11a
3
(K11a
3
)
1
Arc Sequences
5 1 7 2 3 10 4 11 6 8 9
Solving Sequence
1,5
2 3 6
4,9
11 8 7 10
c
1
c
2
c
5
c
4
c
11
c
8
c
7
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
5
u
4
+ 2u
3
u
2
+ u 1, b 1, u
4
u
3
+ 2u
2
+ a u + 1i
I
u
2
= ha
4
5a
3
+ 11a
2
10a + 4, a
3
+ 3a
2
+ 2b 3a, a
3
5a
2
+ 9a + 2u 4i
I
u
3
= hu
66
+ 4u
65
+ ··· + 14u + 1,
2.00992 × 10
19
u
65
1.27810 × 10
20
u
64
+ ··· + 6.92121 × 10
19
a + 5.30233 × 10
20
,
4.73407 × 10
19
u
65
1.78062 × 10
20
u
64
+ ··· + 6.92121 × 10
19
b 9.60242 × 10
19
i
There are 3 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
5
u
4
+ 2u
3
u
2
+ u 1, b 1, u
4
u
3
+ 2u
2
+ a u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
4
u
2
1
u
4
u
3
+ u
2
+ 1
a
4
=
u
u
3
+ u
a
9
=
u
4
+ u
3
2u
2
+ u 1
1
a
11
=
u
4
+ u
3
2u
2
+ u
1
a
8
=
1
0
a
7
=
u
4
u
2
1
u
4
u
3
+ u
2
+ 1
a
10
=
u
4
+ u
3
2u
2
+ u 1
1
a
10
=
u
4
+ u
3
2u
2
+ u 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
4
+ 5u
3
4u
2
+ 3
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.339110 0.822375I
a = 0.428550 1.039275I
b = 1.00000
1.31583 + 1.53058I 8.47842 + 1.00973I
u = 0.339110 + 0.822375I
a = 0.428550 + 1.039275I
b = 1.00000
1.31583 1.53058I 8.47842 1.00973I
u = 0.455697 1.200152I
a = 0.276511 0.728237I
b = 1.00000
4.22763 4.40083I 2.41100 + 1.19010I
u = 0.455697 + 1.200152I
a = 0.276511 + 0.728237I
b = 1.00000
4.22763 + 4.40083I 2.41100 1.19010I
u = 0.766826
a = 1.30408
b = 1.00000
0.756147 1.86516
3
II.
I
u
2
= ha
4
5a
3
+ 11a
2
10a + 4, a
3
+ 3a
2
+ 2b 3a, a
3
5a
2
+ 9a + 2u 4i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
2
a
3
+
5
2
a
2
9
2
a + 2
a
2
=
1
1
2
a
3
+
5
2
a
2
9
2
a + 3
a
3
=
1
2
a
3
5
2
a
2
+
9
2
a 2
1
2
a
3
+
5
2
a
2
9
2
a + 3
a
6
=
1
0
a
4
=
1
2
a
3
5
2
a
2
+
9
2
a 2
1
2
a
3
+
5
2
a
2
9
2
a + 3
a
9
=
a
1
2
a
3
3
2
a
2
+
3
2
a
a
11
=
a
3
4a
2
+ 5a 1
1
2
a
3
+
3
2
a
2
3
2
a + 1
a
8
=
1
2
a
3
5
2
a
2
+
7
2
a 2
1
2
a
3
+
3
2
a
2
3
2
a + 1
a
7
=
1
2
a
3
5
2
a
2
+
7
2
a 2
1
2
a
3
+
3
2
a
2
3
2
a + 1
a
10
=
1
2
a
3
3
2
a
2
+
5
2
a
1
2
a
3
3
2
a
2
+
3
2
a
a
10
=
1
2
a
3
3
2
a
2
+
5
2
a
1
2
a
3
3
2
a
2
+
3
2
a
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5a
3
+ 22a
2
42a + 28
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.690983 0.535233I
b = 0.618034
0.98696 2.02988I 4.50000 + 9.27358I
u = 0.500000 0.866025I
a = 0.690983 + 0.535233I
b = 0.618034
0.98696 + 2.02988I 4.50000 9.27358I
u = 0.500000 0.866025I
a = 1.80902 1.40126I
b = 1.61803
8.88264 + 2.02988I 4.50000 + 2.34537I
u = 0.500000 + 0.866025I
a = 1.80902 + 1.40126I
b = 1.61803
8.88264 2.02988I 4.50000 2.34537I
5
III. I
u
3
=
hu
66
+4u
65
+· · ·+14u+1, 2.01×10
19
u
65
1.28×10
20
u
64
+· · ·+6.92×10
19
a+
5.30 × 10
20
, 4.73 × 10
19
u
65
1.78 × 10
20
u
64
+ · · · + 6.92× 10
19
b 9.60 × 10
19
i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
5
2u
3
u
u
5
+ u
3
+ u
a
4
=
u
u
3
+ u
a
9
=
0.290400u
65
+ 1.84665u
64
+ ··· 17.1503u 7.66099
0.683994u
65
+ 2.57271u
64
+ ··· + 8.88365u + 1.38739
a
11
=
0.477604u
65
0.313880u
64
+ ··· 9.01518u 4.46864
1.76402u
65
+ 7.20916u
64
+ ··· + 25.9654u + 2.45043
a
8
=
0.352013u
65
2.74835u
64
+ ··· 30.8461u 5.72790
1.33994u
65
5.22709u
64
+ ··· 16.3365u 0.873918
a
7
=
0.996758u
65
2.80838u
64
+ ··· 21.4371u 5.07963
1.17865u
65
+ 4.19945u
64
+ ··· + 8.87499u + 0.996758
a
10
=
0.773579u
65
2.81719u
64
+ ··· 35.3229u 9.01284
0.443910u
65
+ 1.66335u
64
+ ··· + 5.63834u + 1.19827
a
10
=
0.773579u
65
2.81719u
64
+ ··· 35.3229u 9.01284
0.443910u
65
+ 1.66335u
64
+ ··· + 5.63834u + 1.19827
(ii) Obstruction class = 1
(iii) Cusp Shapes =
88536532098082237941
69212056571253139646
u
65
346942669555547012397
69212056571253139646
u
64
+ ··· +
2682685309410699867097
69212056571253139646
u +
417517236906079889612
34606028285626569823
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.895716
a = 1.28200
b = 1.26436
0.335750 9.51519
u = 0.860042 0.369782I
a = 2.14032 + 0.32890I
b = 1.47569 + 0.33459I
6.19769 10.10887I 7.50011 + 5.44756I
u = 0.860042 + 0.369782I
a = 2.14032 0.32890I
b = 1.47569 0.33459I
6.19769 + 10.10887I 7.50011 5.44756I
u = 0.779756 0.337981I
a = 0.457305 + 0.348225I
b = 0.360992 0.860365I
0.29837 5.77580I 4.51138 + 5.26923I
u = 0.779756 + 0.337981I
a = 0.457305 0.348225I
b = 0.360992 + 0.860365I
0.29837 + 5.77580I 4.51138 5.26923I
u = 0.721013 0.363719I
a = 2.95924 0.12833I
b = 1.41518 0.16379I
3.08928 3.39261I 6.58379 + 2.75306I
u = 0.721013 + 0.363719I
a = 2.95924 + 0.12833I
b = 1.41518 + 0.16379I
3.08928 + 3.39261I 6.58379 2.75306I
u = 0.712768 0.146826I
a = 0.827880 0.176126I
b = 0.256675 + 0.450186I
2.30399 1.13049I 0.90491 + 1.23607I
u = 0.712768 + 0.146826I
a = 0.827880 + 0.176126I
b = 0.256675 0.450186I
2.30399 + 1.13049I 0.90491 1.23607I
u = 0.626648 0.353158I
a = 1.406461 + 0.110890I
b = 0.808652 + 0.620740I
1.72098 0.65765I 7.06140 + 0.81107I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.626648 + 0.353158I
a = 1.406461 0.110890I
b = 0.808652 0.620740I
1.72098 + 0.65765I 7.06140 0.81107I
u = 0.611501 1.147435I
a = 1.49873 1.96189I
b = 1.48463 0.36648I
3.8588 + 15.5500I 4.71947 9.16271I
u = 0.611501 + 1.147435I
a = 1.49873 + 1.96189I
b = 1.48463 + 0.36648I
3.8588 15.5500I 4.71947 + 9.16271I
u = 0.583954 0.630467I
a = 2.47273 1.26841I
b = 1.58923 0.11013I
9.82684 + 1.74748I 11.13070 + 1.76182I
u = 0.583954 + 0.630467I
a = 2.47273 + 1.26841I
b = 1.58923 + 0.11013I
9.82684 1.74748I 11.13070 1.76182I
u = 0.575045 1.130155I
a = 1.128293 + 0.629597I
b = 0.356915 + 0.930272I
2.03842 + 10.86452I 1.54019 8.71423I
u = 0.575045 + 1.130155I
a = 1.128293 0.629597I
b = 0.356915 0.930272I
2.03842 10.86452I 1.54019 + 8.71423I
u = 0.563876 1.106243I
a = 1.90945 + 1.84534I
b = 1.46658 + 0.20344I
0.91936 + 8.30279I 3.32445 6.69356I
u = 0.563876 + 1.106243I
a = 1.90945 1.84534I
b = 1.46658 0.20344I
0.91936 8.30279I 3.32445 + 6.69356I
u = 0.536106 1.092065I
a = 0.109685 + 0.492756I
b = 0.901979 0.712422I
0.40126 + 5.25319I 3.02303 4.98117I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.536106 + 1.092065I
a = 0.109685 0.492756I
b = 0.901979 + 0.712422I
0.40126 5.25319I 3.02303 + 4.98117I
u = 0.511878 0.976459I
a = 1.64988 1.32112I
b = 1.66385 + 0.06802I
8.80229 + 2.61597I 2.87447 9.70920I
u = 0.511878 + 0.976459I
a = 1.64988 + 1.32112I
b = 1.66385 0.06802I
8.80229 2.61597I 2.87447 + 9.70920I
u = 0.492275 1.150201I
a = 0.949530 0.599065I
b = 0.409340 0.495814I
5.17298 + 5.63394I 3.84541 5.50303I
u = 0.492275 + 1.150201I
a = 0.949530 + 0.599065I
b = 0.409340 + 0.495814I
5.17298 5.63394I 3.84541 + 5.50303I
u = 0.444892 1.249164I
a = 0.162825 0.849550I
b = 1.218659 0.058791I
3.53690 + 4.73542I 7.02954 5.28929I
u = 0.444892 + 1.249164I
a = 0.162825 + 0.849550I
b = 1.218659 + 0.058791I
3.53690 4.73542I 7.02954 + 5.28929I
u = 0.363610 1.151705I
a = 0.015002 + 0.368525I
b = 0.207558 + 0.642919I
6.04161 + 2.45340I 4.67162 2.64189I
u = 0.363610 + 1.151705I
a = 0.015002 0.368525I
b = 0.207558 0.642919I
6.04161 2.45340I 4.67162 + 2.64189I
u = 0.323236 1.068023I
a = 0.71514 + 1.43717I
b = 1.056093 + 0.536731I
1.87944 + 1.86021I 0.04861 1.68702I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.323236 + 1.068023I
a = 0.71514 1.43717I
b = 1.056093 0.536731I
1.87944 1.86021I 0.04861 + 1.68702I
u = 0.246513 1.075850I
a = 0.702183 0.085805I
b = 1.331250 0.286376I
1.20633 0.98148I 0.151618 + 0.428423I
u = 0.246513 + 1.075850I
a = 0.702183 + 0.085805I
b = 1.331250 + 0.286376I
1.20633 + 0.98148I 0.151618 0.428423I
u = 0.221525 1.144870I
a = 0.415305 0.704627I
b = 0.220112 0.840668I
4.39164 2.99363I 2.26986 + 3.15952I
u = 0.221525 + 1.144870I
a = 0.415305 + 0.704627I
b = 0.220112 + 0.840668I
4.39164 + 2.99363I 2.26986 3.15952I
u = 0.156166 1.218032I
a = 0.085968 + 0.372608I
b = 1.40723 + 0.32906I
0.79611 7.18319I 2.04293 + 4.55834I
u = 0.156166 + 1.218032I
a = 0.085968 0.372608I
b = 1.40723 0.32906I
0.79611 + 7.18319I 2.04293 4.55834I
u = 0.104581
a = 6.15002
b = 0.755340
1.11358 9.06931
u = 0.263556 1.157405I
a = 0.512295 0.085483I
b = 1.378617 0.146836I
3.75032 + 0.51941I 5.15368 0.25339I
u = 0.263556 + 1.157405I
a = 0.512295 + 0.085483I
b = 1.378617 + 0.146836I
3.75032 0.51941I 5.15368 + 0.25339I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.350315 0.758669I
a = 0.903762 0.185239I
b = 0.0960512 0.0497974I
0.23109 1.44442I 1.44757 + 4.95270I
u = 0.350315 + 0.758669I
a = 0.903762 + 0.185239I
b = 0.0960512 + 0.0497974I
0.23109 + 1.44442I 1.44757 4.95270I
u = 0.375168 1.011287I
a = 1.006085 + 0.919127I
b = 0.091003 + 0.416704I
1.05484 1.47223I 0.676926 + 0.494281I
u = 0.375168 + 1.011287I
a = 1.006085 0.919127I
b = 0.091003 0.416704I
1.05484 + 1.47223I 0.676926 0.494281I
u = 0.447737 0.886933I
a = 1.92832 + 1.33209I
b = 0.960667 + 0.075689I
1.35276 1.84672I 28.2718 + 21.5804I
u = 0.447737 + 0.886933I
a = 1.92832 1.33209I
b = 0.960667 0.075689I
1.35276 + 1.84672I 28.2718 21.5804I
u = 0.527326 1.065787I
a = 0.79696 1.22027I
b = 0.329323 0.680569I
0.11788 5.06683I 2.93249 + 6.29989I
u = 0.527326 + 1.065787I
a = 0.79696 + 1.22027I
b = 0.329323 + 0.680569I
0.11788 + 5.06683I 2.93249 6.29989I
u = 0.553417 1.013291I
a = 1.98755 2.54381I
b = 1.277913 0.085579I
2.55061 3.21838I 4.95585 + 1.62471I
u = 0.553417 + 1.013291I
a = 1.98755 + 2.54381I
b = 1.277913 + 0.085579I
2.55061 + 3.21838I 4.95585 1.62471I
11
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.563535 1.134559I
a = 0.88677 + 2.16349I
b = 1.43841 + 0.26048I
5.80147 8.50464I 6.13006 + 5.96733I
u = 0.563535 + 1.134559I
a = 0.88677 2.16349I
b = 1.43841 0.26048I
5.80147 + 8.50464I 6.13006 5.96733I
u = 0.578683 0.421784I
a = 0.654555 + 0.516153I
b = 0.475930 + 0.593283I
1.99483 + 0.60906I 7.48413 1.51323I
u = 0.578683 + 0.421784I
a = 0.654555 0.516153I
b = 0.475930 0.593283I
1.99483 0.60906I 7.48413 + 1.51323I
u = 0.593695 0.930839I
a = 0.633018 0.417893I
b = 0.444003 + 0.428029I
0.99829 1.41928I 2.88835 4.30612I
u = 0.593695 + 0.930839I
a = 0.633018 + 0.417893I
b = 0.444003 0.428029I
0.99829 + 1.41928I 2.88835 + 4.30612I
u = 0.633698 0.542548I
a = 3.20205 + 0.42108I
b = 1.320011 0.014248I
3.94155 1.44668I 7.42047 + 3.11484I
u = 0.633698 + 0.542548I
a = 3.20205 0.42108I
b = 1.320011 + 0.014248I
3.94155 + 1.44668I 7.42047 3.11484I
u = 0.668703 0.641624I
a = 0.453771 + 0.163173I
b = 0.430801 0.625032I
1.84919 3.47096I 5.53731 + 7.57944I
u = 0.668703 + 0.641624I
a = 0.453771 0.163173I
b = 0.430801 + 0.625032I
1.84919 + 3.47096I 5.53731 7.57944I
12
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.711073 0.974927I
a = 1.58757 + 1.06886I
b = 1.44305 0.18150I
6.98766 + 0.95393I 6.71287 0.96612I
u = 0.711073 + 0.974927I
a = 1.58757 1.06886I
b = 1.44305 + 0.18150I
6.98766 0.95393I 6.71287 + 0.96612I
u = 0.763032 0.307439I
a = 1.74149 0.72557I
b = 1.46443 0.20883I
8.22126 + 3.50783I 9.72944 1.69161I
u = 0.763032 + 0.307439I
a = 1.74149 + 0.72557I
b = 1.46443 + 0.20883I
8.22126 3.50783I 9.72944 + 1.69161I
u = 0.801015 0.661180I
a = 2.30437 + 0.57137I
b = 1.45947 + 0.22980I
7.91868 6.59447I 8.06911 + 6.00646I
u = 0.801015 + 0.661180I
a = 2.30437 0.57137I
b = 1.45947 0.22980I
7.91868 + 6.59447I 8.06911 6.00646I
13
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
5
u
4
+ ··· + u 1)(u
66
+ 4u
65
+ ··· + 14u + 1)
c
2
(u
2
+ u + 1)
2
(u
5
+ 3u
4
+ ··· u 1)(u
66
+ 32u
65
+ ··· 86u + 1)
c
3
u
4
(u
5
+ u
4
+ ··· + u 1)(u
66
+ 2u
65
+ ··· 16u 16)
c
4
(u
2
u + 1)
2
(u
5
+ u
4
+ ··· + u + 1)(u
66
+ 4u
65
+ ··· + 14u + 1)
c
5
(u
2
+ u + 1)
2
(u
5
u
4
2u
3
+ u
2
+ u + 1)
(u
66
+ 4u
65
+ ··· 4020u + 977)
c
6
u
5
(u
2
u 1)
2
(u
66
+ 3u
65
+ ··· + 96u + 32)
c
7
u
4
(u
5
u
4
+ ··· + u + 1)(u
66
+ 2u
65
+ ··· 16u 16)
c
8
(u + 1)
5
(u
2
u 1)
2
(u
66
+ 8u
65
+ ··· 12u 1)
c
9
u
5
(u
2
+ u 1)
2
(u
66
+ 3u
65
+ ··· + 96u + 32)
c
10
, c
11
(u 1)
5
(u
2
+ u 1)
2
(u
66
+ 8u
65
+ ··· 12u 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
2
(y
5
+ 3y
4
+ ··· y 1)(y
66
+ 32y
65
+ ··· 86y + 1)
c
2
(y
2
+ y + 1)
2
(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)
(y
66
+ 8y
65
+ ··· 8342y + 1)
c
3
, c
7
y
4
(y
5
5y
4
+ ··· y 1)(y
66
30y
65
+ ··· 2688y + 256)
c
5
(y
2
+ y + 1)
2
(y
5
5y
4
+ 8y
3
3y
2
y 1)
(y
66
16y
65
+ ··· 97788750y + 954529)
c
6
, c
9
y
5
(y
2
3y + 1)
2
(y
66
39y
65
+ ··· 7680y + 1024)
c
8
(y 1)
5
(y
2
3y + 1)
2
(y
66
64y
65
+ ··· 92y + 1)
c
10
(y 1)
5
(y
2
3y + 1)
2
(y
66
64y
65
+ ··· 92y + 1)
c
11
(y 1)
5
(y
2
3y + 1)
2
(y
66
64y
65
+ ··· 92y + 1)
15