11a
316
(K11a
316
)
1
Arc Sequences
9 6 1 11 8 2 10 3 7 4 5
Solving Sequence
2,6
3
7,9
10 1 4 8 5 11
c
2
c
6
c
9
c
1
c
3
c
8
c
5
c
11
c
4
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u 1, 5a 3u 4, 5b 2u 1i
I
u
2
= hu
62
+ 2u
61
+ ··· u + 1, 1.84297 × 10
92
u
61
+ 4.82420 × 10
92
u
60
+ ··· + 8.95123 × 10
92
b + 3.45436 × 10
92
,
8.94450 × 10
92
u
61
+ 1.54943 × 10
93
u
60
+ ··· + 8.95123 × 10
92
a 8.71238 × 10
92
i
There are 2 irreducible components with 64 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, 5a 3u 4, 5b 2u 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u + 1
a
7
=
u
u
a
9
=
3
5
u +
4
5
2
5
u +
1
5
a
10
=
8
5
u +
4
5
3
5
u +
1
5
a
1
=
1
5
u +
7
5
0.2
a
4
=
u
1
5
4
5
u +
4
5
a
8
=
3
5
u +
4
5
2
5
u +
1
5
a
5
=
2
5
u
3
5
4
5
u
1
5
a
11
=
2
5
u +
9
5
u
3
5
a
11
=
2
5
u +
9
5
u
3
5
(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 = 1.61803
a = 0.170820
b = 0.447214
7.23771 36.2997
u = 0.618034
a = 1.17082
b = 0.447214
0.657974 4.10031
3
II.
I
u
2
= hu
62
+2u
61
+· · ·u+1, 1.84×10
92
u
61
+4.82×10
92
u
60
+· · ·+8.95×10
92
b+
3.45 × 10
92
, 8.94 × 10
92
u
61
+ 1.55 × 10
93
u
60
+ · · · + 8.95 × 10
92
a 8.71 × 10
92
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
u
u
a
9
=
0.999248u
61
1.73097u
60
+ ··· + 7.69342u + 0.973316
0.205890u
61
0.538943u
60
+ ··· + 3.64802u 0.385909
a
10
=
1.05698u
61
1.86841u
60
+ ··· + 9.03892u + 0.832957
0.148160u
61
0.401509u
60
+ ··· + 2.30253u 0.245549
a
1
=
0.246699u
61
1.81043u
60
+ ··· + 0.744103u + 2.83504
0.707800u
61
+ 2.51840u
60
+ ··· 1.83028u + 1.52032
a
4
=
1.05477u
61
0.547610u
60
+ ··· 2.96099u + 2.37771
0.820020u
61
0.0638714u
60
+ ··· + 1.79809u 1.19452
a
8
=
0.959644u
61
1.67641u
60
+ ··· + 5.31217u + 1.09170
0.173212u
61
0.489459u
60
+ ··· + 3.58377u 0.361261
a
5
=
0.358021u
61
0.474970u
60
+ ··· + 1.00388u 1.17136
1.24252u
61
+ 2.05157u
60
+ ··· 0.491103u 0.185903
a
11
=
0.357662u
61
0.0949908u
60
+ ··· + 3.57658u + 2.71513
0.804722u
61
0.507984u
60
+ ··· 1.17519u + 1.21328
a
11
=
0.357662u
61
0.0949908u
60
+ ··· + 3.57658u + 2.71513
0.804722u
61
0.507984u
60
+ ··· 1.17519u + 1.21328
(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 = 1.226348 0.108993I
a = 0.0661823 + 0.0471175I
b = 0.929307 0.272262I
2.26101 1.03760I 3.34506 + 2.53811I
u = 1.226348 + 0.108993I
a = 0.0661823 0.0471175I
b = 0.929307 + 0.272262I
2.26101 + 1.03760I 3.34506 2.53811I
u = 1.113744 0.125687I
a = 0.089861 + 0.137808I
b = 1.010022 0.625330I
0.48145 10.21060I 1.39697 + 7.43753I
u = 1.113744 + 0.125687I
a = 0.089861 0.137808I
b = 1.010022 + 0.625330I
0.48145 + 10.21060I 1.39697 7.43753I
u = 0.658314 0.044052I
a = 0.542092 0.296560I
b = 0.561727 0.990219I
4.39333 + 5.67407I 6.68227 4.64377I
u = 0.658314 + 0.044052I
a = 0.542092 + 0.296560I
b = 0.561727 + 0.990219I
4.39333 5.67407I 6.68227 + 4.64377I
u = 0.65526 1.45235I
a = 0.470916 0.489439I
b = 0.837326 0.219954I
3.38499 + 3.70672I 5.40950 3.23486I
u = 0.65526 + 1.45235I
a = 0.470916 + 0.489439I
b = 0.837326 + 0.219954I
3.38499 3.70672I 5.40950 + 3.23486I
u = 0.550805 0.317440I
a = 0.250174 0.088130I
b = 0.153184 0.532409I
1.008560 0.666322I 7.37810 + 3.95165I
u = 0.550805 + 0.317440I
a = 0.250174 + 0.088130I
b = 0.153184 + 0.532409I
1.008560 + 0.666322I 7.37810 3.95165I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.54540 1.46800I
a = 0.926837 0.284383I
b = 1.115702 0.565795I
6.81266 5.56177I 4.66163 + 5.86539I
u = 0.54540 + 1.46800I
a = 0.926837 + 0.284383I
b = 1.115702 + 0.565795I
6.81266 + 5.56177I 4.66163 5.86539I
u = 0.50790 1.42710I
a = 1.343186 0.183181I
b = 1.41397 0.80252I
7.18631 7.01551I 2.90347 + 3.14427I
u = 0.50790 + 1.42710I
a = 1.343186 + 0.183181I
b = 1.41397 + 0.80252I
7.18631 + 7.01551I 2.90347 3.14427I
u = 0.49535 1.40480I
a = 1.59941 0.06233I
b = 1.57693 0.99221I
4.3271 15.8582I 0.74909 + 8.45650I
u = 0.49535 + 1.40480I
a = 1.59941 + 0.06233I
b = 1.57693 + 0.99221I
4.3271 + 15.8582I 0.74909 8.45650I
u = 0.466688 0.056845I
a = 0.016943 + 0.880133I
b = 0.936524 + 0.341014I
2.34840 + 0.48442I 4.81285 + 0.96835I
u = 0.466688 + 0.056845I
a = 0.016943 0.880133I
b = 0.936524 0.341014I
2.34840 0.48442I 4.81285 0.96835I
u = 0.320326 1.224200I
a = 1.91360 0.25677I
b = 0.900571 + 0.644914I
0.80126 9.32026I 0.56370 + 8.17638I
u = 0.320326 + 1.224200I
a = 1.91360 + 0.25677I
b = 0.900571 0.644914I
0.80126 + 9.32026I 0.56370 8.17638I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.277759 1.120725I
a = 1.47905 + 0.20875I
b = 0.914383 + 0.159828I
1.45921 2.58594I 1.55168 + 3.98149I
u = 0.277759 + 1.120725I
a = 1.47905 0.20875I
b = 0.914383 0.159828I
1.45921 + 2.58594I 1.55168 3.98149I
u = 0.266596 0.292059I
a = 0.41328 + 3.31809I
b = 1.058607 0.761511I
3.63336 1.04794I 2.34672 + 0.35139I
u = 0.266596 + 0.292059I
a = 0.41328 3.31809I
b = 1.058607 + 0.761511I
3.63336 + 1.04794I 2.34672 0.35139I
u = 0.245626 1.196569I
a = 1.86241 + 0.22802I
b = 1.109762 + 0.416696I
1.22471 2.22328I 1.72830 + 3.36065I
u = 0.245626 + 1.196569I
a = 1.86241 0.22802I
b = 1.109762 0.416696I
1.22471 + 2.22328I 1.72830 3.36065I
u = 0.168925 1.058642I
a = 0.71321 + 1.50228I
b = 0.83421 1.30269I
1.00664 + 2.96825I 0.76468 1.87544I
u = 0.168925 + 1.058642I
a = 0.71321 1.50228I
b = 0.83421 + 1.30269I
1.00664 2.96825I 0.76468 + 1.87544I
u = 0.118148 1.022893I
a = 3.66595 + 1.82260I
b = 3.75706 0.98838I
0.03983 2.79266I 11.94643 + 2.94472I
u = 0.118148 + 1.022893I
a = 3.66595 1.82260I
b = 3.75706 + 0.98838I
0.03983 + 2.79266I 11.94643 2.94472I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.090353 1.257009I
a = 1.23556 + 0.99864I
b = 0.822239 + 0.355328I
8.09797 2.26515I 10.09980 + 3.27421I
u = 0.090353 + 1.257009I
a = 1.23556 0.99864I
b = 0.822239 0.355328I
8.09797 + 2.26515I 10.09980 3.27421I
u = 0.055369 1.146576I
a = 1.186711 0.506164I
b = 1.06745 + 1.54986I
4.30759 + 0.97665I 0.516718 1.197110I
u = 0.055369 + 1.146576I
a = 1.186711 + 0.506164I
b = 1.06745 1.54986I
4.30759 0.97665I 0.516718 + 1.197110I
u = 0.061581 1.260213I
a = 0.880801 + 1.072764I
b = 0.592279 + 0.347209I
4.49072 1.35832I 5.77140 + 0.97290I
u = 0.061581 + 1.260213I
a = 0.880801 1.072764I
b = 0.592279 0.347209I
4.49072 + 1.35832I 5.77140 0.97290I
u = 0.113786 1.261433I
a = 1.50072 + 0.99812I
b = 0.973051 + 0.319989I
3.91657 + 5.96640I 4.70782 6.24297I
u = 0.113786 + 1.261433I
a = 1.50072 0.99812I
b = 0.973051 0.319989I
3.91657 5.96640I 4.70782 + 6.24297I
u = 0.185559 1.050580I
a = 1.34463 + 1.23126I
b = 1.32068 0.83350I
3.55015 + 0.20409I 5.98628 + 1.19975I
u = 0.185559 + 1.050580I
a = 1.34463 1.23126I
b = 1.32068 + 0.83350I
3.55015 0.20409I 5.98628 1.19975I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.188577 0.364096I
a = 0.20443 + 3.96154I
b = 0.84113 1.17897I
0.11388 2.17095I 4.83160 + 3.47030I
u = 0.188577 + 0.364096I
a = 0.20443 3.96154I
b = 0.84113 + 1.17897I
0.11388 + 2.17095I 4.83160 3.47030I
u = 0.207449
a = 3.77069
b = 0.573655
1.30954 9.93956
u = 0.299504 1.211219I
a = 1.88201 0.08991I
b = 0.957244 + 0.548690I
3.96361 + 5.62230I 4.59110 7.16807I
u = 0.299504 + 1.211219I
a = 1.88201 + 0.08991I
b = 0.957244 0.548690I
3.96361 5.62230I 4.59110 + 7.16807I
u = 0.341543 0.277716I
a = 0.11973 + 2.81257I
b = 1.31181 0.54651I
0.57816 + 4.41700I 2.90527 4.08329I
u = 0.341543 + 0.277716I
a = 0.11973 2.81257I
b = 1.31181 + 0.54651I
0.57816 4.41700I 2.90527 + 4.08329I
u = 0.394127 1.133119I
a = 1.214880 0.299863I
b = 0.470685 + 0.349300I
4.26849 + 2.33228I 5.41476 2.55374I
u = 0.394127 + 1.133119I
a = 1.214880 + 0.299863I
b = 0.470685 0.349300I
4.26849 2.33228I 5.41476 + 2.55374I
u = 0.46376 1.46899I
a = 1.064702 + 0.158675I
b = 1.06320 0.97967I
1.16526 + 6.77623I 1.26226 6.46265I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.46376 + 1.46899I
a = 1.064702 0.158675I
b = 1.06320 + 0.97967I
1.16526 6.77623I 1.26226 + 6.46265I
u = 0.50118 1.41253I
a = 1.50871 0.13372I
b = 1.52945 0.90219I
9.4690 + 11.6120I 5.02297 7.15875I
u = 0.50118 + 1.41253I
a = 1.50871 + 0.13372I
b = 1.52945 + 0.90219I
9.4690 11.6120I 5.02297 + 7.15875I
u = 0.589382 0.058868I
a = 0.388375 0.370506I
b = 0.586967 0.761456I
0.49280 2.22863I 1.88873 + 4.27642I
u = 0.589382 + 0.058868I
a = 0.388375 + 0.370506I
b = 0.586967 + 0.761456I
0.49280 + 2.22863I 1.88873 4.27642I
u = 0.59848 1.46855I
a = 0.690335 0.423759I
b = 0.981364 0.361154I
8.77447 + 0.73219I 8.26642 0.24787I
u = 0.59848 + 1.46855I
a = 0.690335 + 0.423759I
b = 0.981364 + 0.361154I
8.77447 0.73219I 8.26642 + 0.24787I
u = 0.801731 0.247306I
a = 0.346425 0.007335I
b = 0.099675 0.837866I
7.01139 + 2.01626I 9.35384 3.40733I
u = 0.801731 + 0.247306I
a = 0.346425 + 0.007335I
b = 0.099675 + 0.837866I
7.01139 2.01626I 9.35384 + 3.40733I
u = 1.152512 0.122537I
a = 0.0728364 + 0.0995513I
b = 1.001116 0.478447I
4.62237 + 5.84398I 3.48358 6.22366I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.152512 + 0.122537I
a = 0.0728364 0.0995513I
b = 1.001116 + 0.478447I
4.62237 5.84398I 3.48358 + 6.22366I
u = 1.71345
a = 0.0745109
b = 0.536613
7.12923 37.6282
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(5u
2
1)(10u
62
+ 14u
61
+ ··· + 200076u 56034)
c
2
(u
2
+ u 1)(u
62
+ 2u
61
+ ··· u + 1)
c
3
(u
2
+ 3u + 1)(u
62
+ 6u
61
+ ··· 795u 117)
c
4
(u
2
u 1)(u
62
+ 2u
61
+ ··· 3u + 1)
c
5
(5u
2
+ 5u + 1)(5u
62
+ 14u
61
+ ··· 61755u + 8377)
c
6
(u
2
u 1)(u
62
+ 2u
61
+ ··· u + 1)
c
7
(u + 1)
2
(u
62
+ 3u
61
+ ··· + 214u 25)
c
8
u
2
(u
62
+ u
61
+ ··· + 280u 100)
c
9
(u 1)
2
(u
62
+ 3u
61
+ ··· + 214u 25)
c
10
(u
2
+ u 1)(u
62
+ 2u
61
+ ··· 3u + 1)
c
11
(u
2
+ u 1)(u
62
+ 2u
61
+ ··· 3u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(5y 1)
2
(900y
62
47124y
61
+ ··· 556472892600y + 28258282400)
c
2
, c
6
(y
2
3y + 1)(y
62
+ 42y
61
+ ··· 25y + 1)
c
3
(y
2
7y + 1)(y
62
+ 18y
61
+ ··· 440613y + 13689)
c
4
(y
2
3y + 1)(y
62
54y
61
+ ··· 25y + 1)
c
5
(25y
2
15y + 1)
(1050y
62
12012y
61
+ ··· 54282378400y + 2947313418)
c
7
, c
9
(y 1)
2
(y
62
49y
61
+ ··· 22596y + 625)
c
8
y
2
(y
62
15y
61
+ ··· 413000y + 10000)
c
10
(y
2
3y + 1)(y
62
54y
61
+ ··· 25y + 1)
c
11
(y
2
3y + 1)(y
62
54y
61
+ ··· 25y + 1)
12