11n
52
(K11n
52
)
1
Arc Sequences
4 1 8 2 10 11 4 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
34
6u
33
+ ··· + 4u 1, 12u
33
+ 61u
32
+ ··· + 8a 5, 43u
33
+ 219u
32
+ ··· + 16b 47i
There are 2 irreducible components with 39 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 9.63839
u = 1.00000
a = 0.455697 1.200152I
b = 0.05352 3.02220I
7.51750 + 4.40083I 13.17182 3.02310I
u = 1.00000
a = 0.455697 + 1.200152I
b = 0.05352 + 3.02220I
7.51750 4.40083I 13.17182 + 3.02310I
u = 1.00000
a = 0.339110 0.822375I
b = 0.528134 1.303899I
1.97403 1.53058I 5.00899 + 6.23673I
u = 1.00000
a = 0.339110 + 0.822375I
b = 0.528134 + 1.303899I
1.97403 + 1.53058I 5.00899 6.23673I
3
II. I
u
2
= hu
34
6u
33
+ · · · + 4u 1, 12u
33
+ 61u
32
+ · · · + 8a
5, 43u
33
+ 219u
32
+ · · · + 16b 47i
(i) Arc colorings
a
5
=
0
u
a
10
=
3
2
u
33
61
8
u
32
+ ···
35
4
u +
5
8
2.68750u
33
13.6875u
32
+ ··· 10.5625u + 2.93750
a
2
=
1
0
a
6
=
14.6875u
33
80.1875u
32
+ ··· 59.0625u + 24.4375
15.6875u
33
86.1875u
32
+ ··· 65.3125u + 28.1875
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
9
=
107
4
u
33
+
287
2
u
32
+ ··· + 102u
177
4
25.7500u
33
+ 137.500u
32
+ ··· + 95.7500u 40.5000
a
11
=
0.0625000u
33
+ 0.312500u
32
+ ··· 1.81250u 0.0625000
0.0625000u
33
+ 0.312500u
32
+ ··· 0.812500u 0.0625000
a
8
=
107
4
u
33
+
287
2
u
32
+ ··· + 102u
177
4
14.2500u
33
+ 76.2500u
32
+ ··· + 54.5000u 23.5000
a
7
=
125
4
u
33
+
675
4
u
32
+ ··· +
241
2
u 52
75
4
u
33
+
203
2
u
32
+ ··· + 73u
125
4
a
7
=
125
4
u
33
+
675
4
u
32
+ ··· +
241
2
u 52
75
4
u
33
+
203
2
u
32
+ ··· + 73u
125
4
(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 = 1.307533 0.065436I
a = 0.299142 + 0.884522I
b = 0.26581 + 3.82507I
5.63834 3.94702I 6.39479 + 3.36113I
u = 1.307533 + 0.065436I
a = 0.299142 0.884522I
b = 0.26581 3.82507I
5.63834 + 3.94702I 6.39479 3.36113I
u = 1.21971
a = 0.485550
b = 1.14126
2.64346 1.59533
u = 1.034701 0.173788I
a = 0.041112 0.660333I
b = 1.02029 1.27410I
2.09473 0.78471I 6.87662 2.65408I
u = 1.034701 + 0.173788I
a = 0.041112 + 0.660333I
b = 1.02029 + 1.27410I
2.09473 + 0.78471I 6.87662 + 2.65408I
u = 0.867707 0.523486I
a = 0.911057 + 0.742984I
b = 0.039980 + 0.922704I
4.35229 + 1.60461I 8.26502 1.09622I
u = 0.867707 + 0.523486I
a = 0.911057 0.742984I
b = 0.039980 0.922704I
4.35229 1.60461I 8.26502 + 1.09622I
u = 0.706452 0.661902I
a = 0.433006 1.202722I
b = 0.50525 1.99068I
3.70744 6.19607I 6.22304 + 6.67245I
u = 0.706452 + 0.661902I
a = 0.433006 + 1.202722I
b = 0.50525 + 1.99068I
3.70744 + 6.19607I 6.22304 6.67245I
u = 0.658221 0.529258I
a = 0.715183 0.315200I
b = 0.152256 0.692515I
0.99883 1.83078I 2.95289 + 3.76618I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.658221 + 0.529258I
a = 0.715183 + 0.315200I
b = 0.152256 + 0.692515I
0.99883 + 1.83078I 2.95289 3.76618I
u = 0.257061 0.435953I
a = 0.71364 + 1.38793I
b = 0.711858 + 0.021585I
0.37642 1.53920I 0.52977 + 5.14051I
u = 0.257061 + 0.435953I
a = 0.71364 1.38793I
b = 0.711858 0.021585I
0.37642 + 1.53920I 0.52977 5.14051I
u = 0.151137 0.378084I
a = 1.72188 + 1.72529I
b = 0.342124 + 1.171360I
0.33123 + 1.99737I 1.04171 3.94659I
u = 0.151137 + 0.378084I
a = 1.72188 1.72529I
b = 0.342124 1.171360I
0.33123 1.99737I 1.04171 + 3.94659I
u = 0.598522
a = 1.34216
b = 0.901374
3.18504 0.914630
u = 0.649218 0.049959I
a = 0.55537 1.91402I
b = 0.21433 2.31941I
6.76337 + 4.50518I 1.87945 4.07859I
u = 0.649218 + 0.049959I
a = 0.55537 + 1.91402I
b = 0.21433 + 2.31941I
6.76337 4.50518I 1.87945 + 4.07859I
u = 0.692953 1.024118I
a = 1.015907 0.151443I
b = 1.65138 0.21047I
2.49823 5.83735I 3.10039 + 3.72465I
u = 0.692953 + 1.024118I
a = 1.015907 + 0.151443I
b = 1.65138 + 0.21047I
2.49823 + 5.83735I 3.10039 3.72465I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.719802 0.838712I
a = 0.642453 + 0.855692I
b = 1.12515 + 1.72788I
0.59034 + 2.20193I 5.39762 2.89255I
u = 0.719802 + 0.838712I
a = 0.642453 0.855692I
b = 1.12515 1.72788I
0.59034 2.20193I 5.39762 + 2.89255I
u = 0.749373 0.980750I
a = 0.203215 + 0.635392I
b = 0.0406697 + 0.1141947I
5.05465 0.88184I 0.0341976 0.1167760I
u = 0.749373 + 0.980750I
a = 0.203215 0.635392I
b = 0.0406697 0.1141947I
5.05465 + 0.88184I 0.0341976 + 0.1167760I
u = 0.919309 0.963610I
a = 0.736247 + 0.034227I
b = 0.242674 0.367171I
7.32303 + 1.01150I 0.462803 + 0.538404I
u = 0.919309 + 0.963610I
a = 0.736247 0.034227I
b = 0.242674 + 0.367171I
7.32303 1.01150I 0.462803 0.538404I
u = 0.986984 0.934448I
a = 0.074663 0.754191I
b = 0.43827 1.77159I
7.11131 + 5.93371I 0.19300 5.69756I
u = 0.986984 + 0.934448I
a = 0.074663 + 0.754191I
b = 0.43827 + 1.77159I
7.11131 5.93371I 0.19300 + 5.69756I
u = 1.058347 0.773720I
a = 0.740255 0.603346I
b = 0.46407 2.48780I
0.44863 + 3.88868I 6.63703 2.26154I
u = 1.058347 + 0.773720I
a = 0.740255 + 0.603346I
b = 0.46407 + 2.48780I
0.44863 3.88868I 6.63703 + 2.26154I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.088073 0.833628I
a = 0.571374 + 0.237996I
b = 1.007891 + 0.420890I
3.98565 + 7.54944I 1.73478 4.55602I
u = 1.088073 + 0.833628I
a = 0.571374 0.237996I
b = 1.007891 0.420890I
3.98565 7.54944I 1.73478 + 4.55602I
u = 1.127075 0.824983I
a = 0.184448 + 0.911740I
b = 0.09128 + 3.62230I
1.13195 + 12.58770I 4.92167 7.87699I
u = 1.127075 + 0.824983I
a = 0.184448 0.911740I
b = 0.09128 3.62230I
1.13195 12.58770I 4.92167 + 7.87699I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
34
+ 6u
33
+ ··· 4u 1)
c
2
(u + 1)
5
(u
34
+ 10u
33
+ ··· + 2u
2
+ 1)
c
3
, c
7
u
5
(u
34
+ u
33
+ ··· 32u + 32)
c
4
(u + 1)
5
(u
34
+ 6u
33
+ ··· 4u 1)
c
5
(u
5
u
4
+ ··· + u 1)(u
34
+ 2u
33
+ ··· 2u 1)
c
6
(u
5
+ u
4
+ ··· + u 1)(u
34
+ 2u
33
+ ··· 58u 17)
c
8
(u
5
u
4
+ ··· + u + 1)(u
34
+ 2u
33
+ ··· 58u 17)
c
9
(u
5
+ u
4
+ ··· + u + 1)(u
34
+ 2u
33
+ ··· 2u 1)
c
10
(u
5
+ 3u
4
+ ··· u 1)(u
34
+ 18u
33
+ ··· + 2u + 1)
c
11
(u
5
u
4
+ ··· + u + 1)(u
34
+ 2u
33
+ ··· 2u + 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
34
10y
33
+ ··· + 2y
2
+ 1)
c
2
(y 1)
5
(y
34
+ 34y
33
+ ··· + 4y + 1)
c
3
, c
7
y
5
(y
34
33y
33
+ ··· 11776y + 1024)
c
5
(y
5
+ 3y
4
+ ··· y 1)(y
34
+ 18y
33
+ ··· + 2y + 1)
c
6
, c
8
(y
5
5y
4
+ ··· y 1)(y
34
22y
33
+ ··· + 682y + 289)
c
9
(y
5
+ 3y
4
+ ··· y 1)(y
34
+ 18y
33
+ ··· + 2y + 1)
c
10
(y
5
y
4
+ ··· + 3y 1)(y
34
2y
33
+ ··· 18y + 1)
c
11
(y
5
5y
4
+ ··· y 1)(y
34
+ 38y
33
+ ··· + 2y + 1)
10