11a
22
(K11a
22
)
1
Arc Sequences
4 1 7 2 10 3 6 11 5 8 9
Solving Sequence
5,10 2,6
4 1 3 7 9 11 8
c
5
c
4
c
1
c
2
c
6
c
9
c
11
c
8
c
3
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
2
5, u 1, b + 2a 1i
I
u
2
= hu
6
+ u
5
u
4
2u
3
+ u + 1, a, u
4
+ u
3
+ b ui
I
u
3
= hu
58
+ 4u
57
+ ··· + 14u + 1,
1.39321 × 10
15
u
57
9.52584 × 10
14
u
56
+ ··· + 4.71171 × 10
15
b + 1.69695 × 10
15
,
2.84174 × 10
16
u
57
9.16487 × 10
16
u
56
+ ··· + 9.42343 × 10
15
a 9.46213 × 10
16
i
There are 3 irreducible components with 66 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
2
5, u 1, b + 2a 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
1
2
b +
1
2
b
a
2
=
1
0
a
6
=
1
2
b
3
2
1
2
b
3
2
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
7
=
1
2
b
3
2
1
2
b
3
2
a
9
=
1
2
b +
1
2
1
2
b
1
2
a
11
=
1
2
b
3
2
2
a
8
=
1
2
b
3
2
1
2
b
3
2
a
8
=
1
2
b
3
2
1
2
b
3
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 21
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.618034
b = 2.23607
2.63189 21.0000
u = 1.00000
a = 1.61803
b = 2.23607
10.5276 21.0000
3
II. I
u
2
= hu
6
+ u
5
u
4
2u
3
+ u + 1, a, u
4
+ u
3
+ b ui
(i) Arc colorings
a
5
=
0
u
a
10
=
0
u
4
u
3
+ u
a
2
=
1
0
a
6
=
0
u
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
7
=
u
2
1
u
4
a
9
=
0
u
4
u
3
+ u
a
11
=
u
2
+ 1
u
4
u
3
u
2
+ u
a
8
=
u
2
1
u
2
a
8
=
u
2
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
4
+ 2u
3
5u
2
6u 7
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.073950 0.558752I
a = 0
b = 0.108378 0.818891I
1.64493 5.69302I 8.89162 + 3.92918I
u = 1.073950 + 0.558752I
a = 0
b = 0.108378 + 0.818891I
1.64493 + 5.69302I 8.89162 3.92918I
u = 0.428243 0.664531I
a = 0
b = 0.659772 0.298454I
0.245672 + 0.924305I 3.44826 0.47256I
u = 0.428243 + 0.664531I
a = 0
b = 0.659772 + 0.298454I
0.245672 0.924305I 3.44826 + 0.47256I
u = 1.002193 0.295542I
a = 0
b = 0.23185 + 1.65564I
3.53554 + 0.92430I 13.66012 2.42665I
u = 1.002193 + 0.295542I
a = 0
b = 0.23185 1.65564I
3.53554 0.92430I 13.66012 + 2.42665I
5
III. I
u
3
=
hu
58
+4u
57
+· · ·+14u+1, 1.39×10
15
u
57
9.53×10
14
u
56
+· · ·+4.71×10
15
b+
1.70× 10
15
, 2.84× 10
16
u
57
9.16 ×10
16
u
56
+· · · +9.42 ×10
15
a 9.46× 10
16
i
(i) Arc colorings
a
5
=
0
u
a
10
=
3.01561u
57
+ 9.72562u
56
+ ··· + 57.2992u + 10.0411
0.295690u
57
+ 0.202174u
56
+ ··· 13.3752u 0.360156
a
2
=
1
0
a
6
=
3.31495u
57
+ 10.2517u
56
+ ··· + 41.8810u + 6.27939
1.40308u
57
+ 4.74267u
56
+ ··· + 14.1209u + 1.31428
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
7
=
3.45270u
57
+ 10.0429u
56
+ ··· + 36.6766u + 6.19087
4.99480u
57
+ 16.8962u
56
+ ··· + 71.0985u + 5.67974
a
9
=
3.01561u
57
+ 9.72562u
56
+ ··· + 57.2992u + 10.0411
1.38901u
57
5.47739u
56
+ ··· 43.0751u 2.69698
a
11
=
0.984390u
57
3.27438u
56
+ ··· 22.7008u 4.95894
3.34793u
57
+ 12.0585u
56
+ ··· + 75.4648u + 5.37039
a
8
=
0.984390u
57
+ 3.27438u
56
+ ··· + 22.7008u + 4.95894
2.67819u
57
+ 9.36965u
56
+ ··· + 47.9552u + 3.99489
a
8
=
0.984390u
57
+ 3.27438u
56
+ ··· + 22.7008u + 4.95894
2.67819u
57
+ 9.36965u
56
+ ··· + 47.9552u + 3.99489
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
1606550278673315
4711714481599631
u
57
+
5531154945267638
4711714481599631
u
56
+ ··· +
44269337858367694
4711714481599631
u
40813840963650864
4711714481599631
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.232896 0.551373I
a = 1.191314 + 0.023058I
b = 3.17323 0.25328I
11.2723 14.6533I 10.80965 + 8.32591I
u = 1.232896 + 0.551373I
a = 1.191314 0.023058I
b = 3.17323 + 0.25328I
11.2723 + 14.6533I 10.80965 8.32591I
u = 1.197641 0.519290I
a = 1.036157 + 0.135628I
b = 3.17838 + 0.13192I
4.65387 10.14069I 8.73196 + 8.19985I
u = 1.197641 + 0.519290I
a = 1.036157 0.135628I
b = 3.17838 0.13192I
4.65387 + 10.14069I 8.73196 8.19985I
u = 1.194329 0.375553I
a = 1.135619 + 0.718552I
b = 2.56192 + 0.29434I
13.34932 0.72037I 12.98227 + 3.28415I
u = 1.194329 + 0.375553I
a = 1.135619 0.718552I
b = 2.56192 0.29434I
13.34932 + 0.72037I 12.98227 3.28415I
u = 1.194314 0.490939I
a = 0.267452 1.038563I
b = 0.538797 1.260271I
7.20694 7.37757I 10.28947 + 5.67068I
u = 1.194314 + 0.490939I
a = 0.267452 + 1.038563I
b = 0.538797 + 1.260271I
7.20694 + 7.37757I 10.28947 5.67068I
u = 1.173328 0.465541I
a = 0.969758 0.396886I
b = 2.92285 + 0.04001I
5.37580 4.41103I 10.20284 + 3.41827I
u = 1.173328 + 0.465541I
a = 0.969758 + 0.396886I
b = 2.92285 0.04001I
5.37580 + 4.41103I 10.20284 3.41827I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.102792 0.531562I
a = 0.177602 + 0.507340I
b = 0.334880 + 0.621835I
0.69171 6.13139I 1.89157 + 6.52197I
u = 1.102792 + 0.531562I
a = 0.177602 0.507340I
b = 0.334880 0.621835I
0.69171 + 6.13139I 1.89157 6.52197I
u = 0.989582 0.681314I
a = 0.913811 0.272536I
b = 1.145487 0.652819I
2.57248 7.45358I 8.30714 + 7.17302I
u = 0.989582 + 0.681314I
a = 0.913811 + 0.272536I
b = 1.145487 + 0.652819I
2.57248 + 7.45358I 8.30714 7.17302I
u = 0.876500 0.566132I
a = 0.956835 0.051894I
b = 1.42795 + 0.12456I
2.08472 4.56768I 1.81782 + 7.20311I
u = 0.876500 + 0.566132I
a = 0.956835 + 0.051894I
b = 1.42795 0.12456I
2.08472 + 4.56768I 1.81782 7.20311I
u = 0.870407
a = 1.94634
b = 2.21434
10.2057 4.64730
u = 0.844388 0.455991I
a = 0.157109 1.054390I
b = 0.439546 + 0.311722I
1.05945 3.12017I 5.50127 + 4.47326I
u = 0.844388 + 0.455991I
a = 0.157109 + 1.054390I
b = 0.439546 0.311722I
1.05945 + 3.12017I 5.50127 4.47326I
u = 0.711409 0.407289I
a = 1.175998 + 0.016113I
b = 1.75724 + 0.23127I
0.629418 0.623631I 5.36343 + 3.54196I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.711409 + 0.407289I
a = 1.175998 0.016113I
b = 1.75724 0.23127I
0.629418 + 0.623631I 5.36343 3.54196I
u = 0.638300 0.592248I
a = 0.012372 + 0.960639I
b = 0.0336808 + 0.0333848I
2.76529 0.01343I 1.057430 0.032902I
u = 0.638300 + 0.592248I
a = 0.012372 0.960639I
b = 0.0336808 0.0333848I
2.76529 + 0.01343I 1.057430 + 0.032902I
u = 0.573182 0.818987I
a = 0.454202 0.976673I
b = 0.060919 0.315107I
1.33556 + 1.90763I 7.20457 1.48336I
u = 0.573182 + 0.818987I
a = 0.454202 + 0.976673I
b = 0.060919 + 0.315107I
1.33556 1.90763I 7.20457 + 1.48336I
u = 0.318010 0.669454I
a = 0.835222 0.119144I
b = 0.204987 0.054879I
1.56893 + 1.49125I 1.52605 1.85258I
u = 0.318010 + 0.669454I
a = 0.835222 + 0.119144I
b = 0.204987 + 0.054879I
1.56893 1.49125I 1.52605 + 1.85258I
u = 0.163994 0.921606I
a = 0.38814 1.60154I
b = 0.359654 + 0.512136I
8.03060 + 9.32855I 7.95440 5.34516I
u = 0.163994 + 0.921606I
a = 0.38814 + 1.60154I
b = 0.359654 0.512136I
8.03060 9.32855I 7.95440 + 5.34516I
u = 0.149661 0.820870I
a = 0.15413 + 1.48993I
b = 0.077737 0.785361I
1.54879 + 5.22529I 5.49197 5.46614I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.149661 + 0.820870I
a = 0.15413 1.48993I
b = 0.077737 + 0.785361I
1.54879 5.22529I 5.49197 + 5.46614I
u = 0.106334
a = 5.36182
b = 0.762327
1.11333 8.96695
u = 0.091411 0.780425I
a = 1.57026 0.01002I
b = 0.545497 + 0.173834I
3.98695 + 2.71614I 7.03008 2.60179I
u = 0.091411 + 0.780425I
a = 1.57026 + 0.01002I
b = 0.545497 0.173834I
3.98695 2.71614I 7.03008 + 2.60179I
u = 0.037554 0.696936I
a = 0.17786 1.62233I
b = 0.438391 + 0.736708I
2.18920 + 0.10128I 7.14023 + 0.24730I
u = 0.037554 + 0.696936I
a = 0.17786 + 1.62233I
b = 0.438391 0.736708I
2.18920 0.10128I 7.14023 0.24730I
u = 0.160069 0.784365I
a = 0.18638 + 1.89429I
b = 0.638001 0.338647I
9.34785 3.11893I 9.46209 + 0.65327I
u = 0.160069 + 0.784365I
a = 0.18638 1.89429I
b = 0.638001 + 0.338647I
9.34785 + 3.11893I 9.46209 0.65327I
u = 0.755123 0.252808I
a = 0.806647 + 0.086757I
b = 0.293637 1.085070I
1.14303 + 1.20148I 5.78859 5.55533I
u = 0.755123 + 0.252808I
a = 0.806647 0.086757I
b = 0.293637 + 1.085070I
1.14303 1.20148I 5.78859 + 5.55533I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.784274 0.573350I
a = 1.083998 + 0.658643I
b = 0.208953 + 1.259873I
6.51259 + 2.28009I 10.96046 3.19134I
u = 0.784274 + 0.573350I
a = 1.083998 0.658643I
b = 0.208953 1.259873I
6.51259 2.28009I 10.96046 + 3.19134I
u = 0.926447 0.125757I
a = 0.199327 + 0.446703I
b = 0.61663 2.78731I
3.24157 + 0.49714I 10.0374 + 15.0443I
u = 0.926447 + 0.125757I
a = 0.199327 0.446703I
b = 0.61663 + 2.78731I
3.24157 0.49714I 10.0374 15.0443I
u = 1.061185 0.329222I
a = 0.072871 0.438850I
b = 0.593523 0.500009I
2.16889 + 1.07216I 4.64986 0.67610I
u = 1.061185 + 0.329222I
a = 0.072871 + 0.438850I
b = 0.593523 + 0.500009I
2.16889 1.07216I 4.64986 + 0.67610I
u = 1.09673
a = 0.500310
b = 1.88968
2.28699 3.94785
u = 1.173675 0.441088I
a = 0.976629 0.150330I
b = 3.14419 0.62683I
5.55187 + 4.00580I 10.40862 4.06457I
u = 1.173675 + 0.441088I
a = 0.976629 + 0.150330I
b = 3.14419 + 0.62683I
5.55187 4.00580I 10.40862 + 4.06457I
u = 1.185228 0.515535I
a = 1.209413 + 0.042403I
b = 3.07922 + 0.81594I
12.3578 + 7.9362I 12.05500 4.43772I
11
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.185228 + 0.515535I
a = 1.209413 0.042403I
b = 3.07922 0.81594I
12.3578 7.9362I 12.05500 + 4.43772I
u = 1.203611 0.410456I
a = 0.289876 + 0.945812I
b = 1.09131 + 1.07260I
7.78075 + 1.40297I 11.36161 0.62111I
u = 1.203611 + 0.410456I
a = 0.289876 0.945812I
b = 1.09131 1.07260I
7.78075 1.40297I 11.36161 + 0.62111I
u = 1.220351 0.368821I
a = 0.860274 + 0.408376I
b = 2.84061 + 0.31923I
5.71379 1.22939I 10.76366 + 2.88408I
u = 1.220351 + 0.368821I
a = 0.860274 0.408376I
b = 2.84061 0.31923I
5.71379 + 1.22939I 10.76366 2.88408I
u = 1.302724 0.354277I
a = 0.949730 0.662491I
b = 2.44022 0.55485I
12.71920 4.94560I 12.13992 + 3.22050I
u = 1.302724 + 0.354277I
a = 0.949730 + 0.662491I
b = 2.44022 + 0.55485I
12.71920 + 4.94560I 12.13992 3.22050I
u = 1.31323
a = 0.969547
b = 1.97267
8.12166 10.1036
12
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u
6
+ u
5
+ ··· + u + 1)(u
58
+ 4u
57
+ ··· + 14u + 1)
c
2
(u + 1)
2
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
(u
58
+ 32u
57
+ ··· + 94u + 1)
c
3
u
2
(u
6
u
5
+ ··· u + 1)(u
58
+ 2u
57
+ ··· 20u + 4)
c
4
(u + 1)
2
(u
6
u
5
+ ··· u + 1)(u
58
+ 4u
57
+ ··· + 14u + 1)
c
5
u
6
(u
2
+ u 1)(u
58
+ 2u
57
+ ··· + 128u + 64)
c
6
u
2
(u
6
+ u
5
+ ··· + u + 1)(u
58
+ 2u
57
+ ··· 20u + 4)
c
7
u
2
(u
6
3u
5
+ ··· u + 1)(u
58
+ 18u
57
+ ··· + 360u + 16)
c
8
(u 1)
6
(u
2
+ u 1)(u
58
+ 8u
57
+ ··· + 4u + 1)
c
9
u
6
(u
2
u 1)(u
58
+ 2u
57
+ ··· + 128u + 64)
c
10
, c
11
(u + 1)
6
(u
2
u 1)(u
58
+ 8u
57
+ ··· + 4u + 1)
13
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
2
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
58
32y
57
+ ··· 94y + 1)
c
2
(y 1)
2
(y
6
+ y
5
+ ··· + 3y + 1)(y
58
8y
57
+ ··· 7838y + 1)
c
3
, c
6
y
2
(y
6
3y
5
+ ··· y + 1)(y
58
18y
57
+ ··· 360y + 16)
c
5
, c
9
y
6
(y
2
3y + 1)(y
58
42y
57
+ ··· 8192y + 4096)
c
7
y
2
(y
6
+ y
5
+ ··· + 3y + 1)(y
58
+ 42y
57
+ ··· 33056y + 256)
c
8
, c
10
, c
11
(y 1)
6
(y
2
3y + 1)(y
58
60y
57
+ ··· 36y + 1)
14