11a
26
(K11a
26
)
1
Arc Sequences
4 1 7 2 8 3 10 11 6 5 9
Solving Sequence
9,11 1,4
2 5 8 6 10 7 3
c
11
c
1
c
4
c
8
c
5
c
10
c
7
c
3
c
2
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
6
+ 5u
5
+ 9u
4
+ 8u
3
+ 5u
2
+ 2u + 1, u
2
+ b + u + 1, u
5
5u
4
9u
3
8u
2
+ a 5u 1i
I
u
2
= hu
84
+ 6u
83
+ ··· + 1561804899u + 3587089741,
6.96275 × 10
906
u
83
3.59284 × 10
907
u
82
+ ··· + 2.21651 × 10
916
b + 8.34747 × 10
916
,
1.09672 × 10
917
u
83
6.21827 × 10
917
u
82
+ ··· + 7.95083 × 10
925
a + 1.62906 × 10
926
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
= hu
6
+ 5u
5
+ 9u
4
+ 8u
3
+ 5u
2
+ 2u + 1, u
2
+ b + u + 1, u
5
5u
4
9u
3
8u
2
+ a 5u 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
u
5
+ 5u
4
+ 9u
3
+ 8u
2
+ 5u + 1
u
2
u 1
a
1
=
u
5
+ 5u
4
+ 9u
3
+ 8u
2
+ 5u + 1
1
a
4
=
1
0
a
2
=
u
5
+ 5u
4
+ 9u
3
+ 8u
2
+ 5u + 2
1
a
5
=
u
5
5u
4
9u
3
8u
2
5u 1
1
a
8
=
u
5
+ 5u
4
+ 9u
3
+ 8u
2
+ 4u
u
3
2u
2
u 1
a
6
=
u
5
4u
4
4u
3
+ u
2
+ 2u + 1
0
a
10
=
u
2
+ 3u + 2
u
a
7
=
u
5
4u
4
4u
3
+ u
2
+ 2u + 1
0
a
3
=
1
0
a
3
=
1
0
(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 = 2.00219 0.29554I
a = 0.511198 0.072152I
b = 2.91924 0.88792I
0.245672 + 0.924305I 0.60470 + 5.55069I
u = 2.00219 + 0.29554I
a = 0.511198 + 0.072152I
b = 2.91924 + 0.88792I
0.245672 0.924305I 0.60470 5.55069I
u = 0.571757 0.664531I
a = 0.256017 0.864704I
b = 0.313547 0.095369I
3.53554 + 0.92430I 6.31051 0.25702I
u = 0.571757 + 0.664531I
a = 0.256017 + 0.864704I
b = 0.313547 + 0.095369I
3.53554 0.92430I 6.31051 + 0.25702I
u = 0.073950 0.558752I
a = 1.23279 1.75889I
b = 0.767214 + 0.641391I
1.64493 5.69302I 0.29418 + 8.33058I
u = 0.073950 + 0.558752I
a = 1.23279 + 1.75889I
b = 0.767214 0.641391I
1.64493 + 5.69302I 0.29418 8.33058I
3
II. I
u
2
= hu
84
+ 6u
83
+ · · · + 1561804899u + 3587089741, 6.96 × 10
906
u
83
3.59 × 10
907
u
82
+ · · · + 2.22 × 10
916
b + 8.35 × 10
916
, 1.10 × 10
917
u
83
6.22 × 10
917
u
82
+ · · · + 7.95 × 10
925
a + 1.63 × 10
926
i
(i) Arc colorings
a
9
=
0
u
a
11
=
1.37937 × 10
9
u
83
+ 7.82091 × 10
9
u
82
+ ··· + 39.0214u 2.04892
3.14131 × 10
10
u
83
+ 1.62094 × 10
9
u
82
+ ··· + 7.49739u 3.76603
a
1
=
1.37937 × 10
9
u
83
+ 7.82091 × 10
9
u
82
+ ··· + 39.0214u 2.04892
4.05143 × 10
10
u
83
+ 2.07768 × 10
9
u
82
+ ··· + 11.7342u 5.39937
a
4
=
1
0
a
2
=
9.74231 × 10
10
u
83
+ 5.74322 × 10
9
u
82
+ ··· + 27.2872u + 3.35045
4.05143 × 10
10
u
83
+ 2.07768 × 10
9
u
82
+ ··· + 11.7342u 5.39937
a
5
=
1.81690 × 10
9
u
83
1.14665 × 10
8
u
82
+ ··· 22.8390u 16.9127
2.37862 × 10
10
u
83
+ 1.60319 × 10
9
u
82
+ ··· 3.22961u + 5.88961
a
8
=
2.59530 × 10
9
u
83
+ 1.48044 × 10
8
u
82
+ ··· + 59.8636u 4.41038
1.26310 × 10
10
u
83
+ 6.05163 × 10
10
u
82
+ ··· + 2.44592u 2.91131
a
6
=
9.84030 × 10
10
u
83
+ 6.61943 × 10
9
u
82
+ ··· + 7.24873u + 23.8347
2.07889 × 10
10
u
83
+ 1.26811 × 10
9
u
82
+ ··· + 0.353804u + 1.35130
a
10
=
3.20206 × 10
9
u
83
1.83565 × 10
8
u
82
+ ··· 80.2881u + 5.95772
5.61230 × 10
10
u
83
+ 3.36801 × 10
9
u
82
+ ··· + 9.06355u + 1.54310
a
7
=
2.69064 × 10
9
u
83
1.59639 × 10
8
u
82
+ ··· 60.6392u 8.18780
9.80274 × 10
11
u
83
+ 6.83784 × 10
10
u
82
+ ··· 1.88172u + 2.14865
a
3
=
2.06850 × 10
9
u
83
1.18343 × 10
8
u
82
+ ··· 40.2076u + 2.73457
6.43725 × 10
11
u
83
+ 4.78536 × 10
10
u
82
+ ··· + 4.23933u + 3.42707
a
3
=
2.06850 × 10
9
u
83
1.18343 × 10
8
u
82
+ ··· 40.2076u + 2.73457
6.43725 × 10
11
u
83
+ 4.78536 × 10
10
u
82
+ ··· + 4.23933u + 3.42707
(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 = 2.34318 1.98452I
a = 0.285124 + 0.208852I
b = 1.93832 1.60811I
3.90317 + 1.17062I 12.4798 + 11.4287I
u = 2.34318 + 1.98452I
a = 0.285124 0.208852I
b = 1.93832 + 1.60811I
3.90317 1.17062I 12.4798 11.4287I
u = 1.38124 0.84433I
a = 0.778747 + 0.129701I
b = 2.39484 1.74796I
0.55551 + 1.40163I 7.12458 7.33605I
u = 1.38124 + 0.84433I
a = 0.778747 0.129701I
b = 2.39484 + 1.74796I
0.55551 1.40163I 7.12458 + 7.33605I
u = 1.10081
a = 0.212859
b = 0.923788
2.49567 2.13157
u = 1.03376 2.79254I
a = 0.016269 0.480041I
b = 1.82480 + 2.54399I
0.85835 11.16100I 0.29756 + 7.65446I
u = 1.03376 + 2.79254I
a = 0.016269 + 0.480041I
b = 1.82480 2.54399I
0.85835 + 11.16100I 0.29756 7.65446I
u = 1.024740 0.823535I
a = 0.643709 + 0.717352I
b = 0.17774 + 1.95417I
1.04730 11.43986I 1.53294 + 9.69928I
u = 1.024740 + 0.823535I
a = 0.643709 0.717352I
b = 0.17774 1.95417I
1.04730 + 11.43986I 1.53294 9.69928I
u = 0.95125 1.87656I
a = 0.137399 0.436289I
b = 0.213353 + 0.941217I
0.71473 2.85212I 5.7739 + 13.2498I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.95125 + 1.87656I
a = 0.137399 + 0.436289I
b = 0.213353 0.941217I
0.71473 + 2.85212I 5.7739 13.2498I
u = 0.94440 3.13399I
a = 0.083917 + 0.265501I
b = 1.86561 3.07562I
0.506438 0.641182I 14.9092 + 1.9149I
u = 0.94440 + 3.13399I
a = 0.083917 0.265501I
b = 1.86561 + 3.07562I
0.506438 + 0.641182I 14.9092 1.9149I
u = 0.79869 1.73150I
a = 0.496398 0.504482I
b = 0.28396 + 1.85531I
2.49717 + 6.15202I 1.87556 3.69911I
u = 0.79869 + 1.73150I
a = 0.496398 + 0.504482I
b = 0.28396 1.85531I
2.49717 6.15202I 1.87556 + 3.69911I
u = 0.75936 3.87303I
a = 0.014782 + 0.275724I
b = 0.32954 3.47752I
2.03656 6.17754I 8.18813 5.12456I
u = 0.75936 + 3.87303I
a = 0.014782 0.275724I
b = 0.32954 + 3.47752I
2.03656 + 6.17754I 8.18813 + 5.12456I
u = 0.73250 1.36223I
a = 0.529364 0.359741I
b = 0.184541 0.797648I
3.75795 + 2.95090I 3.86338 6.17860I
u = 0.73250 + 1.36223I
a = 0.529364 + 0.359741I
b = 0.184541 + 0.797648I
3.75795 2.95090I 3.86338 + 6.17860I
u = 0.730752
a = 1.13005
b = 0.799472
2.21927 13.1201
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.696984 1.155762I
a = 0.255677 0.727330I
b = 0.662432 + 0.196012I
1.52052 3.66155I 1.03794 + 11.92767I
u = 0.696984 + 1.155762I
a = 0.255677 + 0.727330I
b = 0.662432 0.196012I
1.52052 + 3.66155I 1.03794 11.92767I
u = 0.64359 1.64587I
a = 0.112478 + 0.917298I
b = 1.14985 1.58458I
6.53306 0.44104I 9.66197 + 2.06639I
u = 0.64359 + 1.64587I
a = 0.112478 0.917298I
b = 1.14985 + 1.58458I
6.53306 + 0.44104I 9.66197 2.06639I
u = 0.630086 0.240993I
a = 0.487309 + 1.033355I
b = 0.481539 + 0.904725I
1.24805 1.52698I 2.33035 + 1.80956I
u = 0.630086 + 0.240993I
a = 0.487309 1.033355I
b = 0.481539 0.904725I
1.24805 + 1.52698I 2.33035 1.80956I
u = 0.520063 0.210515I
a = 0.272014 + 0.088034I
b = 1.041878 0.283398I
2.25525 1.15492I 4.97585 + 0.17312I
u = 0.520063 + 0.210515I
a = 0.272014 0.088034I
b = 1.041878 + 0.283398I
2.25525 + 1.15492I 4.97585 0.17312I
u = 0.48012 2.56967I
a = 0.214661 + 0.570743I
b = 1.67879 2.48822I
4.05037 8.85752I 4.78522 + 9.32963I
u = 0.48012 + 2.56967I
a = 0.214661 0.570743I
b = 1.67879 + 2.48822I
4.05037 + 8.85752I 4.78522 9.32963I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.467417 1.178967I
a = 0.925808 + 0.829825I
b = 0.615407 1.190193I
5.99281 + 3.52843I 8.38773 4.67888I
u = 0.467417 + 1.178967I
a = 0.925808 0.829825I
b = 0.615407 + 1.190193I
5.99281 3.52843I 8.38773 + 4.67888I
u = 0.348433 0.726741I
a = 0.376432 + 0.249259I
b = 0.878713 0.161003I
0.95967 1.37657I 3.31883 + 4.67522I
u = 0.348433 + 0.726741I
a = 0.376432 0.249259I
b = 0.878713 + 0.161003I
0.95967 + 1.37657I 3.31883 4.67522I
u = 0.328964 0.885158I
a = 0.230704 0.896598I
b = 0.950636 + 0.327724I
1.87887 1.58264I 12.5915 + 7.5531I
u = 0.328964 + 0.885158I
a = 0.230704 + 0.896598I
b = 0.950636 0.327724I
1.87887 + 1.58264I 12.5915 7.5531I
u = 0.221705 0.333848I
a = 1.44988 + 2.31998I
b = 0.219423 + 0.744236I
3.23379 3.44018I 4.16368 + 6.94949I
u = 0.221705 + 0.333848I
a = 1.44988 2.31998I
b = 0.219423 0.744236I
3.23379 + 3.44018I 4.16368 6.94949I
u = 0.219641 1.199669I
a = 0.103473 + 0.652999I
b = 0.54016 1.40640I
1.30476 + 4.74367I 0.87707 10.18383I
u = 0.219641 + 1.199669I
a = 0.103473 0.652999I
b = 0.54016 + 1.40640I
1.30476 4.74367I 0.87707 + 10.18383I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.117763 0.398260I
a = 2.22882 2.15404I
b = 0.831304 + 0.769288I
1.61526 + 4.99229I 0.56825 + 2.61815I
u = 0.117763 + 0.398260I
a = 2.22882 + 2.15404I
b = 0.831304 0.769288I
1.61526 4.99229I 0.56825 2.61815I
u = 0.090393 0.382994I
a = 0.53057 3.48645I
b = 1.032347 + 0.646574I
0.05021 8.32055I 0.19913 + 6.65496I
u = 0.090393 + 0.382994I
a = 0.53057 + 3.48645I
b = 1.032347 0.646574I
0.05021 + 8.32055I 0.19913 6.65496I
u = 0.06355 1.77162I
a = 0.191007 + 0.935616I
b = 1.43033 2.35560I
5.01931 4.78307I 6.84816 + 3.76159I
u = 0.06355 + 1.77162I
a = 0.191007 0.935616I
b = 1.43033 + 2.35560I
5.01931 + 4.78307I 6.84816 3.76159I
u = 0.037571 0.851636I
a = 0.585286 1.039266I
b = 0.590589 + 0.358003I
2.46714 + 6.77503I 3.60903 10.85243I
u = 0.037571 + 0.851636I
a = 0.585286 + 1.039266I
b = 0.590589 0.358003I
2.46714 6.77503I 3.60903 + 10.85243I
u = 0.059516 0.508367I
a = 0.587103 0.041696I
b = 0.166434 0.759187I
1.66501 1.71486I 1.361742 + 0.047755I
u = 0.059516 + 0.508367I
a = 0.587103 + 0.041696I
b = 0.166434 + 0.759187I
1.66501 + 1.71486I 1.361742 0.047755I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.140548 0.220372I
a = 1.18880 1.36239I
b = 1.017217 + 0.526359I
0.73011 6.65250I 2.66432 + 3.51460I
u = 0.140548 + 0.220372I
a = 1.18880 + 1.36239I
b = 1.017217 0.526359I
0.73011 + 6.65250I 2.66432 3.51460I
u = 0.273741 0.353023I
a = 0.38359 2.18005I
b = 0.438839 + 0.342719I
3.09506 1.90596I 1.46467 + 5.46358I
u = 0.273741 + 0.353023I
a = 0.38359 + 2.18005I
b = 0.438839 0.342719I
3.09506 + 1.90596I 1.46467 5.46358I
u = 0.343119 1.208189I
a = 0.676091 0.553902I
b = 0.370343 0.213962I
3.77732 + 0.83071I 3.35550 3.35286I
u = 0.343119 + 1.208189I
a = 0.676091 + 0.553902I
b = 0.370343 + 0.213962I
3.77732 0.83071I 3.35550 + 3.35286I
u = 0.449059 0.015890I
a = 0.13726 2.63357I
b = 0.330993 1.372059I
1.35757 6.09736I 1.65591 + 6.32757I
u = 0.449059 + 0.015890I
a = 0.13726 + 2.63357I
b = 0.330993 + 1.372059I
1.35757 + 6.09736I 1.65591 6.32757I
u = 0.55186 1.84751I
a = 0.304876 0.620116I
b = 0.56366 + 1.70016I
8.69358 1.42500I 6.64770 + 1.92511I
u = 0.55186 + 1.84751I
a = 0.304876 + 0.620116I
b = 0.56366 1.70016I
8.69358 + 1.42500I 6.64770 1.92511I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.57738 1.52979I
a = 0.355396 0.586558I
b = 0.046744 + 0.786366I
4.58405 3.81524I 5.31453 + 8.50688I
u = 0.57738 + 1.52979I
a = 0.355396 + 0.586558I
b = 0.046744 0.786366I
4.58405 + 3.81524I 5.31453 8.50688I
u = 0.62532 2.78108I
a = 0.100349 0.518709I
b = 1.34511 + 2.82657I
3.1034 + 17.9006I 0.66653 10.33126I
u = 0.62532 + 2.78108I
a = 0.100349 + 0.518709I
b = 1.34511 2.82657I
3.1034 17.9006I 0.66653 + 10.33126I
u = 0.63022 1.58325I
a = 0.599277 0.630498I
b = 0.33859 + 1.56764I
5.73608 12.12140I 3.70292 + 6.82930I
u = 0.63022 + 1.58325I
a = 0.599277 + 0.630498I
b = 0.33859 1.56764I
5.73608 + 12.12140I 3.70292 6.82930I
u = 0.70524 2.14783I
a = 0.016726 0.514386I
b = 0.29092 + 1.50578I
2.02563 + 9.85203I 0.78173 9.71467I
u = 0.70524 + 2.14783I
a = 0.016726 + 0.514386I
b = 0.29092 1.50578I
2.02563 9.85203I 0.78173 + 9.71467I
u = 0.800319 0.323216I
a = 1.22815 0.71093I
b = 1.25589 1.58849I
2.30506 + 1.39249I 3.00390 + 0.61304I
u = 0.800319 + 0.323216I
a = 1.22815 + 0.71093I
b = 1.25589 + 1.58849I
2.30506 1.39249I 3.00390 0.61304I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.93514 1.98038I
a = 0.129649 0.625477I
b = 1.11436 + 1.52418I
8.89963 + 7.63957I 5.79608 6.64543I
u = 0.93514 + 1.98038I
a = 0.129649 + 0.625477I
b = 1.11436 1.52418I
8.89963 7.63957I 5.79608 + 6.64543I
u = 0.95815 2.19894I
a = 0.351337 + 0.470972I
b = 0.18218 2.96141I
2.02169 + 0.07920I 5.08615 0.51793I
u = 0.95815 + 2.19894I
a = 0.351337 0.470972I
b = 0.18218 + 2.96141I
2.02169 0.07920I 5.08615 + 0.51793I
u = 0.998636 0.466828I
a = 0.511950 + 0.047799I
b = 0.991011 + 0.176738I
2.75270 + 0.12135I 4.31662 2.02203I
u = 0.998636 + 0.466828I
a = 0.511950 0.047799I
b = 0.991011 0.176738I
2.75270 0.12135I 4.31662 + 2.02203I
u = 1.001240 0.317132I
a = 0.983372 + 0.100045I
b = 0.947429 0.641623I
1.89935 0.79593I 4.72564 0.80285I
u = 1.001240 + 0.317132I
a = 0.983372 0.100045I
b = 0.947429 + 0.641623I
1.89935 + 0.79593I 4.72564 + 0.80285I
u = 1.18299 6.27662I
a = 0.019067 + 0.155293I
b = 0.61459 5.84083I
0.170119 + 1.192108I 4.9918 + 29.6831I
u = 1.18299 + 6.27662I
a = 0.019067 0.155293I
b = 0.61459 + 5.84083I
0.170119 1.192108I 4.9918 29.6831I
12
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.20276 2.77634I
a = 0.021610 + 0.467789I
b = 2.33118 2.34167I
1.46677 + 3.41872I 3.42085 6.92160I
u = 1.20276 + 2.77634I
a = 0.021610 0.467789I
b = 2.33118 + 2.34167I
1.46677 3.41872I 3.42085 + 6.92160I
u = 1.240809 0.433341I
a = 0.314154 + 0.763110I
b = 0.64209 + 1.85505I
3.10217 + 5.44544I 3.85549 6.81268I
u = 1.240809 + 0.433341I
a = 0.314154 0.763110I
b = 0.64209 1.85505I
3.10217 5.44544I 3.85549 + 6.81268I
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
, c
6
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
7
(u
6
+ u
5
+ ··· + u + 1)(u
84
+ 14u
83
+ ··· + 2u + 1)
c
8
(u
6
u
5
+ ··· u + 1)(u
84
+ 2u
83
+ ··· + 14u + 1)
c
9
(u
6
u
5
+ ··· u + 1)(u
84
+ 2u
83
+ ··· 418u + 367)
c
10
(u
6
3u
5
+ ··· u + 1)(u
84
+ 6u
83
+ ··· 1166u 101)
c
11
(u
6
+ u
5
+ ··· + u + 1)(u
84
+ 2u
83
+ ··· + 14u + 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
, c
6
y
6
(y
84
39y
83
+ ··· 61440y + 4096)
c
5
(y
6
+ y
5
+ ··· + 3y + 1)(y
84
+ 14y
83
+ ··· + 6y + 1)
c
7
(y
6
3y
5
+ ··· y + 1)(y
84
6y
83
+ ··· 14y + 1)
c
8
(y
6
3y
5
+ ··· y + 1)(y
84
54y
83
+ ··· 14y + 1)
c
9
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
84
66y
83
+ ··· + 5678926y + 134689)
c
10
(y
6
+ y
5
+ ··· + 3y + 1)(y
84
82y
83
+ ··· 878594y + 10201)
c
11
(y
6
3y
5
+ ··· y + 1)(y
84
54y
83
+ ··· 14y + 1)
15