11a
50
(K11a
50
)
1
Arc Sequences
5 1 7 2 4 10 3 11 6 9 8
Solving Sequence
1,5
2 3 4
6,8
7 11 9 10
c
1
c
2
c
4
c
5
c
7
c
11
c
8
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
6
a
5
a
4
+ 3a
2
2a + 1, a
5
+ a
4
+ a
3
3a
2
+ 5b + 2a + 2, 2a
5
+ 3a
4
+ 3a
3
+ a
2
9a + 5u + 6i
I
u
2
= hu
47
4u
46
+ ··· + 6u 1, 2u
46
+ 9u
45
+ ··· + 4b + 5, u
46
+ 18u
45
+ ··· + 4a 25i
There are 2 irreducible components with 53 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
a
5
a
4
+ 3a
2
2a + 1, a
5
+ a
4
+ a
3
3a
2
+ 5b + 2a +
2, 2a
5
+ 3a
4
+ 3a
3
+ a
2
9a + 5u + 6i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
2
5
a
5
3
5
a
4
+ ··· +
9
5
a
6
5
a
2
=
1
2
5
a
5
3
5
a
4
+ ··· +
9
5
a
1
5
a
3
=
2
5
a
5
+
3
5
a
4
+ ···
9
5
a +
6
5
2
5
a
5
3
5
a
4
+ ··· +
9
5
a
1
5
a
4
=
2
5
a
5
+
3
5
a
4
+ ···
9
5
a +
6
5
2
5
a
5
3
5
a
4
+ ··· +
9
5
a
1
5
a
6
=
1
0
a
8
=
a
1
5
a
5
1
5
a
4
+ ···
2
5
a
2
5
a
7
=
a
1
5
a
5
1
5
a
4
+ ···
2
5
a
2
5
a
11
=
2
5
a
5
+
2
5
a
4
+ ··· +
4
5
a +
4
5
2
5
a
5
2
5
a
4
+ ···
4
5
a +
1
5
a
9
=
a
5
a
3
a
2
+ 2a
1
5
a
5
1
5
a
4
+ ···
2
5
a
2
5
a
10
=
6
5
a
5
+
1
5
a
4
+ ··· +
12
5
a +
2
5
1
5
a
5
1
5
a
4
+ ···
2
5
a
2
5
a
10
=
6
5
a
5
+
1
5
a
4
+ ··· +
12
5
a +
2
5
1
5
a
5
1
5
a
4
+ ···
2
5
a
2
5
(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 = 0.500000 0.866025I
a = 1.024478 0.839835I
b = 0.215080 + 1.307141I
3.02413 + 4.85801I 2.74410 7.22587I
u = 0.500000 + 0.866025I
a = 1.024478 + 0.839835I
b = 0.215080 1.307141I
3.02413 4.85801I 2.74410 + 7.22587I
u = 0.500000 0.866025I
a = 0.284920 0.493496I
b = 0.569840
1.11345 + 2.02988I 12.72167 5.84990I
u = 0.500000 + 0.866025I
a = 0.284920 + 0.493496I
b = 0.569840
1.11345 2.02988I 12.72167 + 5.84990I
u = 0.500000 + 0.866025I
a = 1.239557 0.467306I
b = 0.215080 + 1.307141I
3.02413 + 0.79824I 4.03424 + 1.64667I
u = 0.500000 0.866025I
a = 1.239557 + 0.467306I
b = 0.215080 1.307141I
3.02413 0.79824I 4.03424 1.64667I
3
II. I
u
2
=
hu
47
4u
46
+· · ·+6u1, 2u
46
+9u
45
+· · ·+4b+5, u
46
+18u
45
+· · ·+4a25i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
6
=
u
3
u
5
+ u
3
+ u
a
8
=
1
4
u
46
9
2
u
45
+ ··· 17u +
25
4
1
2
u
46
9
4
u
45
+ ··· +
9
4
u
5
4
a
7
=
7
4
u
46
5
2
u
45
+ ··· + 2u +
7
4
9
2
u
46
+
41
4
u
45
+ ··· +
35
4
u
7
4
a
11
=
1
4
u
46
+
3
4
u
45
+ ··· +
13
4
u + 2
1
4
u
46
u
45
+ ···
5
2
u +
1
4
a
9
=
2u
46
+
15
2
u
45
+ ··· + 20u + 1
5
4
u
46
+
13
4
u
45
+ ···
11
4
u +
1
2
a
10
=
7
4
u
46
+
33
4
u
45
+ ··· +
107
4
u
1
2
5
4
u
46
+
9
4
u
45
+ ···
31
4
u +
3
2
a
10
=
7
4
u
46
+
33
4
u
45
+ ··· +
107
4
u
1
2
5
4
u
46
+
9
4
u
45
+ ···
31
4
u +
3
2
(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 = 0.867038 0.019562I
a = 0.12989 1.85582I
b = 0.08456 + 1.60423I
9.34836 + 3.22875I 0.48816 2.52460I
u = 0.867038 + 0.019562I
a = 0.12989 + 1.85582I
b = 0.08456 1.60423I
9.34836 3.22875I 0.48816 + 2.52460I
u = 0.810352 0.881677I
a = 1.20304 + 2.40238I
b = 0.06174 1.58029I
8.49103 0.19218I 2.01859 0.12637I
u = 0.810352 + 0.881677I
a = 1.20304 2.40238I
b = 0.06174 + 1.58029I
8.49103 + 0.19218I 2.01859 + 0.12637I
u = 0.803055 0.903733I
a = 0.99990 2.49478I
b = 0.11755 + 1.60103I
8.42204 + 6.23223I 2.23245 5.02146I
u = 0.803055 + 0.903733I
a = 0.99990 + 2.49478I
b = 0.11755 1.60103I
8.42204 6.23223I 2.23245 + 5.02146I
u = 0.663428 0.780790I
a = 1.00646 + 1.05901I
b = 0.205400 0.577345I
1.047096 + 0.807076I 4.48198 + 0.15159I
u = 0.663428 + 0.780790I
a = 1.00646 1.05901I
b = 0.205400 + 0.577345I
1.047096 0.807076I 4.48198 0.15159I
u = 0.656947 0.912090I
a = 0.22666 1.51107I
b = 0.392245 + 0.675540I
0.63898 + 4.31334I 6.53825 6.48689I
u = 0.656947 + 0.912090I
a = 0.22666 + 1.51107I
b = 0.392245 0.675540I
0.63898 4.31334I 6.53825 + 6.48689I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.563257 0.159760I
a = 0.608331 0.292239I
b = 0.248810 + 0.689997I
1.45889 + 1.89863I 0.82802 4.86862I
u = 0.563257 + 0.159760I
a = 0.608331 + 0.292239I
b = 0.248810 0.689997I
1.45889 1.89863I 0.82802 + 4.86862I
u = 0.461388 0.956277I
a = 0.665200 0.526100I
b = 0.316069 0.253513I
0.63663 + 1.64887I 2.14725 + 2.29266I
u = 0.461388 + 0.956277I
a = 0.665200 + 0.526100I
b = 0.316069 + 0.253513I
0.63663 1.64887I 2.14725 2.29266I
u = 0.322894 1.152853I
a = 1.126819 0.062685I
b = 0.03457 1.51727I
5.51392 + 0.84918I 3.71085 1.13770I
u = 0.322894 + 1.152853I
a = 1.126819 + 0.062685I
b = 0.03457 + 1.51727I
5.51392 0.84918I 3.71085 + 1.13770I
u = 0.311898 0.787865I
a = 0.956078 0.122134I
b = 0.1036155 0.0693880I
0.33163 + 1.48922I 3.21291 4.41196I
u = 0.311898 + 0.787865I
a = 0.956078 + 0.122134I
b = 0.1036155 + 0.0693880I
0.33163 1.48922I 3.21291 + 4.41196I
u = 0.294048 1.158778I
a = 1.041503 + 0.000140I
b = 0.17743 + 1.57625I
5.33133 + 7.16658I 4.32444 6.19083I
u = 0.294048 + 1.158778I
a = 1.041503 0.000140I
b = 0.17743 1.57625I
5.33133 7.16658I 4.32444 + 6.19083I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.199822 1.009777I
a = 0.834043 + 0.291155I
b = 0.625995 + 0.626906I
2.01497 + 4.25844I 10.22284 8.38293I
u = 0.199822 + 1.009777I
a = 0.834043 0.291155I
b = 0.625995 0.626906I
2.01497 4.25844I 10.22284 + 8.38293I
u = 0.026264 0.834708I
a = 1.139078 + 0.577019I
b = 0.739974 0.342805I
2.87835 0.31776I 14.00580 + 0.89851I
u = 0.026264 + 0.834708I
a = 1.139078 0.577019I
b = 0.739974 + 0.342805I
2.87835 + 0.31776I 14.00580 0.89851I
u = 0.143780
a = 3.43014
b = 0.444877
0.906933 11.3954
u = 0.236667 0.699953I
a = 2.00141 + 0.89361I
b = 0.27531 1.39228I
2.56997 3.95764I 7.31001 + 0.68586I
u = 0.236667 + 0.699953I
a = 2.00141 0.89361I
b = 0.27531 + 1.39228I
2.56997 + 3.95764I 7.31001 0.68586I
u = 0.262896 0.609439I
a = 2.19053 0.65507I
b = 0.102273 + 1.283388I
2.83383 + 1.80935I 5.60137 4.89150I
u = 0.262896 + 0.609439I
a = 2.19053 + 0.65507I
b = 0.102273 1.283388I
2.83383 1.80935I 5.60137 + 4.89150I
u = 0.749711 0.881463I
a = 0.133762 + 0.692661I
b = 1.094818 + 0.057901I
1.38780 2.84463I 6.42020 + 2.87095I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.749711 + 0.881463I
a = 0.133762 0.692661I
b = 1.094818 0.057901I
1.38780 + 2.84463I 6.42020 2.87095I
u = 0.760355 0.975109I
a = 0.78027 + 1.38920I
b = 0.832040 0.941509I
3.97968 9.12021I 4.70207 + 8.49829I
u = 0.760355 + 0.975109I
a = 0.78027 1.38920I
b = 0.832040 + 0.941509I
3.97968 + 9.12021I 4.70207 8.49829I
u = 0.796590 1.043644I
a = 1.65368 + 1.73183I
b = 0.26885 1.69481I
12.7501 13.4737I 1.97906 + 7.86996I
u = 0.796590 + 1.043644I
a = 1.65368 1.73183I
b = 0.26885 + 1.69481I
12.7501 + 13.4737I 1.97906 7.86996I
u = 0.797400 0.942409I
a = 0.817901 0.783072I
b = 0.484733 + 0.494859I
6.27252 4.67969I 0.19550 + 3.17611I
u = 0.797400 + 0.942409I
a = 0.817901 + 0.783072I
b = 0.484733 0.494859I
6.27252 + 4.67969I 0.19550 3.17611I
u = 0.807992 1.034532I
a = 1.69856 1.55368I
b = 0.14248 + 1.55761I
13.2416 6.9328I 1.14544 + 3.17565I
u = 0.807992 + 1.034532I
a = 1.69856 + 1.55368I
b = 0.14248 1.55761I
13.2416 + 6.9328I 1.14544 3.17565I
u = 0.810606 0.776375I
a = 0.565835 0.632629I
b = 0.750225 + 1.009345I
4.58669 + 3.21526I 3.05328 3.33895I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.810606 + 0.776375I
a = 0.565835 + 0.632629I
b = 0.750225 1.009345I
4.58669 3.21526I 3.05328 + 3.33895I
u = 0.830386 0.839820I
a = 0.137676 + 0.366174I
b = 0.439191 0.593639I
6.59048 1.40114I 0.38925 + 2.24691I
u = 0.830386 + 0.839820I
a = 0.137676 0.366174I
b = 0.439191 + 0.593639I
6.59048 + 1.40114I 0.38925 2.24691I
u = 0.927893 0.738669I
a = 0.28332 1.95602I
b = 0.22878 + 1.70233I
13.7092 + 7.1190I 0.57186 3.20529I
u = 0.927893 + 0.738669I
a = 0.28332 + 1.95602I
b = 0.22878 1.70233I
13.7092 7.1190I 0.57186 + 3.20529I
u = 0.928006 0.759035I
a = 0.06191 + 1.86044I
b = 0.10905 1.59219I
14.10936 + 0.53726I 0.02243 + 1.50138I
u = 0.928006 + 0.759035I
a = 0.06191 1.86044I
b = 0.10905 + 1.59219I
14.10936 0.53726I 0.02243 1.50138I
9
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
47
+ 4u
46
+ ··· + 6u + 1)
c
2
, c
5
(u
2
+ u + 1)
3
(u
47
+ 14u
46
+ ··· + 38u 1)
c
3
, c
7
u
6
(u
47
+ u
46
+ ··· + 96u 64)
c
4
(u
2
u + 1)
3
(u
47
+ 4u
46
+ ··· + 6u + 1)
c
6
(u
3
+ u
2
1)
2
(u
47
+ 3u
46
+ ··· u + 1)
c
8
(u
3
u
2
+ 2u 1)
2
(u
47
+ 11u
46
+ ··· + u + 1)
c
9
(u
3
u
2
+ 1)
2
(u
47
+ 3u
46
+ ··· u + 1)
c
10
, c
11
(u
3
+ u
2
+ 2u + 1)
2
(u
47
+ 11u
46
+ ··· + u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
3
(y
47
+ 14y
46
+ ··· + 38y 1)
c
2
, c
5
(y
2
+ y + 1)
3
(y
47
+ 42y
46
+ ··· + 2346y 1)
c
3
, c
7
y
6
(y
47
+ 35y
46
+ ··· 23552y 4096)
c
6
, c
9
(y
3
y
2
+ 2y 1)
2
(y
47
11y
46
+ ··· + y 1)
c
8
, c
10
, c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
47
+ 53y
46
+ ··· + 41y 1)
11