11a
275
(K11a
275
)
1
Arc Sequences
5 7 1 10 2 9 3 11 6 4 8
Solving Sequence
6,9 2,7
3 8 10 5 1 4 11
c
6
c
2
c
7
c
9
c
5
c
1
c
4
c
11
c
3
, c
8
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
16
+ 4u
15
+ 4u
14
3u
13
9u
12
9u
11
4u
10
+ 6u
9
+ 13u
8
+ 11u
7
+ 4u
6
8u
5
13u
4
5u
3
+ 4u
2
+ 4u + 1,
u
15
+ u
14
+ ··· + 7b 9, 61u
15
208u
14
+ ··· + 7a 116i
I
u
2
= hu
80
u
79
+ ··· + 6u 1, 4.17432 × 10
182
u
79
+ 9.96041 × 10
181
u
78
+ ··· + 1.39633 × 10
181
b 5.52448 × 10
182
,
1.24614 × 10
183
u
79
3.07904 × 10
182
u
78
+ ··· + 1.39633 × 10
181
a + 1.67471 × 10
183
i
There are 2 irreducible components with 96 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
15
+· · ·+4u+1, u
15
+u
14
+· · ·+7b9, 61u
15
208u
14
+· · ·+7a116i
(i) Arc colorings
a
6
=
0
u
a
9
=
8.71429u
15
+ 29.7143u
14
+ ··· + 43.4286u + 16.5714
1
7
u
15
1
7
u
14
+ ··· +
19
7
u +
9
7
a
2
=
1
0
a
7
=
18
7
u
15
+
32
7
u
14
+ ··· +
50
7
u
43
7
4
7
u
15
+
11
7
u
14
+ ···
20
7
u
15
7
a
3
=
15
7
u
15
+
71
7
u
14
+ ··· +
65
7
u +
75
7
16
7
u
15
+
65
7
u
14
+ ··· +
11
7
u +
10
7
a
8
=
20
7
u
15
+
48
7
u
14
+ ··· +
152
7
u +
37
7
13
7
u
15
+
62
7
u
14
+ ···
107
7
u
26
7
a
10
=
8.71429u
15
+ 29.7143u
14
+ ··· + 43.4286u + 16.5714
9
7
u
15
+
30
7
u
14
+ ··· +
102
7
u +
45
7
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
15
7
u
15
+
71
7
u
14
+ ··· +
72
7
u +
68
7
9
7
u
15
+
44
7
u
14
+ ···
10
7
u +
3
7
a
11
=
8.71429u
15
31.7143u
14
+ ··· 22.4286u 9.57143
4.71429u
15
20.7143u
14
+ ··· + 4.57143u + 0.428571
a
11
=
8.71429u
15
31.7143u
14
+ ··· 22.4286u 9.57143
4.71429u
15
20.7143u
14
+ ··· + 4.57143u + 0.428571
(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 = 1.70034 0.18352I
a = 0.094675 + 0.232751I
b = 0.433638 0.737020I
8.21157 + 3.18915I 16.8553 0.1010I
u = 1.70034 + 0.18352I
a = 0.094675 0.232751I
b = 0.433638 + 0.737020I
8.21157 3.18915I 16.8553 + 0.1010I
u = 1.114263 0.514050I
a = 1.138075 + 0.514998I
b = 1.208658 0.510688I
1.36811 5.87068I 3.73603 + 6.05900I
u = 1.114263 + 0.514050I
a = 1.138075 0.514998I
b = 1.208658 + 0.510688I
1.36811 + 5.87068I 3.73603 6.05900I
u = 0.606505 0.489801I
a = 1.115326 + 0.047548I
b = 2.13324 + 0.31490I
3.58696 6.81151I 11.0601 + 13.1790I
u = 0.606505 + 0.489801I
a = 1.115326 0.047548I
b = 2.13324 0.31490I
3.58696 + 6.81151I 11.0601 13.1790I
u = 0.522619 0.189004I
a = 0.46878 3.33017I
b = 0.196791 0.073543I
3.98234 + 2.11939I 1.081156 0.586483I
u = 0.522619 + 0.189004I
a = 0.46878 + 3.33017I
b = 0.196791 + 0.073543I
3.98234 2.11939I 1.081156 + 0.586483I
u = 0.252552 1.034956I
a = 1.000407 0.544203I
b = 1.181516 0.299233I
4.17484 + 1.01269I 2.56085 0.73282I
u = 0.252552 + 1.034956I
a = 1.000407 + 0.544203I
b = 1.181516 + 0.299233I
4.17484 1.01269I 2.56085 + 0.73282I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.260836 1.037797I
a = 0.585606 0.323554I
b = 2.00369 0.49582I
2.13903 1.01254I 1.87145 + 5.73964I
u = 0.260836 + 1.037797I
a = 0.585606 + 0.323554I
b = 2.00369 + 0.49582I
2.13903 + 1.01254I 1.87145 5.73964I
u = 0.933085 0.164926I
a = 0.811766 + 0.813283I
b = 0.813621 0.297282I
1.48457 + 2.42366I 12.58854 4.41901I
u = 0.933085 + 0.164926I
a = 0.811766 0.813283I
b = 0.813621 + 0.297282I
1.48457 2.42366I 12.58854 + 4.41901I
u = 1.002355 0.178823I
a = 0.267272 0.931683I
b = 2.67918 + 0.39597I
1.67108 + 0.41329I 23.4089 + 3.7817I
u = 1.002355 + 0.178823I
a = 0.267272 + 0.931683I
b = 2.67918 0.39597I
1.67108 0.41329I 23.4089 3.7817I
4
II. I
u
2
=
hu
80
u
79
+· · ·+6u1, 4.17×10
182
u
79
+9.96×10
181
u
78
+· · ·+1.40×10
181
b
5.52×10
182
, 1.25×10
183
u
79
3.08×10
182
u
78
+· · ·+1.40×10
181
a+1.67×10
183
i
(i) Arc colorings
a
6
=
0
u
a
9
=
89.2437u
79
+ 22.0510u
78
+ ··· + 544.648u 119.936
29.8950u
79
7.13328u
78
+ ··· 188.900u + 39.5643
a
2
=
1
0
a
7
=
49.9530u
79
10.1178u
78
+ ··· 298.128u + 58.7911
7.98809u
79
2.13996u
78
+ ··· 52.1496u + 10.1055
a
3
=
17.1292u
79
+ 2.11056u
78
+ ··· + 95.1710u 14.7175
11.0890u
79
+ 2.94419u
78
+ ··· + 71.2321u 13.3223
a
8
=
54.7051u
79
13.6367u
78
+ ··· 339.531u + 72.2928
4.83266u
79
+ 1.09306u
78
+ ··· 18.3818u + 4.28124
a
10
=
89.2437u
79
+ 22.0510u
78
+ ··· + 544.648u 119.936
21.0206u
79
+ 5.13656u
78
+ ··· + 125.012u 27.6284
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
10.6398u
79
+ 1.10347u
78
+ ··· + 56.9203u 8.85042
22.3895u
79
+ 5.39795u
78
+ ··· + 139.416u 27.6512
a
11
=
69.4000u
79
15.3467u
78
+ ··· 423.670u + 87.6568
16.5205u
79
+ 4.50873u
78
+ ··· + 100.299u 23.2041
a
11
=
69.4000u
79
15.3467u
78
+ ··· 423.670u + 87.6568
16.5205u
79
+ 4.50873u
78
+ ··· + 100.299u 23.2041
(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 = 1.86404 0.21206I
a = 0.033610 0.389234I
b = 0.338451 + 0.468699I
7.72237 + 3.39581I 2.62157 5.42993I
u = 1.86404 + 0.21206I
a = 0.033610 + 0.389234I
b = 0.338451 0.468699I
7.72237 3.39581I 2.62157 + 5.42993I
u = 1.365604 0.175084I
a = 0.651242 0.341051I
b = 0.047194 0.204442I
10.27104 4.13973I 15.7867 + 1.8019I
u = 1.365604 + 0.175084I
a = 0.651242 + 0.341051I
b = 0.047194 + 0.204442I
10.27104 + 4.13973I 15.7867 1.8019I
u = 1.32951
a = 0.724080
b = 0.0352726
6.48924 14.1025
u = 1.244741 0.562971I
a = 0.699773 0.686631I
b = 1.71717 + 0.46640I
1.63511 10.27853I 6.83143 + 8.38677I
u = 1.244741 + 0.562971I
a = 0.699773 + 0.686631I
b = 1.71717 0.46640I
1.63511 + 10.27853I 6.83143 8.38677I
u = 1.221698 0.524298I
a = 0.606933 + 0.660408I
b = 1.03724 1.74870I
5.77402 + 0.98954I 11.12740 0.57373I
u = 1.221698 + 0.524298I
a = 0.606933 0.660408I
b = 1.03724 + 1.74870I
5.77402 0.98954I 11.12740 + 0.57373I
u = 1.219073 0.431198I
a = 0.695672 0.794061I
b = 1.19475 + 1.39911I
2.57891 5.15325I 9.93239 + 4.60505I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.219073 + 0.431198I
a = 0.695672 + 0.794061I
b = 1.19475 1.39911I
2.57891 + 5.15325I 9.93239 4.60505I
u = 1.208242 0.650185I
a = 0.645927 + 0.536884I
b = 1.58353 0.37164I
3.92912 5.75666I 2.36580 + 3.05869I
u = 1.208242 + 0.650185I
a = 0.645927 0.536884I
b = 1.58353 + 0.37164I
3.92912 + 5.75666I 2.36580 3.05869I
u = 1.199522 0.436737I
a = 0.776521 + 0.735599I
b = 1.57919 1.45389I
7.51244 9.85286I 12.10596 + 7.10888I
u = 1.199522 + 0.436737I
a = 0.776521 0.735599I
b = 1.57919 + 1.45389I
7.51244 + 9.85286I 12.10596 7.10888I
u = 1.074304 0.512049I
a = 1.119503 0.674580I
b = 1.31849 + 0.58216I
0.33774 4.80042I 8.57856 + 3.44500I
u = 1.074304 + 0.512049I
a = 1.119503 + 0.674580I
b = 1.31849 0.58216I
0.33774 + 4.80042I 8.57856 3.44500I
u = 1.059497 0.561243I
a = 1.257694 + 0.356390I
b = 1.299155 0.546716I
0.29351 6.24678I 10.51712 + 6.83067I
u = 1.059497 + 0.561243I
a = 1.257694 0.356390I
b = 1.299155 + 0.546716I
0.29351 + 6.24678I 10.51712 6.83067I
u = 0.967920 0.489832I
a = 0.838299 + 0.257212I
b = 0.927575 + 0.226556I
2.35882 5.51067I 9.84350 + 5.87992I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.967920 + 0.489832I
a = 0.838299 0.257212I
b = 0.927575 0.226556I
2.35882 + 5.51067I 9.84350 5.87992I
u = 0.728847 0.455431I
a = 0.287732 0.642319I
b = 0.636043 + 0.267204I
1.91146 1.96925I 2.72689 + 3.92118I
u = 0.728847 + 0.455431I
a = 0.287732 + 0.642319I
b = 0.636043 0.267204I
1.91146 + 1.96925I 2.72689 3.92118I
u = 0.684548 0.092570I
a = 0.95594 + 1.30363I
b = 0.542804 + 0.021856I
0.81827 + 2.05131I 3.05692 + 0.85893I
u = 0.684548 + 0.092570I
a = 0.95594 1.30363I
b = 0.542804 0.021856I
0.81827 2.05131I 3.05692 0.85893I
u = 0.447980 0.630904I
a = 0.058243 0.425268I
b = 1.81159 0.50025I
3.20065 6.05158I 6.61264 + 3.93826I
u = 0.447980 + 0.630904I
a = 0.058243 + 0.425268I
b = 1.81159 + 0.50025I
3.20065 + 6.05158I 6.61264 3.93826I
u = 0.442076 0.614277I
a = 1.29575 + 1.42306I
b = 0.896350 + 0.209188I
2.20909 + 0.33519I 6.58348 + 0.96753I
u = 0.442076 + 0.614277I
a = 1.29575 1.42306I
b = 0.896350 0.209188I
2.20909 0.33519I 6.58348 0.96753I
u = 0.440321 0.673105I
a = 0.96121 1.50834I
b = 0.645856 0.257557I
2.09582 + 1.45533I 7.26748 0.30058I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.440321 + 0.673105I
a = 0.96121 + 1.50834I
b = 0.645856 + 0.257557I
2.09582 1.45533I 7.26748 + 0.30058I
u = 0.400724 1.016076I
a = 1.004915 0.480578I
b = 1.235420 + 0.530364I
6.51711 0.28172I 0.56456 + 1.39014I
u = 0.400724 + 1.016076I
a = 1.004915 + 0.480578I
b = 1.235420 0.530364I
6.51711 + 0.28172I 0.56456 1.39014I
u = 0.220687 0.347615I
a = 0.13254 + 1.83276I
b = 0.306392 + 0.250146I
0.75850 + 1.84706I 5.49749 3.01488I
u = 0.220687 + 0.347615I
a = 0.13254 1.83276I
b = 0.306392 0.250146I
0.75850 1.84706I 5.49749 + 3.01488I
u = 0.190591 0.919605I
a = 1.36331 + 0.50542I
b = 1.367216 0.218965I
4.86269 + 4.86500I 2.31634 4.47073I
u = 0.190591 + 0.919605I
a = 1.36331 0.50542I
b = 1.367216 + 0.218965I
4.86269 4.86500I 2.31634 + 4.47073I
u = 0.046673 0.232573I
a = 3.06219 + 1.49670I
b = 1.031254 + 0.653277I
1.09182 + 1.88602I 3.93284 6.84227I
u = 0.046673 + 0.232573I
a = 3.06219 1.49670I
b = 1.031254 0.653277I
1.09182 1.88602I 3.93284 + 6.84227I
u = 0.038672 1.233929I
a = 0.568324 + 0.274814I
b = 2.04739 + 0.47108I
1.84806 + 0.31109I 8.59302 + 2.40672I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.038672 + 1.233929I
a = 0.568324 0.274814I
b = 2.04739 0.47108I
1.84806 0.31109I 8.59302 2.40672I
u = 0.199019 0.681360I
a = 1.41536 0.55430I
b = 1.61591 0.46580I
3.86467 + 5.85230I 9.22119 4.58171I
u = 0.199019 + 0.681360I
a = 1.41536 + 0.55430I
b = 1.61591 + 0.46580I
3.86467 5.85230I 9.22119 + 4.58171I
u = 0.252459 0.114986I
a = 2.00420 2.60228I
b = 1.52744 + 0.38935I
3.58794 6.02169I 10.86973 + 2.67201I
u = 0.252459 + 0.114986I
a = 2.00420 + 2.60228I
b = 1.52744 0.38935I
3.58794 + 6.02169I 10.86973 2.67201I
u = 0.306276 1.309435I
a = 0.764041 0.546983I
b = 1.61842 0.28687I
0.18620 9.95806I 6.40665 + 6.67717I
u = 0.306276 + 1.309435I
a = 0.764041 + 0.546983I
b = 1.61842 + 0.28687I
0.18620 + 9.95806I 6.40665 6.67717I
u = 0.452023
a = 0.481863
b = 0.356676
0.718352 14.2619
u = 0.468858 0.776453I
a = 0.267160 0.932666I
b = 1.004335 0.132186I
4.43027 + 1.02077I 11.34703 + 0.00095I
u = 0.468858 + 0.776453I
a = 0.267160 + 0.932666I
b = 1.004335 + 0.132186I
4.43027 1.02077I 11.34703 0.00095I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.59005 1.44645I
a = 0.512441 + 0.476887I
b = 1.68585 + 0.00681I
3.03457 2.75755I 6.68824 + 5.61340I
u = 0.59005 + 1.44645I
a = 0.512441 0.476887I
b = 1.68585 0.00681I
3.03457 + 2.75755I 6.68824 5.61340I
u = 0.651728 0.201624I
a = 1.12767 1.17204I
b = 1.27325 + 0.89025I
0.84491 + 2.13326I 4.59887 2.62126I
u = 0.651728 + 0.201624I
a = 1.12767 + 1.17204I
b = 1.27325 0.89025I
0.84491 2.13326I 4.59887 + 2.62126I
u = 0.757995 0.007392I
a = 0.39244 + 2.56773I
b = 0.216581 0.835406I
3.44325 2.33828I 12.28624 + 5.22657I
u = 0.757995 + 0.007392I
a = 0.39244 2.56773I
b = 0.216581 + 0.835406I
3.44325 + 2.33828I 12.28624 5.22657I
u = 0.934597 0.108810I
a = 0.235798 + 1.047611I
b = 3.35764 1.68301I
1.54698 + 0.32425I 2.5271 + 43.4640I
u = 0.934597 + 0.108810I
a = 0.235798 1.047611I
b = 3.35764 + 1.68301I
1.54698 0.32425I 2.5271 43.4640I
u = 0.957483 0.269471I
a = 0.364357 0.136869I
b = 0.664030 + 1.041517I
3.66318 + 0.26666I 14.9328 0.9814I
u = 0.957483 + 0.269471I
a = 0.364357 + 0.136869I
b = 0.664030 1.041517I
3.66318 0.26666I 14.9328 + 0.9814I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.966164 0.359794I
a = 0.409170 + 0.730726I
b = 1.337836 0.269958I
0.328167 + 0.843678I 6.69440 1.81465I
u = 0.966164 + 0.359794I
a = 0.409170 0.730726I
b = 1.337836 + 0.269958I
0.328167 0.843678I 6.69440 + 1.81465I
u = 1.099083 0.188046I
a = 0.175105 0.666022I
b = 1.52341 1.25756I
2.33888 + 0.32984I 8.07472 + 4.98162I
u = 1.099083 + 0.188046I
a = 0.175105 + 0.666022I
b = 1.52341 + 1.25756I
2.33888 0.32984I 8.07472 4.98162I
u = 1.111850 0.790562I
a = 0.834333 + 0.313848I
b = 1.64948 1.19626I
5.89450 + 4.83270I 12.21221 5.22669I
u = 1.111850 + 0.790562I
a = 0.834333 0.313848I
b = 1.64948 + 1.19626I
5.89450 4.83270I 12.21221 + 5.22669I
u = 1.152845 0.428505I
a = 0.980754 0.731105I
b = 0.1022159 0.0077917I
6.52297 + 9.40235I 11.44040 7.08794I
u = 1.152845 + 0.428505I
a = 0.980754 + 0.731105I
b = 0.1022159 + 0.0077917I
6.52297 9.40235I 11.44040 + 7.08794I
u = 1.200571 0.409251I
a = 0.540865 + 0.890907I
b = 0.960507 0.145389I
0.524621 0.718329I 5.35435 + 2.41953I
u = 1.200571 + 0.409251I
a = 0.540865 0.890907I
b = 0.960507 + 0.145389I
0.524621 + 0.718329I 5.35435 2.41953I
12
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.209872 0.311520I
a = 0.767349 0.755587I
b = 0.662326 + 0.031360I
1.00831 + 3.62362I 6.29566 3.68166I
u = 1.209872 + 0.311520I
a = 0.767349 + 0.755587I
b = 0.662326 0.031360I
1.00831 3.62362I 6.29566 + 3.68166I
u = 1.214423 0.397022I
a = 0.774614 + 0.442145I
b = 0.075278 0.196081I
2.84331 + 4.24599I 9.21561 3.58842I
u = 1.214423 + 0.397022I
a = 0.774614 0.442145I
b = 0.075278 + 0.196081I
2.84331 4.24599I 9.21561 + 3.58842I
u = 1.273862 0.525082I
a = 0.341271 0.682552I
b = 0.476308 + 0.283423I
6.89106 0.74404I 11.94215 + 0.32385I
u = 1.273862 + 0.525082I
a = 0.341271 + 0.682552I
b = 0.476308 0.283423I
6.89106 + 0.74404I 11.94215 0.32385I
u = 1.32553 0.70779I
a = 0.868725 + 0.542951I
b = 1.63282 1.15912I
3.4640 + 16.9486I 9.01069 8.95671I
u = 1.32553 + 0.70779I
a = 0.868725 0.542951I
b = 1.63282 + 1.15912I
3.4640 16.9486I 9.01069 + 8.95671I
u = 1.33184 0.78181I
a = 0.783075 0.488278I
b = 1.60618 + 1.22134I
0.29094 + 10.51255I 7.02099 7.24029I
u = 1.33184 + 0.78181I
a = 0.783075 + 0.488278I
b = 1.60618 1.22134I
0.29094 10.51255I 7.02099 + 7.24029I
13
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
16
+ 4u
15
+ ··· + 4u + 1)(u
80
+ u
79
+ ··· 6u 1)
c
2
(u
16
+ 6u
13
+ ··· 2u + 1)(u
80
+ u
79
+ ··· 3416u 3721)
c
3
(u
16
+ 6u
15
+ ··· + 12u + 1)(u
80
+ 3u
79
+ ··· 35212u 5777)
c
4
(u
16
4u
15
+ ··· + 4u + 1)(u
80
+ 5u
79
+ ··· 306u 527)
c
5
(u
16
4u
15
+ ··· 4u + 1)(u
80
+ u
79
+ ··· 6u 1)
c
6
(u
16
3u
15
+ ··· + u + 1)(u
80
+ 4u
79
+ ··· 27u + 34)
c
7
(u
16
6u
13
+ ··· + 2u + 1)(u
80
+ u
79
+ ··· 3416u 3721)
c
8
(u
16
+ 3u
15
+ ··· u + 1)(u
80
+ 2u
79
+ ··· 181u + 31)
c
9
(u
16
+ 3u
15
+ ··· u + 1)(u
80
+ 4u
79
+ ··· 27u + 34)
c
10
(u
16
+ 4u
15
+ ··· 4u + 1)(u
80
+ 5u
79
+ ··· 306u 527)
c
11
(u
16
3u
15
+ ··· + u + 1)(u
80
+ 2u
79
+ ··· 181u + 31)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
16
8y
15
+ ··· 8y + 1)(y
80
43y
79
+ ··· 4y + 1)
c
2
(y
16
36y
14
+ ··· 2y + 1)
(y
80
47y
79
+ ··· 72209726y + 13845841)
c
3
(y
16
2y
15
+ ··· 16y + 1)
(y
80
25y
79
+ ··· 1723258088y + 33373729)
c
4
, c
10
(y
16
40y
14
+ ··· + 2y + 1)(y
80
+ 37y
79
+ ··· + 5061478y + 277729)
c
6
, c
9
(y
16
+ 13y
15
+ ··· + 13y + 1)(y
80
+ 46y
79
+ ··· 6577y + 1156)
c
7
(y
16
36y
14
+ ··· 2y + 1)
(y
80
47y
79
+ ··· 72209726y + 13845841)
c
8
, c
11
(y
16
+ 5y
15
+ ··· + 7y + 1)(y
80
+ 46y
79
+ ··· 21229y + 961)
15