11a
52
(K11a
52
)
1
Arc Sequences
5 1 11 2 9 4 3 10 6 8 7
Solving Sequence
5,9
6
2,10
1 3 4 7 8 11
c
5
c
9
c
1
c
2
c
4
c
6
c
8
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, a u 1, b + 3u + 1i
I
u
2
= hu
70
+ 2u
69
+ ··· + 7u + 1, 4.29315 × 10
70
u
69
+ 9.25186 × 10
70
u
68
+ ··· + 4.17991 × 10
70
b + 8.65097 × 10
70
,
6.25102 × 10
70
u
69
1.20643 × 10
71
u
68
+ ··· + 4.17991 × 10
70
a 3.98376 × 10
71
i
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
= hu
2
+ u + 1, a u 1, b + 3u + 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
u + 1
3u 1
a
6
=
u + 1
2u 1
a
2
=
1
0
a
10
=
0
u
a
1
=
1
u 1
a
3
=
u
u
a
4
=
u
u
a
7
=
u + 2
2u 2
a
8
=
u + 1
2u 1
a
11
=
u 1
u + 1
a
11
=
u 1
u + 1
(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 = 0.500000 0.866025I
b = 0.50000 + 2.59808I
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.50000 2.59808I
1.64493 2.02988I 3.00000 + 3.46410I
3
II. I
u
2
=
hu
70
+ 2u
69
+ · · · +7u + 1, 4.29 × 10
70
u
69
+ 9.25 × 10
70
u
68
+ · · · + 4.18 × 10
70
b +
8.65× 10
70
, 6.25×10
70
u
69
1.21 × 10
71
u
68
+· · · +4.18 × 10
70
a 3.98 ×10
71
i
(i) Arc colorings
a
5
=
0
u
a
9
=
1.49549u
69
+ 2.88627u
68
+ ··· + 11.1491u + 9.53072
1.02709u
69
2.21341u
68
+ ··· 4.32788u 2.06965
a
6
=
1.48730u
69
+ 2.86554u
68
+ ··· + 1.33091u + 8.39096
0.657446u
69
1.05978u
68
+ ··· 4.77620u 2.02891
a
2
=
1
0
a
10
=
0.273410u
69
0.519559u
68
+ ··· + 0.135853u + 0.316491
0.0101833u
69
0.253892u
68
+ ··· + 1.28544u 0.0243111
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
4
=
u
u
a
7
=
1.52549u
69
+ 3.20894u
68
+ ··· + 3.18319u + 8.53702
0.695638u
69
1.40318u
68
+ ··· 6.62848u 2.17497
a
8
=
1.32926u
69
+ 2.65160u
68
+ ··· 1.32082u + 6.55666
0.448125u
69
0.872997u
68
+ ··· 6.11574u 2.04777
a
11
=
1.90147u
69
3.00687u
68
+ ··· 17.2798u 7.54591
0.445769u
69
+ 0.536167u
68
+ ··· + 2.09334u + 1.10540
a
11
=
1.90147u
69
3.00687u
68
+ ··· 17.2798u 7.54591
0.445769u
69
+ 0.536167u
68
+ ··· + 2.09334u + 1.10540
(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 = 0.954425 0.553357I
a = 1.150122 + 0.165836I
b = 0.866121 0.472624I
7.36308 10.70735I 3.01709 + 6.14665I
u = 0.954425 + 0.553357I
a = 1.150122 0.165836I
b = 0.866121 + 0.472624I
7.36308 + 10.70735I 3.01709 6.14665I
u = 0.933958 0.585997I
a = 0.166743 1.054817I
b = 0.229351 + 0.766410I
8.09011 4.47407I 4.24940 + 1.33305I
u = 0.933958 + 0.585997I
a = 0.166743 + 1.054817I
b = 0.229351 0.766410I
8.09011 + 4.47407I 4.24940 1.33305I
u = 0.776298 0.488443I
a = 1.160648 + 0.295201I
b = 0.786256 0.488134I
0.03545 5.59321I 0.93885 + 5.88908I
u = 0.776298 + 0.488443I
a = 1.160648 0.295201I
b = 0.786256 + 0.488134I
0.03545 + 5.59321I 0.93885 5.88908I
u = 0.725223 1.096767I
a = 0.886974 0.091424I
b = 0.885386 0.136936I
6.51009 + 10.55953I 2.16375 5.53235I
u = 0.725223 + 1.096767I
a = 0.886974 + 0.091424I
b = 0.885386 + 0.136936I
6.51009 10.55953I 2.16375 + 5.53235I
u = 0.720698 0.648573I
a = 0.457752 + 0.676305I
b = 0.223376 0.557342I
3.13610 1.75909I 5.49516 + 1.80706I
u = 0.720698 + 0.648573I
a = 0.457752 0.676305I
b = 0.223376 + 0.557342I
3.13610 + 1.75909I 5.49516 1.80706I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.719263 1.118333I
a = 0.080090 0.972006I
b = 0.33627 + 3.01349I
5.6142 + 16.8243I 0.71174 10.03515I
u = 0.719263 + 1.118333I
a = 0.080090 + 0.972006I
b = 0.33627 3.01349I
5.6142 16.8243I 0.71174 + 10.03515I
u = 0.697500 0.803718I
a = 0.178031 + 1.124480I
b = 0.187669 1.075527I
6.85272 + 3.28181I 6.45434 3.67109I
u = 0.697500 + 0.803718I
a = 0.178031 1.124480I
b = 0.187669 + 1.075527I
6.85272 3.28181I 6.45434 + 3.67109I
u = 0.689342 0.894211I
a = 1.035608 0.003596I
b = 0.791092 + 0.192095I
6.57916 + 2.04356I 5.98703 2.65335I
u = 0.689342 + 0.894211I
a = 1.035608 + 0.003596I
b = 0.791092 0.192095I
6.57916 2.04356I 5.98703 + 2.65335I
u = 0.685303 0.740870I
a = 1.282462 + 0.185307I
b = 0.336677 + 0.039372I
6.10986 3.15378I 5.18638 + 2.28891I
u = 0.685303 + 0.740870I
a = 1.282462 0.185307I
b = 0.336677 0.039372I
6.10986 + 3.15378I 5.18638 2.28891I
u = 0.664262 0.939808I
a = 0.030828 + 1.144787I
b = 0.24220 2.62804I
5.50755 + 8.37088I 3.57481 8.42575I
u = 0.664262 + 0.939808I
a = 0.030828 1.144787I
b = 0.24220 + 2.62804I
5.50755 8.37088I 3.57481 + 8.42575I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.661839 0.998275I
a = 0.592526 0.385144I
b = 0.462170 + 0.535095I
2.09416 + 7.06733I 3.10109 7.21627I
u = 0.661839 + 0.998275I
a = 0.592526 + 0.385144I
b = 0.462170 0.535095I
2.09416 7.06733I 3.10109 + 7.21627I
u = 0.642942 1.069942I
a = 0.250713 + 0.894107I
b = 0.09210 2.87809I
1.65551 + 10.94813I 3.48280 9.65202I
u = 0.642942 + 1.069942I
a = 0.250713 0.894107I
b = 0.09210 + 2.87809I
1.65551 10.94813I 3.48280 + 9.65202I
u = 0.581386 0.947779I
a = 0.768796 0.774844I
b = 0.32121 + 1.82192I
1.93471 + 3.15779I 1.32353 9.16120I
u = 0.581386 + 0.947779I
a = 0.768796 + 0.774844I
b = 0.32121 1.82192I
1.93471 3.15779I 1.32353 + 9.16120I
u = 0.546606 0.763209I
a = 0.599370 1.121498I
b = 0.04744 + 2.10166I
1.30105 + 1.37775I 4.35855 + 4.78657I
u = 0.546606 + 0.763209I
a = 0.599370 + 1.121498I
b = 0.04744 2.10166I
1.30105 1.37775I 4.35855 4.78657I
u = 0.225350 0.984742I
a = 0.212970 1.060490I
b = 0.72523 + 2.48015I
3.99310 + 2.18696I 8.57359 2.22609I
u = 0.225350 + 0.984742I
a = 0.212970 + 1.060490I
b = 0.72523 2.48015I
3.99310 2.18696I 8.57359 + 2.22609I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.172499 0.029720I
a = 4.63826 1.73204I
b = 0.486621 + 0.483308I
1.46844 + 0.34695I 5.88246 0.51749I
u = 0.172499 + 0.029720I
a = 4.63826 + 1.73204I
b = 0.486621 0.483308I
1.46844 0.34695I 5.88246 + 0.51749I
u = 0.036237 1.176333I
a = 0.166737 + 0.841040I
b = 0.27535 2.82107I
5.59225 3.69663I 9.44807 + 5.64116I
u = 0.036237 + 1.176333I
a = 0.166737 0.841040I
b = 0.27535 + 2.82107I
5.59225 + 3.69663I 9.44807 5.64116I
u = 0.006007 0.938840I
a = 0.583105 0.275897I
b = 0.419174 0.345013I
1.97330 1.67417I 2.89034 + 3.89687I
u = 0.006007 + 0.938840I
a = 0.583105 + 0.275897I
b = 0.419174 + 0.345013I
1.97330 + 1.67417I 2.89034 3.89687I
u = 0.093994 0.803000I
a = 0.69744 + 1.23795I
b = 1.19255 1.55778I
1.51661 4.29217I 0.31628 + 5.07111I
u = 0.093994 + 0.803000I
a = 0.69744 1.23795I
b = 1.19255 + 1.55778I
1.51661 + 4.29217I 0.31628 5.07111I
u = 0.116249 1.319014I
a = 0.493322 0.775316I
b = 0.01428 + 2.29159I
0.06465 8.51032I 2.47657 + 8.97656I
u = 0.116249 + 1.319014I
a = 0.493322 + 0.775316I
b = 0.01428 2.29159I
0.06465 + 8.51032I 2.47657 8.97656I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.174351 1.283412I
a = 0.674289 0.508970I
b = 0.303026 + 1.195672I
0.52470 2.76607I 1.22752 + 4.10691I
u = 0.174351 + 1.283412I
a = 0.674289 + 0.508970I
b = 0.303026 1.195672I
0.52470 + 2.76607I 1.22752 4.10691I
u = 0.248336 0.844356I
a = 1.016255 + 0.682065I
b = 0.15852 2.43203I
1.83973 + 0.97289I 2.18559 1.15743I
u = 0.248336 + 0.844356I
a = 1.016255 0.682065I
b = 0.15852 + 2.43203I
1.83973 0.97289I 2.18559 + 1.15743I
u = 0.356108 0.054613I
a = 2.09936 + 2.52412I
b = 0.462646 + 0.124379I
3.87537 2.94930I 1.97198 + 2.94182I
u = 0.356108 + 0.054613I
a = 2.09936 2.52412I
b = 0.462646 0.124379I
3.87537 + 2.94930I 1.97198 2.94182I
u = 0.425524 0.842724I
a = 0.433995 0.557687I
b = 1.89420 0.74507I
1.06799 1.58770I 1.27174 + 9.61226I
u = 0.425524 + 0.842724I
a = 0.433995 + 0.557687I
b = 1.89420 + 0.74507I
1.06799 + 1.58770I 1.27174 9.61226I
u = 0.481951 0.889150I
a = 0.317059 0.715177I
b = 1.32678 + 5.33597I
1.27782 2.20300I 28.2888 15.0727I
u = 0.481951 + 0.889150I
a = 0.317059 + 0.715177I
b = 1.32678 5.33597I
1.27782 + 2.20300I 28.2888 + 15.0727I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.578878 0.878433I
a = 0.134382 + 1.109729I
b = 2.70748 2.77752I
3.41570 5.19038I 13.1775 6.7973I
u = 0.578878 + 0.878433I
a = 0.134382 1.109729I
b = 2.70748 + 2.77752I
3.41570 + 5.19038I 13.1775 + 6.7973I
u = 0.583169 0.848078I
a = 1.092123 + 0.269922I
b = 0.65494 + 2.36227I
3.51311 + 0.58844I 14.6540 7.5568I
u = 0.583169 + 0.848078I
a = 1.092123 0.269922I
b = 0.65494 2.36227I
3.51311 0.58844I 14.6540 + 7.5568I
u = 0.583671 1.124247I
a = 0.146048 + 0.647206I
b = 0.36300 2.53812I
1.06806 4.56339I 2.77563 + 10.69902I
u = 0.583671 + 1.124247I
a = 0.146048 0.647206I
b = 0.36300 + 2.53812I
1.06806 + 4.56339I 2.77563 10.69902I
u = 0.626147 0.914671I
a = 0.274202 0.021237I
b = 0.176960 0.315393I
0.59157 2.55234I 2.38321 + 1.51119I
u = 0.626147 + 0.914671I
a = 0.274202 + 0.021237I
b = 0.176960 + 0.315393I
0.59157 + 2.55234I 2.38321 1.51119I
u = 0.627763 0.219249I
a = 0.762541 0.231325I
b = 0.624904 0.241870I
1.45022 0.18071I 7.16773 1.07123I
u = 0.627763 + 0.219249I
a = 0.762541 + 0.231325I
b = 0.624904 + 0.241870I
1.45022 + 0.18071I 7.16773 + 1.07123I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.760996 0.715752I
a = 0.253472 + 0.580095I
b = 0.443597 0.627153I
1.05262 2.72622I 4.23016 + 8.33460I
u = 0.760996 + 0.715752I
a = 0.253472 0.580095I
b = 0.443597 + 0.627153I
1.05262 + 2.72622I 4.23016 8.33460I
u = 0.78729 1.17932I
a = 0.091695 0.844947I
b = 0.34565 + 2.50453I
4.61940 7.83060I 4.38555 + 10.26240I
u = 0.78729 + 1.17932I
a = 0.091695 + 0.844947I
b = 0.34565 2.50453I
4.61940 + 7.83060I 4.38555 10.26240I
u = 0.815235 1.138532I
a = 0.811045 0.159049I
b = 0.154524 + 0.019192I
4.87181 1.93616I 5.33797 + 4.75512I
u = 0.815235 + 1.138532I
a = 0.811045 + 0.159049I
b = 0.154524 0.019192I
4.87181 + 1.93616I 5.33797 4.75512I
u = 1.082267 0.519895I
a = 0.389488 0.919470I
b = 0.050529 + 1.168874I
6.73681 4.83956I 9.12593 + 6.03580I
u = 1.082267 + 0.519895I
a = 0.389488 + 0.919470I
b = 0.050529 1.168874I
6.73681 + 4.83956I 9.12593 6.03580I
u = 1.085196 0.452446I
a = 0.949553 + 0.340609I
b = 0.615057 0.596176I
6.80863 + 1.13129I 9.46306 + 0.26470I
u = 1.085196 + 0.452446I
a = 0.949553 0.340609I
b = 0.615057 + 0.596176I
6.80863 1.13129I 9.46306 0.26470I
11
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
70
+ 2u
69
+ ··· + 7u + 1)
c
2
(u
2
+ u + 1)(u
70
+ 26u
69
+ ··· 61u + 1)
c
3
u
2
(u
70
+ 7u
69
+ ··· 4u + 4)
c
4
(u
2
u + 1)(u
70
+ 2u
69
+ ··· + 7u + 1)
c
5
(u 1)
2
(u
70
+ 3u
69
+ ··· 2u + 1)
c
6
(u
2
u + 1)(u
70
+ 4u
69
+ ··· + 14859u + 4643)
c
7
(u
2
u + 1)(u
70
+ 2u
69
+ ··· 3989u + 641)
c
8
(u 1)
2
(u
70
+ 21u
69
+ ··· + 8u + 1)
c
9
(u + 1)
2
(u
70
+ 3u
69
+ ··· 2u + 1)
c
10
(u + 1)
2
(u
70
+ 21u
69
+ ··· + 8u + 1)
c
11
(u + 1)
2
(u
70
+ 7u
69
+ ··· 4u
2
+ 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)(y
70
+ 26y
69
+ ··· 61y + 1)
c
2
(y
2
+ y + 1)(y
70
+ 38y
69
+ ··· 1741y + 1)
c
3
y
2
(y
70
15y
69
+ ··· 328y + 16)
c
5
(y 1)
2
(y
70
21y
69
+ ··· 8y + 1)
c
6
(y
2
+ y + 1)(y
70
46y
69
+ ··· + 5.12637 × 10
8
y + 2.15574 × 10
7
)
c
7
(y
2
+ y + 1)(y
70
90y
69
+ ··· 1.22379 × 10
7
y + 410881)
c
8
, c
10
(y 1)
2
(y
70
+ 59y
69
+ ··· + 128y + 1)
c
9
(y 1)
2
(y
70
21y
69
+ ··· 8y + 1)
c
11
(y 1)
2
(y
70
9y
69
+ ··· 8y + 1)
13