11a
304
(K11a
304
)
1
Arc Sequences
5 10 1 7 2 9 4 11 6 3 8
Solving Sequence
8,11
9
1,4
3 7 5 2 6 10
c
8
c
11
c
3
c
7
c
4
c
1
c
6
c
10
c
2
, c
5
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
8
u
7
+ 4u
6
3u
5
+ 6u
4
3u
3
+ 3u
2
u + 1, a + 1, u
7
+ 2u
6
5u
5
+ 6u
4
8u
3
+ 5u
2
+ b 3u + 1i
I
u
2
= hu
18
+ u
17
+ ··· 4u + 1, a + 1, 4427u
17
+ 4238u
16
+ ··· + 4846b 789i
I
u
3
= hu
48
+ 5u
47
+ ··· 314u + 61,
1.19355 × 10
93
u
47
+ 6.16000 × 10
93
u
46
+ ··· + 1.64883 × 10
93
b + 4.18982 × 10
94
,
7.44333 × 10
94
u
47
+ 4.07702 × 10
95
u
46
+ ··· + 1.00579 × 10
95
a + 9.36425 × 10
95
i
There are 3 irreducible components with 74 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
8
u
7
+4u
6
3u
5
+6u
4
3u
3
+3u
2
u+1, a+1, u
7
+2u
6
+· · ·+b+1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
u
7
2u
6
+ 5u
5
6u
4
+ 8u
3
5u
2
+ 3u 1
a
9
=
u
u
7
+ u
6
3u
5
+ 2u
4
2u
3
+ u 1
a
1
=
1
u
7
2u
6
+ 5u
5
6u
4
+ 8u
3
6u
2
+ 3u 1
a
4
=
1
0
a
3
=
u
7
2u
6
+ 5u
5
6u
4
+ 8u
3
6u
2
+ 3u
u
7
+ 3u
6
6u
5
+ 11u
4
12u
3
+ 13u
2
8u + 3
a
7
=
u
u
a
5
=
u
2
+ 1
u
2
a
2
=
u
7
+ u
6
3u
5
+ 2u
4
3u
3
+ u
2
1
u
7
2u
6
+ 6u
5
7u
4
+ 10u
3
7u
2
+ 4u 1
a
6
=
u
7
+ u
6
4u
5
+ 3u
4
5u
3
+ 2u
2
2u
7
3u
6
+ 8u
5
8u
4
+ 11u
3
7u
2
+ 4u
a
10
=
2u
7
3u
6
+ 9u
5
9u
4
+ 13u
3
8u
2
+ 4u 1
u
7
+ 3u
6
6u
5
+ 11u
4
12u
3
+ 14u
2
8u + 3
a
10
=
2u
7
3u
6
+ 9u
5
9u
4
+ 13u
3
8u
2
+ 4u 1
u
7
+ 3u
6
6u
5
+ 11u
4
12u
3
+ 14u
2
8u + 3
(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 = 0.265930 0.663572I
a = 1.00000
b = 1.62445 0.95813I
7.12528 4.91384I 15.9725 + 5.8373I
u = 0.265930 + 0.663572I
a = 1.00000
b = 1.62445 + 0.95813I
7.12528 + 4.91384I 15.9725 5.8373I
u = 0.239460 1.335293I
a = 1.00000
b = 2.13175 + 0.56567I
1.86433 + 0.65741I 7.52191 0.35368I
u = 0.239460 + 1.335293I
a = 1.00000
b = 2.13175 0.56567I
1.86433 0.65741I 7.52191 + 0.35368I
u = 0.495635 1.261447I
a = 1.00000
b = 2.31639 + 0.06726I
3.58036 + 6.89137I 7.40021 6.36507I
u = 0.495635 + 1.261447I
a = 1.00000
b = 2.31639 0.06726I
3.58036 6.89137I 7.40021 + 6.36507I
u = 0.509756 0.564785I
a = 1.00000
b = 0.676307 0.513532I
1.17049 + 1.46276I 13.60538 3.21811I
u = 0.509756 + 0.564785I
a = 1.00000
b = 0.676307 + 0.513532I
1.17049 1.46276I 13.60538 + 3.21811I
3
II.
I
u
2
= hu
18
+ u
17
+ · · · 4u + 1, a + 1, 4427u
17
+ 4238u
16
+ · · · + 4846b 789i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0.913537u
17
0.874536u
16
+ ··· + 0.241436u + 0.162815
a
9
=
u
0.0390012u
17
+ 0.601940u
16
+ ··· 2.49133u + 0.913537
a
1
=
1
0.913537u
17
0.874536u
16
+ ··· + 0.241436u + 0.162815
a
4
=
1
0
a
3
=
0.913537u
17
0.874536u
16
+ ··· + 0.241436u + 1.16281
1.84585u
17
1.50186u
16
+ ··· 1.96574u 0.151259
a
7
=
u
u
a
5
=
u
2
+ 1
u
2
a
2
=
0.0879076u
17
+ 0.221832u
16
+ ··· 0.647132u 1.37742
1.17623u
17
1.25691u
16
+ ··· + 0.905283u 0.0907965
a
6
=
0.292612u
17
+ 0.592860u
16
+ ··· 1.21275u + 0.562939
0.776310u
17
+ 1.21956u
16
+ ··· 0.827487u + 0.659926
a
10
=
0.309946u
17
+ 0.249278u
16
+ ··· + 1.12443u 0.419934
1.60111u
17
+ 0.745770u
16
+ ··· + 4.80025u 0.316756
a
10
=
0.309946u
17
+ 0.249278u
16
+ ··· + 1.12443u 0.419934
1.60111u
17
+ 0.745770u
16
+ ··· + 4.80025u 0.316756
(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 = 0.857117 0.325932I
a = 1.00000
b = 0.569887 + 0.875038I
10.72733 + 4.74355I 14.8235 1.3888I
u = 0.857117 + 0.325932I
a = 1.00000
b = 0.569887 0.875038I
10.72733 4.74355I 14.8235 + 1.3888I
u = 0.61297 1.38177I
a = 1.00000
b = 2.67097 + 0.24138I
3.9883 16.2483I 8.61000 + 8.11381I
u = 0.61297 + 1.38177I
a = 1.00000
b = 2.67097 0.24138I
3.9883 + 16.2483I 8.61000 8.11381I
u = 0.531929 0.836030I
a = 1.00000
b = 0.478818 0.947490I
5.93754 0.90931I 12.67105 + 3.59251I
u = 0.531929 + 0.836030I
a = 1.00000
b = 0.478818 + 0.947490I
5.93754 + 0.90931I 12.67105 3.59251I
u = 0.256117 1.267060I
a = 1.00000
b = 2.27184 0.48589I
6.34892 0.61291I 0.81864 1.96169I
u = 0.256117 + 1.267060I
a = 1.00000
b = 2.27184 + 0.48589I
6.34892 + 0.61291I 0.81864 + 1.96169I
u = 0.106940 0.947275I
a = 1.00000
b = 3.55884 + 1.33317I
6.13065 4.90278I 7.01114 + 4.15155I
u = 0.106940 + 0.947275I
a = 1.00000
b = 3.55884 1.33317I
6.13065 + 4.90278I 7.01114 4.15155I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.237777
a = 1.00000
b = 0.516116
0.811801 12.0985
u = 0.406222 1.352414I
a = 1.00000
b = 2.52525 0.20338I
5.38862 + 5.69558I 1.81453 3.55021I
u = 0.406222 + 1.352414I
a = 1.00000
b = 2.52525 + 0.20338I
5.38862 5.69558I 1.81453 + 3.55021I
u = 0.458737 0.547198I
a = 1.00000
b = 0.258817 0.335251I
0.39235 + 1.59654I 2.49522 4.52605I
u = 0.458737 + 0.547198I
a = 1.00000
b = 0.258817 + 0.335251I
0.39235 1.59654I 2.49522 + 4.52605I
u = 0.482845 1.256566I
a = 1.00000
b = 2.64003 + 0.57753I
0.95391 + 8.87623I 7.74572 6.85139I
u = 0.482845 + 1.256566I
a = 1.00000
b = 2.64003 0.57753I
0.95391 8.87623I 7.74572 + 6.85139I
u = 0.796765
a = 1.00000
b = 0.567193
6.40721 14.9219
6
III. I
u
3
=
hu
48
+5u
47
+· · ·314u+61, 1.19×10
93
u
47
+6.16×10
93
u
46
+· · ·+1.65×10
93
b+
4.19 × 10
94
, 7.44 × 10
94
u
47
+ 4.08 × 10
95
u
46
+ · · · + 1.01 × 10
95
a + 9.36 × 10
95
i
(i) Arc colorings
a
8
=
0
u
a
11
=
0.740050u
47
4.05356u
46
+ ··· + 101.257u 9.31038
0.723874u
47
3.73598u
46
+ ··· + 58.6317u 25.4108
a
9
=
0.144862u
47
0.321223u
46
+ ··· + 424.069u 106.348
0.145684u
47
0.378256u
46
+ ··· + 358.711u 66.2412
a
1
=
0.740050u
47
4.05356u
46
+ ··· + 101.257u 9.31038
0.156309u
47
0.781076u
46
+ ··· 7.16508u 3.85883
a
4
=
1
0
a
3
=
0.0403867u
47
+ 0.393829u
46
+ ··· 63.0379u + 8.89542
0.886672u
47
+ 4.56313u
46
+ ··· 94.2005u + 33.7713
a
7
=
u
u
a
5
=
u
2
+ 1
u
2
a
2
=
0.696258u
47
3.70972u
46
+ ··· + 35.4526u + 9.47658
0.632760u
47
3.14751u
46
+ ··· 4.65590u 10.5113
a
6
=
1.04265u
47
5.22867u
46
+ ··· 58.4887u + 3.14813
0.943596u
47
5.23155u
46
+ ··· + 135.512u 41.8484
a
10
=
0.178281u
47
1.13825u
46
+ ··· + 43.9290u + 6.09493
1.05239u
47
5.41675u
46
+ ··· + 93.1989u 28.0182
a
10
=
0.178281u
47
1.13825u
46
+ ··· + 43.9290u + 6.09493
1.05239u
47
5.41675u
46
+ ··· + 93.1989u 28.0182
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 2.28147 2.79272I
a = 0.228767 0.178927I
b = 3.10682 + 2.16956I
1.83976 + 0.26235I 18.4351 + 36.8531I
u = 2.28147 + 2.79272I
a = 0.228767 + 0.178927I
b = 3.10682 2.16956I
1.83976 0.26235I 18.4351 36.8531I
u = 1.291291 0.048766I
a = 0.355413 + 0.898270I
b = 0.051335 0.715852I
8.17000 + 9.69978I 11.19558 6.76089I
u = 1.291291 + 0.048766I
a = 0.355413 0.898270I
b = 0.051335 + 0.715852I
8.17000 9.69978I 11.19558 + 6.76089I
u = 0.800467 0.719023I
a = 0.190830 0.189545I
b = 0.454645 0.824616I
4.43678 0.22592I 13.45627 0.52268I
u = 0.800467 + 0.719023I
a = 0.190830 + 0.189545I
b = 0.454645 + 0.824616I
4.43678 + 0.22592I 13.45627 + 0.52268I
u = 0.72686 1.40262I
a = 0.875338 + 0.166876I
b = 2.46224 0.32677I
1.74071 9.53525I 7.20923 + 7.09915I
u = 0.72686 + 1.40262I
a = 0.875338 0.166876I
b = 2.46224 + 0.32677I
1.74071 + 9.53525I 7.20923 7.09915I
u = 0.596834 0.792322I
a = 0.632821 + 0.697017I
b = 0.177976 + 1.161761I
6.06530 3.64576I 10.43387 + 2.45073I
u = 0.596834 + 0.792322I
a = 0.632821 0.697017I
b = 0.177976 1.161761I
6.06530 + 3.64576I 10.43387 2.45073I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.579065 0.197316I
a = 0.23645 1.78528I
b = 0.108214 0.269994I
1.82625 + 2.21677I 1.73188 4.68950I
u = 0.579065 + 0.197316I
a = 0.23645 + 1.78528I
b = 0.108214 + 0.269994I
1.82625 2.21677I 1.73188 + 4.68950I
u = 0.542205 1.034905I
a = 0.550346 0.600481I
b = 0.242196 0.100812I
3.16425 4.93690I 9.72258 + 5.53812I
u = 0.542205 + 1.034905I
a = 0.550346 + 0.600481I
b = 0.242196 + 0.100812I
3.16425 + 4.93690I 9.72258 5.53812I
u = 0.502747 1.142596I
a = 0.380851 + 0.962562I
b = 0.051335 + 0.715852I
8.17000 9.69978I 11.19558 + 6.76089I
u = 0.502747 + 1.142596I
a = 0.380851 0.962562I
b = 0.051335 0.715852I
8.17000 + 9.69978I 11.19558 6.76089I
u = 0.330498 1.145321I
a = 1.201793 0.093121I
b = 2.25735 + 0.37692I
4.61294 5.64930I 2.87192 + 2.20392I
u = 0.330498 + 1.145321I
a = 1.201793 + 0.093121I
b = 2.25735 0.37692I
4.61294 + 5.64930I 2.87192 2.20392I
u = 0.174573 0.917402I
a = 0.714007 + 0.786439I
b = 0.177976 1.161761I
6.06530 + 3.64576I 10.43387 2.45073I
u = 0.174573 + 0.917402I
a = 0.714007 0.786439I
b = 0.177976 + 1.161761I
6.06530 3.64576I 10.43387 + 2.45073I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.099389 0.960926I
a = 1.41531 + 0.01548I
b = 2.60667 0.70175I
0.47769 2.08395I 4.74669 + 3.16145I
u = 0.099389 + 0.960926I
a = 1.41531 0.01548I
b = 2.60667 + 0.70175I
0.47769 + 2.08395I 4.74669 3.16145I
u = 0.061740 0.876312I
a = 1.17704 1.09617I
b = 0.960843 0.043439I
0.782529 + 1.065845I 2.95105 0.35875I
u = 0.061740 + 0.876312I
a = 1.17704 + 1.09617I
b = 0.960843 + 0.043439I
0.782529 1.065845I 2.95105 + 0.35875I
u = 0.022232 1.047100I
a = 2.71214 + 2.12126I
b = 3.10682 + 2.16956I
1.83976 + 0.26235I 18.4351 + 36.8531I
u = 0.022232 + 1.047100I
a = 2.71214 2.12126I
b = 3.10682 2.16956I
1.83976 0.26235I 18.4351 36.8531I
u = 0.100648 0.894015I
a = 0.494606 1.061476I
b = 1.029254 0.404165I
0.282838 0.252163I 7.62766 + 0.43499I
u = 0.100648 + 0.894015I
a = 0.494606 + 1.061476I
b = 1.029254 + 0.404165I
0.282838 + 0.252163I 7.62766 0.43499I
u = 0.125787 1.361551I
a = 0.706473 + 0.007729I
b = 2.60667 + 0.70175I
0.47769 + 2.08395I 4.74669 3.16145I
u = 0.125787 + 1.361551I
a = 0.706473 0.007729I
b = 2.60667 0.70175I
0.47769 2.08395I 4.74669 + 3.16145I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.215349 1.080451I
a = 0.072906 + 0.550479I
b = 0.108214 0.269994I
1.82625 + 2.21677I 1.73188 4.68950I
u = 0.215349 + 1.080451I
a = 0.072906 0.550479I
b = 0.108214 + 0.269994I
1.82625 2.21677I 1.73188 + 4.68950I
u = 0.289040 0.014513I
a = 2.63784 + 2.62007I
b = 0.454645 0.824616I
4.43678 0.22592I 13.45627 0.52268I
u = 0.289040 + 0.014513I
a = 2.63784 2.62007I
b = 0.454645 + 0.824616I
4.43678 + 0.22592I 13.45627 + 0.52268I
u = 0.365070 1.196053I
a = 0.162895 0.619557I
b = 0.647164 0.281770I
2.69997 + 4.12763I 10.48837 3.99209I
u = 0.365070 + 1.196053I
a = 0.162895 + 0.619557I
b = 0.647164 + 0.281770I
2.69997 4.12763I 10.48837 + 3.99209I
u = 0.402181 1.349061I
a = 1.102352 + 0.210154I
b = 2.46224 + 0.32677I
1.74071 + 9.53525I 7.20923 7.09915I
u = 0.402181 + 1.349061I
a = 1.102352 0.210154I
b = 2.46224 0.32677I
1.74071 9.53525I 7.20923 + 7.09915I
u = 0.50384 1.34566I
a = 0.827124 0.064090I
b = 2.25735 0.37692I
4.61294 + 5.64930I 2.87192 2.20392I
u = 0.50384 + 1.34566I
a = 0.827124 + 0.064090I
b = 2.25735 + 0.37692I
4.61294 5.64930I 2.87192 + 2.20392I
11
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.800491 0.031350I
a = 0.39693 1.50969I
b = 0.647164 + 0.281770I
2.69997 4.12763I 10.48837 + 3.99209I
u = 0.800491 + 0.031350I
a = 0.39693 + 1.50969I
b = 0.647164 0.281770I
2.69997 + 4.12763I 10.48837 3.99209I
u = 0.887919 1.099131I
a = 0.454979 + 0.423721I
b = 0.960843 0.043439I
0.782529 + 1.065845I 2.95105 0.35875I
u = 0.887919 + 1.099131I
a = 0.454979 0.423721I
b = 0.960843 + 0.043439I
0.782529 1.065845I 2.95105 + 0.35875I
u = 0.899195 0.549021I
a = 0.360667 0.774028I
b = 1.029254 + 0.404165I
0.282838 + 0.252163I 7.62766 0.43499I
u = 0.899195 + 0.549021I
a = 0.360667 + 0.774028I
b = 1.029254 0.404165I
0.282838 0.252163I 7.62766 + 0.43499I
u = 0.919842 0.243972I
a = 0.829511 0.905077I
b = 0.242196 + 0.100812I
3.16425 + 4.93690I 9.72258 5.53812I
u = 0.919842 + 0.243972I
a = 0.829511 + 0.905077I
b = 0.242196 0.100812I
3.16425 4.93690I 9.72258 + 5.53812I
12
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
8
4u
6
+ ··· u + 1)(u
18
9u
16
+ ··· 2u 1)
(u
48
+ 3u
47
+ ··· 2258u + 319)
c
2
(u
8
4u
6
+ ··· + u + 1)(u
18
9u
16
+ ··· 2u 1)
(u
48
+ 3u
47
+ ··· 2258u + 319)
c
3
(u
8
2u
6
u
5
+ 16u
4
+ 32u
3
+ 24u
2
+ 8u + 1)
(u
18
+ 17u
17
+ ··· 640u 64)(u
48
8u
47
+ ··· 4u + 1)
c
4
, c
8
(u
8
u
7
+ ··· u + 1)(u
18
+ u
17
+ ··· 4u + 1)
(u
48
+ 5u
47
+ ··· 314u + 61)
c
5
(u
8
4u
6
+ ··· + u + 1)(u
18
9u
16
+ ··· 2u 1)
(u
48
+ 3u
47
+ ··· 2258u + 319)
c
6
(u
8
+ 2u
7
+ 5u
6
+ 2u
5
+ 4u
4
2u
3
+ u
2
u + 1)
(u
18
+ 11u
17
+ ··· + 208u + 16)
(1 2u + 28u
2
72u
3
+ 117u
4
172u
5
+ 180u
6
154u
7
+ 194u
8
192u
9
+ 384u
10
550u
11
+ 788u
12
966u
13
+ 1016u
14
972u
15
+ 807u
16
596u
17
+ 398u
18
222u
19
+ 117u
20
46u
21
+ 18u
22
4u
23
+ u
24
)
2
c
7
, c
11
(u
8
+ u
7
+ ··· + u + 1)(u
18
+ u
17
+ ··· 4u + 1)
(u
48
+ 5u
47
+ ··· 314u + 61)
c
9
(u
8
2u
7
+ 5u
6
2u
5
+ 4u
4
+ 2u
3
+ u
2
+ u + 1)
(u
18
+ 11u
17
+ ··· + 208u + 16)
(1 2u + 28u
2
72u
3
+ 117u
4
172u
5
+ 180u
6
154u
7
+ 194u
8
192u
9
+ 384u
10
550u
11
+ 788u
12
966u
13
+ 1016u
14
972u
15
+ 807u
16
596u
17
+ 398u
18
222u
19
+ 117u
20
46u
21
+ 18u
22
4u
23
+ u
24
)
2
c
10
(u 1)(u
8
4u
6
+ ··· u + 1)(u
18
9u
16
+ ··· 2u 1)
(u
47
+ 4u
46
+ ··· + 1939u 319)
13
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
8
8y
7
+ 28y
6
54y
5
+ 62y
4
45y
3
+ 23y
2
7y + 1)
(y
18
18y
17
+ ··· + 6y + 1)(y
48
37y
47
+ ··· + 318056y + 101761)
c
2
(y
8
8y
7
+ 28y
6
54y
5
+ 62y
4
45y
3
+ 23y
2
7y + 1)
(y
18
18y
17
+ ··· + 6y + 1)(y
48
37y
47
+ ··· + 318056y + 101761)
c
3
(y
8
4y
7
+ 36y
6
17y
5
+ 226y
4
244y
3
+ 96y
2
16y + 1)
(y
18
3y
17
+ ··· 24576y + 4096)(y
48
+ 12y
47
+ ··· + 16y + 1)
c
4
, c
7
, c
8
c
11
(y
8
+ 7y
7
+ 22y
6
+ 39y
5
+ 42y
4
+ 29y
3
+ 15y
2
+ 5y + 1)
(y
18
+ 13y
17
+ ··· 14y + 1)(y
48
+ 31y
47
+ ··· 55164y + 3721)
c
6
, c
9
(y
8
+ 6y
7
+ 25y
6
+ 46y
5
+ 40y
4
+ 18y
3
+ 5y
2
+ y + 1)
(y
18
+ 11y
17
+ ··· 12160y + 256)(y
48
+ 40y
47
+ ··· + 104y + 1)
c
10
(y
8
8y
7
+ 28y
6
54y
5
+ 62y
4
45y
3
+ 23y
2
7y + 1)
(y
18
18y
17
+ ··· + 6y + 1)(y
48
37y
47
+ ··· + 318056y + 101761)
14