010020 0.515425I
a = 0.660566 0.201521I
b = 0.94858 1.16687I
2.81565 4.36823I 8.36975 + 4.25487I
u = 1.010020 + 0.515425I
a = 0.660566 + 0.201521I
b = 0.94858 + 1.16687I
2.81565 + 4.36823I 8.36975 4.25487I
u = 0.887590 0.499549I
a = 0.042921 + 0.947362I
b = 1.80807 + 1.58915I
0.03256 4.08481I 5.93835 + 7.04941I
u = 0.887590 + 0.499549I
a = 0.042921 0.947362I
b = 1.80807 1.58915I
0.03256 + 4.08481I 5.93835 7.04941I
u = 0.797853 0.467895I
a = 0.850077 0.394612I
b = 0.164292 + 1.156811I
0.345703 + 0.068999I 4.55173 0.43344I
u = 0.797853 + 0.467895I
a = 0.850077 + 0.394612I
b = 0.164292 1.156811I
0.345703 0.068999I 4.55173 + 0.43344I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.735196
a = 0.423970
b = 0.519703
1.10322 8.76946
u = 0.501049 0.552756I
a = 1.59068 + 0.57127I
b = 0.787473 0.559176I
4.35980 + 4.63647I 9.51375 2.49759I
u = 0.501049 + 0.552756I
a = 1.59068 0.57127I
b = 0.787473 + 0.559176I
4.35980 4.63647I 9.51375 + 2.49759I
u = 0.481544 0.437537I
a = 0.680252 0.806020I
b = 0.539720 0.237662I
1.341862 + 0.241306I 6.37372 + 0.86588I
u = 0.481544 + 0.437537I
a = 0.680252 + 0.806020I
b = 0.539720 + 0.237662I
1.341862 0.241306I 6.37372 0.86588I
u = 0.351913 0.524199I
a = 1.16170 1.58819I
b = 0.052578 0.297784I
4.77435 3.67797I 10.21618 + 4.79091I
u = 0.351913 + 0.524199I
a = 1.16170 + 1.58819I
b = 0.052578 + 0.297784I
4.77435 + 3.67797I 10.21618 4.79091I
u = 0.113095 0.211783I
a = 3.49764 0.69203I
b = 0.048211 0.599860I
0.492874 1.272412I 5.34668 + 4.93990I
u = 0.113095 + 0.211783I
a = 3.49764 + 0.69203I
b = 0.048211 + 0.599860I
0.492874 + 1.272412I 5.34668 4.93990I
u = 0.331641 0.843858I
a = 1.079254 + 0.851282I
b = 1.100739 + 0.560025I
2.23668 1.18578I 8.34848 + 0.31014I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.331641 + 0.843858I
a = 1.079254 0.851282I
b = 1.100739 0.560025I
2.23668 + 1.18578I 8.34848 0.31014I
u = 0.360963 0.899604I
a = 1.330234 + 0.042740I
b = 1.56297 + 0.51725I
1.13304 9.79846I 6.75502 + 6.40442I
u = 0.360963 + 0.899604I
a = 1.330234 0.042740I
b = 1.56297 0.51725I
1.13304 + 9.79846I 6.75502 6.40442I
u = 0.374311 0.872073I
a = 0.163016 + 0.833425I
b = 0.322740 0.045910I
1.83934 4.96300I 3.50692 + 3.21590I
u = 0.374311 + 0.872073I
a = 0.163016 0.833425I
b = 0.322740 + 0.045910I
1.83934 + 4.96300I 3.50692 3.21590I
u = 0.481444 0.849559I
a = 1.020453 0.086750I
b = 0.489108 0.877868I
4.87488 4.14687I 1.59935 + 4.64549I
u = 0.481444 + 0.849559I
a = 1.020453 + 0.086750I
b = 0.489108 + 0.877868I
4.87488 + 4.14687I 1.59935 4.64549I
u = 0.534420 0.823507I
a = 0.281794 0.880900I
b = 0.632763 0.841927I
5.22376 + 0.38517I 0.51661 2.40952I
u = 0.534420 + 0.823507I
a = 0.281794 + 0.880900I
b = 0.632763 + 0.841927I
5.22376 0.38517I 0.51661 + 2.40952I
u = 0.575942 0.558676I
a = 1.117764 0.711552I
b = 0.642391 1.053019I
0.281369 0.970663I 7.60845 + 0.86547I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.575942 + 0.558676I
a = 1.117764 + 0.711552I
b = 0.642391 + 1.053019I
0.281369 + 0.970663I 7.60845 0.86547I
u = 0.639652 0.752359I
a = 0.648998 + 0.349476I
b = 0.376338 + 0.480154I
3.53541 + 1.62116I 1.50598 2.39002I
u = 0.639652 + 0.752359I
a = 0.648998 0.349476I
b = 0.376338 0.480154I
3.53541 1.62116I 1.50598 + 2.39002I
u = 0.704157 0.772274I
a = 0.466795 + 1.082116I
b = 0.11191 + 2.14669I
1.05507 + 6.16799I 5.26616 6.32160I
u = 0.704157 + 0.772274I
a = 0.466795 1.082116I
b = 0.11191 2.14669I
1.05507 6.16799I 5.26616 + 6.32160I
u = 0.912328 0.663370I
a = 0.912600 0.514369I
b = 0.811656 0.697183I
0.415223 0.749566I 6.20790 + 0.77032I
u = 0.912328 + 0.663370I
a = 0.912600 + 0.514369I
b = 0.811656 + 0.697183I
0.415223 + 0.749566I 6.20790 0.77032I
u = 0.939042 0.333718I
a = 0.856355 + 0.990770I
b = 0.68859 + 2.70408I
7.39058 3.26134I 10.40979 2.26306I
u = 0.939042 + 0.333718I
a = 0.856355 0.990770I
b = 0.68859 2.70408I
7.39058 + 3.26134I 10.40979 + 2.26306I
u = 0.943939 0.369724I
a = 0.738078 0.330946I
b = 1.031512 + 0.146812I
3.90258 + 1.39316I 7.44622 4.95368I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.943939 + 0.369724I
a = 0.738078 + 0.330946I
b = 1.031512 0.146812I
3.90258 1.39316I 7.44622 + 4.95368I
u = 0.982398 0.369460I
a = 0.022506 1.236414I
b = 0.80096 3.11496I
7.66696 + 5.81189I 11.6562 8.3452I
u = 0.982398 + 0.369460I
a = 0.022506 + 1.236414I
b = 0.80096 + 3.11496I
7.66696 5.81189I 11.6562 + 8.3452I
u = 0.982683 0.638281I
a = 0.293509 + 0.545871I
b = 0.159738 + 0.483130I
2.50124 + 3.65316I 3.04825 3.77451I
u = 0.982683 + 0.638281I
a = 0.293509 0.545871I
b = 0.159738 0.483130I
2.50124 3.65316I 3.04825 + 3.77451I
u = 1.039733 0.556055I
a = 0.339957 + 0.849842I
b = 0.32176 + 2.62236I
1.77198 + 5.50921I 11.54486 6.05946I
u = 1.039733 + 0.556055I
a = 0.339957 0.849842I
b = 0.32176 2.62236I
1.77198 5.50921I 11.54486 + 6.05946I
u = 1.063279 0.655308I
a = 0.676447 + 0.193941I
b = 0.296276 0.459721I
3.63203 + 5.13427I 3.00083 2.99523I
u = 1.063279 + 0.655308I
a = 0.676447 0.193941I
b = 0.296276 + 0.459721I
3.63203 5.13427I 3.00083 + 2.99523I
u = 1.098235 0.652944I
a = 0.037272 0.809710I
b = 1.05859 2.28373I
3.02110 + 9.72497I 4.76527 9.27372I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.098235 + 0.652944I
a = 0.037272 + 0.809710I
b = 1.05859 + 2.28373I
3.02110 9.72497I 4.76527 + 9.27372I
u = 1.150515 0.623562I
a = 0.606368 + 0.191190I
b = 1.26411 + 0.77269I
0.49031 + 10.48345I 6.68771 6.96472I
u = 1.150515 + 0.623562I
a = 0.606368 0.191190I
b = 1.26411 0.77269I
0.49031 10.48345I 6.68771 + 6.96472I
u = 1.151996 0.602107I
a = 0.681623 0.700492I
b = 0.36986 2.81662I
4.67059 + 6.53918I 11.45004 4.11624I
u = 1.151996 + 0.602107I
a = 0.681623 + 0.700492I
b = 0.36986 + 2.81662I
4.67059 6.53918I 11.45004 + 4.11624I
u = 1.164334 0.627272I
a = 0.080852 + 0.981912I
b = 0.55077 + 4.07067I
3.5558 + 15.4016I 9.79278 10.02195I
u = 1.164334 + 0.627272I
a = 0.080852 0.981912I
b = 0.55077 4.07067I
3.5558 15.4016I 9.79278 + 10.02195I
11
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
67
+ 6u
66
+ ··· 6u 1)
c
2
(u + 1)
5
(u
67
+ 32u
66
+ ··· 6u + 1)
c
3
, c
7
u
5
(u
67
+ u
66
+ ··· + 96u + 32)
c
4
(u + 1)
5
(u
67
+ 6u
66
+ ··· 6u 1)
c
5
(u
5
u
4
+ ··· + u 1)(u
67
+ 2u
66
+ ··· + 4u
2
1)
c
6
(u
5
+ u
4
+ ··· + u 1)(u
67
+ 2u
66
+ ··· + 78u + 9)
c
8
(u
5
u
4
+ ··· + u + 1)(u
67
+ 2u
66
+ ··· + 78u + 9)
c
9
(u
5
+ u
4
+ ··· + u + 1)(u
67
+ 2u
66
+ ··· + 4u
2
1)
c
10
(u
5
+ 3u
4
+ ··· u 1)(u
67
+ 36u
66
+ ··· + 8u 1)
c
11
(u
5
u
4
+ ··· + u + 1)(u
67
+ 8u
66
+ ··· + 2798u 53)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
67
32y
66
+ ··· 6y 1)
c
2
(y 1)
5
(y
67
+ 12y
66
+ ··· 266y 1)
c
3
, c
7
y
5
(y
67
+ 33y
66
+ ··· 14848y 1024)
c
5
(y
5
+ 3y
4
+ ··· y 1)(y
67
+ 36y
66
+ ··· + 8y 1)
c
6
(y
5
5y
4
+ ··· y 1)(y
67
52y
66
+ ··· 360y 81)
c
8
(y
5
5y
4
+ ··· y 1)(y
67
52y
66
+ ··· 360y 81)
c
9
(y
5
+ 3y
4
+ ··· y 1)(y
67
+ 36y
66
+ ··· + 8y 1)
c
10
(y
5
y
4
+ ··· + 3y 1)(y
67
8y
66
+ ··· + 124y 1)
c
11
(y
5
5y
4
+ ··· y 1)(y
67
+ 8y
66
+ ··· + 6484936y 2809)
13