11a
27
(K11a
27
)
1
Arc Sequences
5 1 11 2 9 3 4 10 6 8 7
Solving Sequence
4,11 3,8
7 1 2 6 10 9 5
c
3
c
7
c
11
c
2
c
6
c
10
c
8
c
5
c
1
, c
4
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b, a + u + 1i
I
u
2
= hu
73
+ 2u
72
+ ··· 784u + 224,
5.41969 × 10
257
u
72
2.81275 × 10
256
u
71
+ ··· + 8.46748 × 10
259
b 2.77370 × 10
259
,
7.43384 × 10
258
u
72
1.34318 × 10
259
u
71
+ ··· + 5.92724 × 10
260
a 1.00426 × 10
262
i
There are 2 irreducible components with 75 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, b, a + u + 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
u 1
0
a
3
=
1
0
a
8
=
0
u
a
7
=
u
u
a
1
=
2u 1
u
a
2
=
u 1
u + 1
a
6
=
0
u
a
10
=
u 1
u
a
9
=
u 1
2u
a
5
=
u + 1
u
a
5
=
u + 1
u
(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
1.64493 2.02988I 9.00000 + 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0
1.64493 + 2.02988I 9.00000 3.46410I
3
II. I
u
2
= hu
73
+ 2u
72
+ · · · 784u + 224, 5.42 × 10
257
u
72
2.81 × 10
256
u
71
+
· · · + 8.47 × 10
259
b 2.77 × 10
259
, 7.43 × 10
258
u
72
1.34 × 10
259
u
71
+ · · · +
5.93 × 10
260
a 1.00 × 10
262
i
(i) Arc colorings
a
4
=
1
0
a
11
=
0.0125418u
72
+ 0.0226611u
71
+ ··· 49.8356u + 16.9432
0.00640059u
72
+ 0.000332183u
71
+ ··· 3.00631u + 0.327571
a
3
=
0.0435585u
72
+ 0.0509612u
71
+ ··· + 99.6072u 21.4440
0.00973873u
72
+ 0.00503448u
71
+ ··· 86.2725u + 20.1008
a
8
=
0
u
a
7
=
u
u
a
1
=
0.0185994u
72
+ 0.0349545u
71
+ ··· 56.8573u + 19.3424
0.0124582u
72
0.0119612u
71
+ ··· + 4.01534u 2.07165
a
2
=
0.0124116u
72
+ 0.0218431u
71
+ ··· + 7.44848u + 0.766620
0.0109037u
72
0.0254418u
71
+ ··· + 11.9417u 4.79598
a
6
=
0.00257307u
72
+ 0.0246508u
71
+ ··· 47.4396u + 15.9214
0.0305982u
72
0.0677537u
71
+ ··· + 22.1281u 11.4149
a
10
=
0.0125418u
72
+ 0.0226611u
71
+ ··· 49.8356u + 16.9432
0.0128835u
72
0.00785097u
71
+ ··· 7.71495u + 0.870220
a
9
=
0.0272494u
72
+ 0.0594532u
71
+ ··· 10.6219u + 10.1246
0.0476397u
72
0.0935743u
71
+ ··· + 10.4863u 10.8669
a
5
=
0.0197337u
72
0.0592553u
71
+ ··· 1.17499u 1.97148
0.00317357u
72
+ 0.0240883u
71
+ ··· 29.9329u + 8.71462
a
5
=
0.0197337u
72
0.0592553u
71
+ ··· 1.17499u 1.97148
0.00317357u
72
+ 0.0240883u
71
+ ··· 29.9329u + 8.71462
(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 = 1.12301 1.36301I
a = 0.868710 0.369423I
b = 0.99821 + 1.22701I
4.1062 + 17.0192I 2.90782 10.28377I
u = 1.12301 + 1.36301I
a = 0.868710 + 0.369423I
b = 0.99821 1.22701I
4.1062 17.0192I 2.90782 + 10.28377I
u = 1.073440 0.377264I
a = 0.155363 + 1.004890I
b = 0.18350 1.64558I
0.261500 + 0.649592I 8.78620 3.09010I
u = 1.073440 + 0.377264I
a = 0.155363 1.004890I
b = 0.18350 + 1.64558I
0.261500 0.649592I 8.78620 + 3.09010I
u = 1.054986 0.922796I
a = 0.920524 + 0.239448I
b = 0.904852 + 0.904984I
2.75520 + 10.95232I 7.44212 9.37085I
u = 1.054986 + 0.922796I
a = 0.920524 0.239448I
b = 0.904852 0.904984I
2.75520 10.95232I 7.44212 + 9.37085I
u = 1.00449 1.19281I
a = 0.692727 + 0.404704I
b = 0.85474 1.25657I
2.75982 + 5.17856I 3.80134 2.28640I
u = 1.00449 + 1.19281I
a = 0.692727 0.404704I
b = 0.85474 + 1.25657I
2.75982 5.17856I 3.80134 + 2.28640I
u = 0.771915 1.016807I
a = 1.097565 0.509906I
b = 1.09209 + 1.38351I
8.19243 + 8.08051I 0.92498 7.17495I
u = 0.771915 + 1.016807I
a = 1.097565 + 0.509906I
b = 1.09209 1.38351I
8.19243 8.08051I 0.92498 + 7.17495I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.75672 1.70120I
a = 0.656800 0.121180I
b = 0.521307 + 0.447048I
2.40217 + 4.04097I 1.16757 7.09941I
u = 0.75672 + 1.70120I
a = 0.656800 + 0.121180I
b = 0.521307 0.447048I
2.40217 4.04097I 1.16757 + 7.09941I
u = 0.727699
a = 1.00090
b = 0.291116
1.41721 6.25417
u = 0.706305 0.439910I
a = 0.920408 + 0.281189I
b = 0.88728 1.14467I
2.29836 + 4.22810I 12.1430 9.3439I
u = 0.706305 + 0.439910I
a = 0.920408 0.281189I
b = 0.88728 + 1.14467I
2.29836 4.22810I 12.1430 + 9.3439I
u = 0.695071 0.959449I
a = 0.004634 + 0.185185I
b = 0.382619 0.469678I
0.75785 + 1.41365I 4.07014 4.97652I
u = 0.695071 + 0.959449I
a = 0.004634 0.185185I
b = 0.382619 + 0.469678I
0.75785 1.41365I 4.07014 + 4.97652I
u = 0.663769 0.462321I
a = 0.70947 + 1.65424I
b = 0.531987 + 0.577235I
2.41837 + 3.22713I 6.50397 8.13076I
u = 0.663769 + 0.462321I
a = 0.70947 1.65424I
b = 0.531987 0.577235I
2.41837 3.22713I 6.50397 + 8.13076I
u = 0.587100 0.723419I
a = 0.198283 0.518162I
b = 0.623940 0.811732I
0.57464 + 2.49330I 4.09751 3.36278I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.587100 + 0.723419I
a = 0.198283 + 0.518162I
b = 0.623940 + 0.811732I
0.57464 2.49330I 4.09751 + 3.36278I
u = 0.548650 1.015824I
a = 1.284021 0.238994I
b = 0.413663 + 0.746665I
2.97910 + 4.50576I 3.61990 4.22524I
u = 0.548650 + 1.015824I
a = 1.284021 + 0.238994I
b = 0.413663 0.746665I
2.97910 4.50576I 3.61990 + 4.22524I
u = 0.401941 0.804946I
a = 0.897223 + 0.360889I
b = 0.955416 + 0.688471I
1.56864 + 4.14463I 2.64822 8.01358I
u = 0.401941 + 0.804946I
a = 0.897223 0.360889I
b = 0.955416 0.688471I
1.56864 4.14463I 2.64822 + 8.01358I
u = 0.31676 3.10066I
a = 0.420952 + 0.129401I
b = 0.525473 0.263881I
3.18426 5.39888I 5.84795 9.57058I
u = 0.31676 + 3.10066I
a = 0.420952 0.129401I
b = 0.525473 + 0.263881I
3.18426 + 5.39888I 5.84795 + 9.57058I
u = 0.31454 3.49405I
a = 0.002735 + 0.185466I
b = 0.020347 0.290364I
1.30311 2.11076I 42.0323 9.8539I
u = 0.31454 + 3.49405I
a = 0.002735 0.185466I
b = 0.020347 + 0.290364I
1.30311 + 2.11076I 42.0323 + 9.8539I
u = 0.211879 0.584703I
a = 0.450705 + 0.007341I
b = 0.575775 0.413622I
0.76514 + 1.25127I 5.38460 5.06088I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.211879 + 0.584703I
a = 0.450705 0.007341I
b = 0.575775 + 0.413622I
0.76514 1.25127I 5.38460 + 5.06088I
u = 0.199476 0.702761I
a = 0.27447 + 1.92998I
b = 0.134422 + 0.525881I
1.11279 1.68465I 4.96907 2.48353I
u = 0.199476 + 0.702761I
a = 0.27447 1.92998I
b = 0.134422 0.525881I
1.11279 + 1.68465I 4.96907 + 2.48353I
u = 0.196172 0.545724I
a = 2.42765 0.23693I
b = 1.62844 0.90039I
2.67029 + 8.34069I 4.38270 11.12769I
u = 0.196172 + 0.545724I
a = 2.42765 + 0.23693I
b = 1.62844 + 0.90039I
2.67029 8.34069I 4.38270 + 11.12769I
u = 0.122145 0.591196I
a = 2.45996 + 0.27818I
b = 0.061707 0.909666I
6.72947 4.09633I 3.14446 + 4.83546I
u = 0.122145 + 0.591196I
a = 2.45996 0.27818I
b = 0.061707 + 0.909666I
6.72947 + 4.09633I 3.14446 4.83546I
u = 0.096382 1.059678I
a = 1.150597 0.231346I
b = 0.234833 0.948799I
6.08025 + 2.66865I 2.56867 3.42709I
u = 0.096382 + 1.059678I
a = 1.150597 + 0.231346I
b = 0.234833 + 0.948799I
6.08025 2.66865I 2.56867 + 3.42709I
u = 0.099814 0.392098I
a = 2.83754 + 0.89871I
b = 1.60921 + 0.66159I
3.09906 2.70236I 2.89632 + 5.58014I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.099814 + 0.392098I
a = 2.83754 0.89871I
b = 1.60921 0.66159I
3.09906 + 2.70236I 2.89632 5.58014I
u = 0.152205 0.282535I
a = 3.71687 0.84191I
b = 1.62722 0.22397I
0.69933 2.58301I 11.91137 + 2.89959I
u = 0.152205 + 0.282535I
a = 3.71687 + 0.84191I
b = 1.62722 + 0.22397I
0.69933 + 2.58301I 11.91137 2.89959I
u = 0.251013 1.024044I
a = 1.59999 + 0.14403I
b = 0.412283 + 0.874957I
4.88061 8.78731I 0.32196 + 9.11237I
u = 0.251013 + 1.024044I
a = 1.59999 0.14403I
b = 0.412283 0.874957I
4.88061 + 8.78731I 0.32196 9.11237I
u = 0.269009 0.408888I
a = 3.05631 1.40140I
b = 0.214868 + 0.856112I
6.25965 2.30457I 2.36678 + 1.26889I
u = 0.269009 + 0.408888I
a = 3.05631 + 1.40140I
b = 0.214868 0.856112I
6.25965 + 2.30457I 2.36678 1.26889I
u = 0.34736 2.81362I
a = 0.454897 0.005069I
b = 0.584992 + 0.148286I
3.36772 0.28806I 8.2957 + 11.4804I
u = 0.34736 + 2.81362I
a = 0.454897 + 0.005069I
b = 0.584992 0.148286I
3.36772 + 0.28806I 8.2957 11.4804I
u = 0.349807 1.049909I
a = 1.039581 + 0.270557I
b = 0.225312 0.818115I
3.74839 + 1.23253I 1.83312 1.85392I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.349807 + 1.049909I
a = 1.039581 0.270557I
b = 0.225312 + 0.818115I
3.74839 1.23253I 1.83312 + 1.85392I
u = 0.352818 0.985323I
a = 0.395836 0.276475I
b = 0.457094 0.889285I
3.66692 0.65369I 3.56200 + 0.87256I
u = 0.352818 + 0.985323I
a = 0.395836 + 0.276475I
b = 0.457094 + 0.889285I
3.66692 + 0.65369I 3.56200 0.87256I
u = 0.55855 2.25389I
a = 0.485906 + 0.242323I
b = 0.421934 0.489041I
2.39394 + 1.31325I 3.63687 + 2.32909I
u = 0.55855 + 2.25389I
a = 0.485906 0.242323I
b = 0.421934 + 0.489041I
2.39394 1.31325I 3.63687 2.32909I
u = 0.599679 0.261791I
a = 0.86479 + 1.79489I
b = 0.685403 + 0.478823I
3.44487 + 1.41386I 10.83320 + 0.26593I
u = 0.599679 + 0.261791I
a = 0.86479 1.79489I
b = 0.685403 0.478823I
3.44487 1.41386I 10.83320 0.26593I
u = 0.702214 1.085511I
a = 0.898238 + 0.518500I
b = 0.94267 1.39992I
8.72144 2.03741I 2.06720 + 1.77059I
u = 0.702214 + 1.085511I
a = 0.898238 0.518500I
b = 0.94267 + 1.39992I
8.72144 + 2.03741I 2.06720 1.77059I
u = 0.704933 0.862455I
a = 0.326461 0.533891I
b = 0.626006 0.900852I
1.24065 7.33132I 0.88010 + 7.59105I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.704933 + 0.862455I
a = 0.326461 + 0.533891I
b = 0.626006 + 0.900852I
1.24065 + 7.33132I 0.88010 7.59105I
u = 0.718334 0.034783I
a = 1.170911 0.016850I
b = 1.09050 + 1.02689I
4.06916 1.66774I 17.5628 + 3.8525I
u = 0.718334 + 0.034783I
a = 1.170911 + 0.016850I
b = 1.09050 1.02689I
4.06916 + 1.66774I 17.5628 3.8525I
u = 0.857855 0.206337I
a = 0.0945515 + 0.0017358I
b = 0.625708 0.391085I
0.16446 + 2.80927I 2.04943 2.04126I
u = 0.857855 + 0.206337I
a = 0.0945515 0.0017358I
b = 0.625708 + 0.391085I
0.16446 2.80927I 2.04943 + 2.04126I
u = 1.00254 1.34079I
a = 0.719486 + 0.325890I
b = 0.90851 1.22268I
5.08354 10.84059I 1.35643 + 5.82505I
u = 1.00254 + 1.34079I
a = 0.719486 0.325890I
b = 0.90851 + 1.22268I
5.08354 + 10.84059I 1.35643 5.82505I
u = 1.004968 0.736127I
a = 0.945284 + 0.222410I
b = 0.942016 + 0.894877I
4.44539 5.47964I 10.52914 + 5.00212I
u = 1.004968 + 0.736127I
a = 0.945284 0.222410I
b = 0.942016 0.894877I
4.44539 + 5.47964I 10.52914 5.00212I
u = 1.096408 0.249087I
a = 0.557924 1.029705I
b = 0.54381 + 1.69143I
0.01661 5.97370I 9.92070 + 8.84627I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.096408 + 0.249087I
a = 0.557924 + 1.029705I
b = 0.54381 1.69143I
0.01661 + 5.97370I 9.92070 8.84627I
u = 1.14109 1.18189I
a = 0.868102 0.447299I
b = 0.96563 + 1.27526I
1.84899 11.18435I 5.28704 + 6.93589I
u = 1.14109 + 1.18189I
a = 0.868102 + 0.447299I
b = 0.96563 1.27526I
1.84899 + 11.18435I 5.28704 6.93589I
12
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
73
+ 2u
72
+ ··· + 7u 1)
c
2
(u
2
+ u + 1)(u
73
+ 32u
72
+ ··· + 131u 1)
c
3
u
2
(u
73
+ 7u
72
+ ··· + 12u + 4)
c
4
(u
2
u + 1)(u
73
+ 2u
72
+ ··· + 7u 1)
c
5
(u + 1)
2
(u
73
+ 3u
72
+ ··· + 8u + 1)
c
6
(u
2
u + 1)(u
73
+ 34u
71
+ ··· 149u 41)
c
7
(u
2
u + 1)(u
73
+ 2u
72
+ ··· 784u + 224)
c
8
(u + 1)
2
(u
73
+ 23u
72
+ ··· 2u + 1)
c
9
(u 1)
2
(u
73
+ 3u
72
+ ··· + 8u + 1)
c
10
(u 1)
2
(u
73
+ 23u
72
+ ··· 2u + 1)
c
11
(u 1)
2
(u
73
+ 7u
72
+ ··· + u
2
+ 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)(y
73
+ 32y
72
+ ··· + 131y 1)
c
2
(y
2
+ y + 1)(y
73
+ 20y
72
+ ··· + 19907y 1)
c
3
y
2
(y
73
+ 15y
72
+ ··· 344y 16)
c
5
1.0000000000000(1y 1.0000000000000)
2
(1.00y
73
23.0y
72
+ ··· 2.00y 1.00)
c
6
(y
2
+ y + 1)(y
73
+ 68y
72
+ ··· 11173y 1681)
c
7
(y
2
+ y + 1)(y
73
+ 84y
72
+ ··· 1858304y 50176)
c
8
, c
10
(y 1)
2
(y
73
+ 57y
72
+ ··· + 254y 1)
c
9
1.0000000000000(1y 1.00000000000000)
2
(1.00y
73
23.0y
72
+ ··· 2.00y 1.00)
c
11
(y 1)
2
(y
73
+ 5y
72
+ ··· 2y 1)
14