11a
24
(K11a
24
)
1
Arc Sequences
4 1 7 2 11 3 10 6 5 8 9
Solving Sequence
1,4
2 3
5,9
11 6 7 8 10
c
1
c
2
c
4
c
11
c
5
c
6
c
8
c
10
c
3
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
6
b
5
b
4
+ 2b
3
b + 1, u 1, b
4
+ 2b
2
b + a 2i
I
u
2
= hu
84
+ 7u
83
+ ··· + 3u + 1, 1.77437 × 10
71
u
83
+ 1.12033 × 10
72
u
82
+ ··· + 2.39346 × 10
70
a 1.61532 × 10
70
,
2.07667 × 10
71
u
83
1.27278 × 10
72
u
82
+ ··· + 4.78692 × 10
70
b 1.59319 × 10
71
i
There are 2 irreducible components with 90 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
= hb
6
b
5
b
4
+ 2b
3
b + 1, u 1, b
4
+ 2b
2
b + a 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
3
=
0
1
a
5
=
1
0
a
9
=
b
4
2b
2
+ b + 2
b
a
11
=
b
5
+ 2b
3
b
2
2b + 1
b
2
a
6
=
0
b
4
a
7
=
0
b
4
a
8
=
b
4
2b
2
+ b + 2
b
5
b
4
2b
3
+ b
2
+ b 1
a
10
=
b
4
2b
2
+ 2
b
a
10
=
b
4
2b
2
+ 2
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3b
5
+ b
4
4b
3
+ 3b
2
+ 8b 4
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.346225 0.393823I
b = 1.002193 0.295542I
3.53554 0.92430I 6.31051 + 0.25702I
u = 1.00000
a = 0.346225 + 0.393823I
b = 1.002193 + 0.295542I
3.53554 + 0.92430I 6.31051 0.25702I
u = 1.00000
a = 2.68739 + 0.76772I
b = 0.428243 0.664531I
0.245672 0.924305I 0.60470 5.55069I
u = 1.00000
a = 2.68739 0.76772I
b = 0.428243 + 0.664531I
0.245672 + 0.924305I 0.60470 + 5.55069I
u = 1.00000
a = 0.658836 0.177500I
b = 1.073950 0.558752I
1.64493 + 5.69302I 0.29418 8.33058I
u = 1.00000
a = 0.658836 + 0.177500I
b = 1.073950 + 0.558752I
1.64493 5.69302I 0.29418 + 8.33058I
3
II. I
u
2
=
hu
84
+7u
83
+· · · +3u + 1, 1.77 × 10
71
u
83
+1.12 × 10
72
u
82
+· · · +2.39 × 10
70
a
1.62 × 10
70
, 2.08× 10
71
u
83
1.27 × 10
72
u
82
+ · · · + 4.79 ×10
70
b 1.59 × 10
71
i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
9
=
7.41340u
83
46.8077u
82
+ ··· + 32.0820u + 0.674890
4.33821u
83
+ 26.5886u
82
+ ··· + 3.62456u + 3.32821
a
11
=
3.26390u
83
19.2317u
82
+ ··· + 6.89193u 6.39737
2.92755u
83
18.7843u
82
+ ··· 3.42332u 3.43162
a
6
=
2.95981u
83
19.0451u
82
+ ··· 27.6514u 5.08421
4.22596u
83
+ 24.4825u
82
+ ··· + 9.63893u + 3.12800
a
7
=
0.297603u
83
0.0843443u
82
+ ··· 17.4398u 1.52108
0.435133u
83
4.60968u
82
+ ··· + 0.0556085u 0.732737
a
8
=
0.847061u
83
1.81334u
82
+ ··· + 50.3618u + 7.69403
14.3007u
83
+ 90.3845u
82
+ ··· + 22.0436u + 13.2590
a
10
=
1.49670u
83
+ 4.15046u
82
+ ··· + 47.9853u + 6.89323
8.36007u
83
+ 54.1537u
82
+ ··· + 13.0488u + 8.52239
a
10
=
1.49670u
83
+ 4.15046u
82
+ ··· + 47.9853u + 6.89323
8.36007u
83
+ 54.1537u
82
+ ··· + 13.0488u + 8.52239
(ii) Obstruction class = 1
(iii) Cusp Shapes = 34.4571u
83
206.279u
82
+ ··· 74.1253u 20.5385
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.198555 0.662000I
a = 0.406604 + 0.948678I
b = 0.688091 0.625478I
2.02563 9.85203I 0.78173 + 9.71467I
u = 1.198555 + 0.662000I
a = 0.406604 0.948678I
b = 0.688091 + 0.625478I
2.02563 + 9.85203I 0.78173 9.71467I
u = 1.195940 0.633491I
a = 1.04328 1.62210I
b = 1.20343 + 1.09002I
3.1034 17.9006I 0.66653 + 10.33126I
u = 1.195940 + 0.633491I
a = 1.04328 + 1.62210I
b = 1.20343 1.09002I
3.1034 + 17.9006I 0.66653 10.33126I
u = 1.149115 0.600939I
a = 1.12380 + 1.67786I
b = 1.35943 1.20365I
1.04730 11.43986I 1.53294 + 9.69928I
u = 1.149115 + 0.600939I
a = 1.12380 1.67786I
b = 1.35943 + 1.20365I
1.04730 + 11.43986I 1.53294 9.69928I
u = 1.135861 0.560749I
a = 0.801659 0.871224I
b = 0.623387 + 0.419149I
0.73011 6.65250I 2.66432 + 3.51460I
u = 1.135861 + 0.560749I
a = 0.801659 + 0.871224I
b = 0.623387 0.419149I
0.73011 + 6.65250I 2.66432 3.51460I
u = 1.097491 0.619257I
a = 1.251626 0.126388I
b = 0.99586 1.61510I
4.05037 8.85752I 4.78522 + 9.32963I
u = 1.097491 + 0.619257I
a = 1.251626 + 0.126388I
b = 0.99586 + 1.61510I
4.05037 + 8.85752I 4.78522 9.32963I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.080767 0.356839I
a = 0.575324 0.230643I
b = 1.015323 + 0.250361I
2.46714 6.77503I 3.60903 + 10.85243I
u = 1.080767 + 0.356839I
a = 0.575324 + 0.230643I
b = 1.015323 0.250361I
2.46714 + 6.77503I 3.60903 10.85243I
u = 1.079069 0.586402I
a = 1.07775 1.16711I
b = 0.496042 + 0.378784I
2.03656 6.17754I 8.18813 5.12456I
u = 1.079069 + 0.586402I
a = 1.07775 + 1.16711I
b = 0.496042 0.378784I
2.03656 + 6.17754I 8.18813 + 5.12456I
u = 1.042265 0.620203I
a = 0.82098 1.77446I
b = 1.71639 + 0.59202I
5.01931 4.78307I 6.84816 + 3.76159I
u = 1.042265 + 0.620203I
a = 0.82098 + 1.77446I
b = 1.71639 0.59202I
5.01931 + 4.78307I 6.84816 3.76159I
u = 1.028210 0.552628I
a = 0.16583 + 1.66492I
b = 0.125700 0.706897I
1.30476 4.74367I 0.87707 + 10.18383I
u = 1.028210 + 0.552628I
a = 0.16583 1.66492I
b = 0.125700 + 0.706897I
1.30476 + 4.74367I 0.87707 10.18383I
u = 0.996208 0.396743I
a = 0.356253 + 0.808044I
b = 1.39742 + 0.51890I
3.77732 0.83071I 3.35550 + 3.35286I
u = 0.996208 + 0.396743I
a = 0.356253 0.808044I
b = 1.39742 0.51890I
3.77732 + 0.83071I 3.35550 3.35286I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.965351 0.572013I
a = 1.39932 0.62947I
b = 0.07156 + 1.56178I
2.30506 1.39249I 3.00390 0.61304I
u = 0.965351 + 0.572013I
a = 1.39932 + 0.62947I
b = 0.07156 1.56178I
2.30506 + 1.39249I 3.00390 + 0.61304I
u = 0.951859 0.248441I
a = 0.579937 0.439950I
b = 1.204871 0.621831I
1.61526 + 4.99229I 0.56825 + 2.61815I
u = 0.951859 + 0.248441I
a = 0.579937 + 0.439950I
b = 1.204871 + 0.621831I
1.61526 4.99229I 0.56825 2.61815I
u = 0.880373 0.849310I
a = 0.849446 + 0.566687I
b = 0.462202 0.987645I
8.69358 + 1.42500I 6.64770 1.92511I
u = 0.880373 + 0.849310I
a = 0.849446 0.566687I
b = 0.462202 + 0.987645I
8.69358 1.42500I 6.64770 + 1.92511I
u = 0.879247 0.308527I
a = 0.503888 + 0.099186I
b = 1.201689 0.064278I
3.09506 1.90596I 1.46467 + 5.46358I
u = 0.879247 + 0.308527I
a = 0.503888 0.099186I
b = 1.201689 + 0.064278I
3.09506 + 1.90596I 1.46467 5.46358I
u = 0.807220 0.862104I
a = 0.054119 1.312931I
b = 0.673682 + 1.081484I
8.89963 7.63957I 5.79608 + 6.64543I
u = 0.807220 + 0.862104I
a = 0.054119 + 1.312931I
b = 0.673682 1.081484I
8.89963 + 7.63957I 5.79608 6.64543I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.653182 0.674145I
a = 0.17331 + 1.46465I
b = 0.49270 1.39917I
3.23379 3.44018I 4.16368 + 6.94949I
u = 0.653182 + 0.674145I
a = 0.17331 1.46465I
b = 0.49270 + 1.39917I
3.23379 + 3.44018I 4.16368 6.94949I
u = 0.547458 0.616819I
a = 1.10301 1.44126I
b = 0.026284 + 0.664009I
2.75270 + 0.12135I 4.31662 2.02203I
u = 0.547458 + 0.616819I
a = 1.10301 + 1.44126I
b = 0.026284 0.664009I
2.75270 0.12135I 4.31662 + 2.02203I
u = 0.535364 0.758601I
a = 0.617447 + 0.547155I
b = 1.52096 0.84531I
6.53306 0.44104I 9.66197 + 2.06639I
u = 0.535364 + 0.758601I
a = 0.617447 0.547155I
b = 1.52096 + 0.84531I
6.53306 + 0.44104I 9.66197 2.06639I
u = 0.446596 0.719406I
a = 2.13603 + 1.64632I
b = 0.459164 0.208972I
3.90317 + 1.17062I 12.4798 + 11.4287I
u = 0.446596 + 0.719406I
a = 2.13603 1.64632I
b = 0.459164 + 0.208972I
3.90317 1.17062I 12.4798 11.4287I
u = 0.445418 0.802301I
a = 0.18751 1.63916I
b = 1.12473 + 1.40081I
5.99281 + 3.52843I 8.38773 4.67888I
u = 0.445418 + 0.802301I
a = 0.18751 + 1.63916I
b = 1.12473 1.40081I
5.99281 3.52843I 8.38773 + 4.67888I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.369678 1.018600I
a = 0.348335 0.586443I
b = 0.471379 + 0.566383I
4.58405 + 3.81524I 5.31453 8.50688I
u = 0.369678 + 1.018600I
a = 0.348335 + 0.586443I
b = 0.471379 0.566383I
4.58405 3.81524I 5.31453 + 8.50688I
u = 0.340933 0.839696I
a = 0.745320 1.174812I
b = 1.17161 + 1.21483I
1.35757 + 6.09736I 1.65591 6.32757I
u = 0.340933 + 0.839696I
a = 0.745320 + 1.174812I
b = 1.17161 1.21483I
1.35757 6.09736I 1.65591 + 6.32757I
u = 0.332631 0.958315I
a = 0.550668 + 1.221892I
b = 1.10438 1.09237I
5.73608 + 12.12140I 3.70292 6.82930I
u = 0.332631 + 0.958315I
a = 0.550668 1.221892I
b = 1.10438 + 1.09237I
5.73608 12.12140I 3.70292 + 6.82930I
u = 0.318606 0.734288I
a = 0.281137 + 0.562104I
b = 0.394347 0.415917I
1.66501 + 1.71486I 1.361742 0.047755I
u = 0.318606 + 0.734288I
a = 0.281137 0.562104I
b = 0.394347 + 0.415917I
1.66501 1.71486I 1.361742 + 0.047755I
u = 0.0030157 0.1073637I
a = 7.33134 5.70225I
b = 0.414534 + 0.524432I
1.89935 + 0.79593I 4.72564 + 0.80285I
u = 0.0030157 + 0.1073637I
a = 7.33134 + 5.70225I
b = 0.414534 0.524432I
1.89935 0.79593I 4.72564 0.80285I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.292041 0.834139I
a = 0.592343 + 0.349327I
b = 0.522623 0.418584I
1.52052 + 3.66155I 1.03794 11.92767I
u = 0.292041 + 0.834139I
a = 0.592343 0.349327I
b = 0.522623 + 0.418584I
1.52052 3.66155I 1.03794 + 11.92767I
u = 0.332881 0.406496I
a = 1.45828 + 1.92502I
b = 0.832442 0.729614I
1.24805 1.52698I 2.33035 + 1.80956I
u = 0.332881 + 0.406496I
a = 1.45828 1.92502I
b = 0.832442 + 0.729614I
1.24805 + 1.52698I 2.33035 1.80956I
u = 0.507684 0.708326I
a = 1.13721 1.28121I
b = 0.914742 + 0.905966I
2.49717 6.15202I 1.87556 + 3.69911I
u = 0.507684 + 0.708326I
a = 1.13721 + 1.28121I
b = 0.914742 0.905966I
2.49717 + 6.15202I 1.87556 3.69911I
u = 0.599364 0.250259I
a = 1.54512 1.37549I
b = 0.412974 + 0.489784I
0.95967 + 1.37657I 3.31883 4.67522I
u = 0.599364 + 0.250259I
a = 1.54512 + 1.37549I
b = 0.412974 0.489784I
0.95967 1.37657I 3.31883 + 4.67522I
u = 0.790504 0.363992I
a = 0.42678 + 2.89820I
b = 1.105909 0.660409I
2.02169 + 0.07920I 5.08615 0.51793I
u = 0.790504 + 0.363992I
a = 0.42678 2.89820I
b = 1.105909 + 0.660409I
2.02169 0.07920I 5.08615 + 0.51793I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.928263 0.443032I
a = 1.82091 0.25704I
b = 0.88835 + 1.21673I
1.46677 + 3.41872I 3.42085 6.92160I
u = 0.928263 + 0.443032I
a = 1.82091 + 0.25704I
b = 0.88835 1.21673I
1.46677 3.41872I 3.42085 + 6.92160I
u = 0.967113 0.342930I
a = 1.64207 + 3.53612I
b = 0.234835 0.269901I
0.170119 + 1.192108I 4.9918 + 29.6831I
u = 0.967113 + 0.342930I
a = 1.64207 3.53612I
b = 0.234835 + 0.269901I
0.170119 1.192108I 4.9918 29.6831I
u = 0.984748 0.690918I
a = 0.186988 0.727061I
b = 0.487211 + 0.270919I
0.71473 + 2.85212I 5.7739 13.2498I
u = 0.984748 + 0.690918I
a = 0.186988 + 0.727061I
b = 0.487211 0.270919I
0.71473 2.85212I 5.7739 + 13.2498I
u = 1.038050 0.485618I
a = 1.01029 2.19498I
b = 1.14543 + 1.02229I
3.10217 + 5.44544I 3.85549 6.81268I
u = 1.038050 + 0.485618I
a = 1.01029 + 2.19498I
b = 1.14543 1.02229I
3.10217 5.44544I 3.85549 + 6.81268I
u = 1.043360 0.207956I
a = 0.81314 4.12005I
b = 0.023184 + 0.394211I
0.506438 + 0.641182I 14.9092 1.9149I
u = 1.043360 + 0.207956I
a = 0.81314 + 4.12005I
b = 0.023184 0.394211I
0.506438 0.641182I 14.9092 + 1.9149I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.056260 0.592575I
a = 0.75805 + 2.01135I
b = 1.07772 0.99600I
0.85835 + 11.16100I 0.29756 7.65446I
u = 1.056260 + 0.592575I
a = 0.75805 2.01135I
b = 1.07772 + 0.99600I
0.85835 11.16100I 0.29756 + 7.65446I
u = 1.084118 0.362868I
a = 0.818446 + 0.605892I
b = 0.377313 0.131812I
2.25525 + 1.15492I 4.97585 0.17312I
u = 1.084118 + 0.362868I
a = 0.818446 0.605892I
b = 0.377313 + 0.131812I
2.25525 1.15492I 4.97585 + 0.17312I
u = 1.161769 0.086552I
a = 2.15204 + 0.06691I
b = 0.752434 0.948844I
0.55551 1.40163I 7.12458 + 7.33605I
u = 1.161769 + 0.086552I
a = 2.15204 0.06691I
b = 0.752434 + 0.948844I
0.55551 + 1.40163I 7.12458 7.33605I
u = 1.18173
a = 0.243735
b = 0.470587
2.49567 2.13157
u = 1.231666 0.215746I
a = 0.259775 0.779735I
b = 1.083564 0.880565I
3.75795 2.95090I 3.86338 + 6.17860I
u = 1.231666 + 0.215746I
a = 0.259775 + 0.779735I
b = 1.083564 + 0.880565I
3.75795 + 2.95090I 3.86338 6.17860I
u = 1.316998 0.412094I
a = 0.161002 + 0.262329I
b = 0.496808 0.115065I
1.87887 + 1.58264I 12.5915 7.5531I
12
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.316998 + 0.412094I
a = 0.161002 0.262329I
b = 0.496808 + 0.115065I
1.87887 1.58264I 12.5915 + 7.5531I
u = 1.347241 0.200389I
a = 0.345598 + 0.267749I
b = 1.029718 + 0.932033I
0.05021 8.32055I 0.19913 + 6.65496I
u = 1.347241 + 0.200389I
a = 0.345598 0.267749I
b = 1.029718 0.932033I
0.05021 + 8.32055I 0.19913 6.65496I
u = 1.40291
a = 0.0166509
b = 0.512511
2.21927 13.1201
13
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
6
(u
84
+ 7u
83
+ ··· + 3u + 1)
c
2
(u + 1)
6
(u
84
+ 41u
83
+ ··· 157u + 1)
c
3
u
6
(u
84
+ u
83
+ ··· + 320u + 64)
c
4
(u + 1)
6
(u
84
+ 7u
83
+ ··· + 3u + 1)
c
5
(u
6
3u
5
+ ··· u + 1)(u
84
+ 6u
83
+ ··· + 2u + 1)
c
6
u
6
(u
84
+ u
83
+ ··· + 320u + 64)
c
7
(u
6
u
5
+ ··· u + 1)(u
84
+ 2u
83
+ ··· + 14u + 1)
c
8
(u
6
3u
5
+ ··· u + 1)(u
84
+ 6u
83
+ ··· 1166u 101)
c
9
(u
6
u
5
+ ··· u + 1)(u
84
+ 2u
83
+ ··· 418u + 367)
c
10
(u
6
+ u
5
+ ··· + u + 1)(u
84
+ 2u
83
+ ··· + 14u + 1)
c
11
(u
6
u
5
+ ··· u + 1)(u
84
+ 14u
83
+ ··· + 2u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
84
41y
83
+ ··· + 157y + 1)
c
2
(y 1)
6
(y
84
+ 11y
83
+ ··· 14895y + 1)
c
3
y
6
(y
84
39y
83
+ ··· 61440y + 4096)
c
5
(y
6
+ y
5
+ ··· + 3y + 1)(y
84
+ 14y
83
+ ··· + 6y + 1)
c
6
y
6
(y
84
39y
83
+ ··· 61440y + 4096)
c
7
(y
6
3y
5
+ ··· y + 1)(y
84
54y
83
+ ··· 14y + 1)
c
8
(y
6
+ y
5
+ ··· + 3y + 1)(y
84
82y
83
+ ··· 878594y + 10201)
c
9
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
84
66y
83
+ ··· + 5678926y + 134689)
c
10
(y
6
3y
5
+ ··· y + 1)(y
84
54y
83
+ ··· 14y + 1)
c
11
(y
6
3y
5
+ ··· y + 1)(y
84
6y
83
+ ··· 14y + 1)
15