11a
19
(K11a
19
)
1
Arc Sequences
4 1 7 2 11 3 10 5 6 8 9
Solving Sequence
1,4
2 3
5,9
8 11 6 7 10
c
1
c
2
c
4
c
8
c
11
c
5
c
6
c
10
c
3
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
5
+ b
4
+ 2b
3
+ b
2
+ b + 1, u 1, b
4
2b
2
+ a 2i
I
u
2
= hu
82
+ 6u
81
+ ··· 8u 1, 9.78643 × 10
68
u
81
+ 4.98123 × 10
69
u
80
+ ··· + 3.91373 × 10
68
a 2.00173 × 10
68
,
5.52146 × 10
69
u
81
2.43339 × 10
70
u
80
+ ··· + 1.56549 × 10
69
b + 3.43259 × 10
69
i
There are 2 irreducible components with 87 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
5
+ b
4
+ 2b
3
+ b
2
+ b + 1, u 1, b
4
2b
2
+ a 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
3
=
0
1
a
5
=
1
0
a
9
=
b
4
+ 2b
2
+ 2
b
a
8
=
b
4
+ 2b
2
b + 2
b
a
11
=
b
4
b
2
+ b
b
2
a
6
=
0
b
4
a
7
=
0
b
4
a
10
=
b
4
+ 2b
2
+ 2
b
4
a
10
=
b
4
+ 2b
2
+ 2
b
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b
4
3b
3
11b
2
2b 11
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 3.52181
b = 0.766826
4.04602 15.9649
u = 1.00000
a = 0.142272 0.509071I
b = 0.455697 1.200152I
7.51750 + 4.40083I 3.44484 1.78781I
u = 1.00000
a = 0.142272 + 0.509071I
b = 0.455697 + 1.200152I
7.51750 4.40083I 3.44484 + 1.78781I
u = 1.00000
a = 0.881366 0.489365I
b = 0.339110 0.822375I
1.97403 1.53058I 3.57269 + 4.45807I
u = 1.00000
a = 0.881366 + 0.489365I
b = 0.339110 + 0.822375I
1.97403 + 1.53058I 3.57269 4.45807I
3
II. I
u
2
=
hu
82
+6u
81
+· · · 8u 1, 9.79×10
68
u
81
+4.98 × 10
69
u
80
+· · · +3.91 × 10
68
a
2.00 × 10
68
, 5.52× 10
69
u
81
2.43 × 10
70
u
80
+ · · · + 1.57 ×10
69
b + 3.43 × 10
69
i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
9
=
2.50054u
81
12.7276u
80
+ ··· + 16.4655u + 0.511465
3.52698u
81
+ 15.5440u
80
+ ··· 10.4604u 2.19266
a
8
=
6.17468u
81
31.9950u
80
+ ··· + 39.1931u + 3.64241
7.33864u
81
+ 30.8977u
80
+ ··· 14.6427u 2.54618
a
11
=
0.393237u
81
2.05370u
80
+ ··· + 5.46852u + 3.40029
2.58681u
81
15.0269u
80
+ ··· + 30.1392u + 5.36022
a
6
=
3.69251u
81
+ 18.0231u
80
+ ··· 19.9154u 0.624870
2.07099u
81
5.98340u
80
+ ··· 5.40373u 0.423689
a
7
=
2.73425u
81
+ 10.6085u
80
+ ··· + 4.73671u + 3.63574
4.68430u
81
+ 29.2185u
80
+ ··· 55.5656u 7.41855
a
10
=
4.37647u
81
24.4869u
80
+ ··· + 40.7059u + 6.37209
7.89799u
81
+ 35.9315u
80
+ ··· 27.7980u 3.82601
a
10
=
4.37647u
81
24.4869u
80
+ ··· + 40.7059u + 6.37209
7.89799u
81
+ 35.9315u
80
+ ··· 27.7980u 3.82601
(ii) Obstruction class = 1
(iii) Cusp Shapes = 11.3752u
81
+ 53.0887u
80
+ ··· 66.0984u 14.2656
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.237191 0.595619I
a = 0.801746 0.451396I
b = 0.559407 0.749631I
3.13741 9.59200I 7.14919 + 9.30743I
u = 1.237191 + 0.595619I
a = 0.801746 + 0.451396I
b = 0.559407 + 0.749631I
3.13741 + 9.59200I 7.14919 9.30743I
u = 1.210675 0.603160I
a = 1.58308 + 1.06220I
b = 1.15795 + 1.15923I
4.0922 17.7863I 5.47301 + 10.24572I
u = 1.210675 + 0.603160I
a = 1.58308 1.06220I
b = 1.15795 1.15923I
4.0922 + 17.7863I 5.47301 10.24572I
u = 1.153023 0.586404I
a = 1.65299 1.15643I
b = 1.26891 1.32756I
0.45408 11.38862I 3.22823 + 9.94815I
u = 1.153023 + 0.586404I
a = 1.65299 + 1.15643I
b = 1.26891 + 1.32756I
0.45408 + 11.38862I 3.22823 9.94815I
u = 1.133184 0.536226I
a = 0.717432 + 0.739753I
b = 1.20723 1.55953I
4.37377 8.54127I 9.27955 + 9.08846I
u = 1.133184 + 0.536226I
a = 0.717432 0.739753I
b = 1.20723 + 1.55953I
4.37377 + 8.54127I 9.27955 9.08846I
u = 1.114119 0.596036I
a = 1.128236 + 0.721911I
b = 0.502767 + 0.529316I
0.34729 6.79546I 2.04718 + 3.52041I
u = 1.114119 + 0.596036I
a = 1.128236 0.721911I
b = 0.502767 0.529316I
0.34729 + 6.79546I 2.04718 3.52041I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.098864 0.495683I
a = 1.28101 + 1.45134I
b = 1.71373 + 0.88660I
5.00012 4.29669I 10.41022 + 3.63392I
u = 1.098864 + 0.495683I
a = 1.28101 1.45134I
b = 1.71373 0.88660I
5.00012 + 4.29669I 10.41022 3.63392I
u = 1.095489 0.542489I
a = 1.72972 + 0.59563I
b = 0.563284 + 0.253505I
2.13971 6.01246I 8.23433 2.70516I
u = 1.095489 + 0.542489I
a = 1.72972 0.59563I
b = 0.563284 0.253505I
2.13971 + 6.01246I 8.23433 + 2.70516I
u = 1.056733 0.265244I
a = 0.397754 0.329757I
b = 0.66718 1.28554I
8.00907 + 3.75643I 8.56422 + 3.52534I
u = 1.056733 + 0.265244I
a = 0.397754 + 0.329757I
b = 0.66718 + 1.28554I
8.00907 3.75643I 8.56422 3.52534I
u = 1.035011 0.523379I
a = 0.057335 1.044841I
b = 0.012552 0.742165I
0.96667 4.63901I 3.87963 + 8.57640I
u = 1.035011 + 0.523379I
a = 0.057335 + 1.044841I
b = 0.012552 + 0.742165I
0.96667 + 4.63901I 3.87963 8.57640I
u = 1.030345 0.222684I
a = 0.147138 + 0.478594I
b = 0.262578 + 1.206059I
7.77694 4.95365I 9.5185 + 11.5933I
u = 1.030345 + 0.222684I
a = 0.147138 0.478594I
b = 0.262578 1.206059I
7.77694 + 4.95365I 9.5185 11.5933I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.023158 0.461176I
a = 1.044851 + 0.190301I
b = 0.49803 + 1.59017I
1.65757 0.93442I 5.02283 + 0.17696I
u = 1.023158 + 0.461176I
a = 1.044851 0.190301I
b = 0.49803 1.59017I
1.65757 + 0.93442I 5.02283 0.17696I
u = 0.898916 0.791063I
a = 1.286018 + 0.131771I
b = 0.996012 + 0.582488I
2.97750 8.50086I 0.82594 + 8.56912I
u = 0.898916 + 0.791063I
a = 1.286018 0.131771I
b = 0.996012 0.582488I
2.97750 + 8.50086I 0.82594 8.56912I
u = 0.870879 0.677783I
a = 0.655354 1.103397I
b = 1.164217 + 0.241792I
4.80113 1.92406I 2.63229 + 2.70065I
u = 0.870879 + 0.677783I
a = 0.655354 + 1.103397I
b = 1.164217 0.241792I
4.80113 + 1.92406I 2.63229 2.70065I
u = 0.728926 0.864243I
a = 0.729650 + 0.743224I
b = 0.839576 0.372013I
3.48456 + 2.48996I 0.90365 4.62403I
u = 0.728926 + 0.864243I
a = 0.729650 0.743224I
b = 0.839576 + 0.372013I
3.48456 2.48996I 0.90365 + 4.62403I
u = 0.725256 0.718006I
a = 1.404793 0.127111I
b = 1.247104 0.531059I
5.22827 3.35824I 3.04121 + 4.47637I
u = 0.725256 + 0.718006I
a = 1.404793 + 0.127111I
b = 1.247104 + 0.531059I
5.22827 + 3.35824I 3.04121 4.47637I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.616890 0.459273I
a = 0.269056 0.141099I
b = 0.08898 1.41568I
0.34685 2.85468I 2.55427 + 7.34325I
u = 0.616890 + 0.459273I
a = 0.269056 + 0.141099I
b = 0.08898 + 1.41568I
0.34685 + 2.85468I 2.55427 7.34325I
u = 0.505284 0.582244I
a = 0.76662 + 1.54271I
b = 0.109566 + 0.615490I
0.603497 + 0.216471I 2.65199 5.07001I
u = 0.505284 + 0.582244I
a = 0.76662 1.54271I
b = 0.109566 0.615490I
0.603497 0.216471I 2.65199 + 5.07001I
u = 0.415240 0.792411I
a = 0.941111 + 0.069935I
b = 0.504549 0.325176I
2.44851 + 1.57419I 1.58009 + 0.44639I
u = 0.415240 + 0.792411I
a = 0.941111 0.069935I
b = 0.504549 + 0.325176I
2.44851 1.57419I 1.58009 0.44639I
u = 0.359004 0.663764I
a = 2.17009 2.04805I
b = 0.382225 0.325898I
0.006489 + 1.319229I 7.9750 + 14.9866I
u = 0.359004 + 0.663764I
a = 2.17009 + 2.04805I
b = 0.382225 + 0.325898I
0.006489 1.319229I 7.9750 14.9866I
u = 0.319570 0.836317I
a = 1.21702 0.75247I
b = 1.22833 + 1.13874I
2.93914 + 6.11947I 0.48338 6.25574I
u = 0.319570 + 0.836317I
a = 1.21702 + 0.75247I
b = 1.22833 1.13874I
2.93914 6.11947I 0.48338 + 6.25574I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.268436 0.946194I
a = 1.30528 + 0.58158I
b = 1.11988 1.06158I
1.22744 + 12.16919I 2.52898 7.09852I
u = 0.268436 + 0.946194I
a = 1.30528 0.58158I
b = 1.11988 + 1.06158I
1.22744 12.16919I 2.52898 + 7.09852I
u = 0.256433 0.716128I
a = 1.58980 0.40698I
b = 0.95054 + 1.39710I
1.85423 + 3.78257I 5.86247 6.02702I
u = 0.256433 + 0.716128I
a = 1.58980 + 0.40698I
b = 0.95054 1.39710I
1.85423 3.78257I 5.86247 + 6.02702I
u = 0.222834 0.492928I
a = 1.37187 + 1.11969I
b = 1.35337 0.51898I
2.65606 + 0.12519I 6.98986 0.10633I
u = 0.222834 + 0.492928I
a = 1.37187 1.11969I
b = 1.35337 + 0.51898I
2.65606 0.12519I 6.98986 + 0.10633I
u = 0.197051
a = 1.34599
b = 1.03636
2.55123 4.11145
u = 0.196895 0.993276I
a = 0.450229 0.302114I
b = 0.502549 + 0.495269I
0.01560 + 3.91593I 3.20651 9.86113I
u = 0.196895 + 0.993276I
a = 0.450229 + 0.302114I
b = 0.502549 0.495269I
0.01560 3.91593I 3.20651 + 9.86113I
u = 0.216654 0.255543I
a = 0.19217 2.23903I
b = 0.453491 + 0.525799I
0.041508 + 1.376627I 0.16544 4.98668I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.216654 + 0.255543I
a = 0.19217 + 2.23903I
b = 0.453491 0.525799I
0.041508 1.376627I 0.16544 + 4.98668I
u = 0.373251 0.739288I
a = 1.42329 0.99546I
b = 0.872546 + 0.953025I
3.71598 6.06156I 4.81312 + 3.61533I
u = 0.373251 + 0.739288I
a = 1.42329 + 0.99546I
b = 0.872546 0.953025I
3.71598 + 6.06156I 4.81312 3.61533I
u = 0.455771 0.389865I
a = 1.66735 + 1.65690I
b = 0.795340 0.768821I
0.28541 1.54658I 0.57343 + 2.06881I
u = 0.455771 + 0.389865I
a = 1.66735 1.65690I
b = 0.795340 + 0.768821I
0.28541 + 1.54658I 0.57343 2.06881I
u = 0.573215 0.805489I
a = 0.575253 + 0.584192I
b = 0.378064 0.513988I
2.85407 + 3.43010I 8.3578 12.3401I
u = 0.573215 + 0.805489I
a = 0.575253 0.584192I
b = 0.378064 + 0.513988I
2.85407 3.43010I 8.3578 + 12.3401I
u = 0.822458 0.162075I
a = 3.05753 + 0.76985I
b = 0.235985 + 0.507922I
2.85067 + 0.96287I 12.24182 + 4.62200I
u = 0.822458 + 0.162075I
a = 3.05753 0.76985I
b = 0.235985 0.507922I
2.85067 0.96287I 12.24182 4.62200I
u = 0.974615 0.396279I
a = 1.45063 0.42945I
b = 0.234307 0.335757I
1.87549 + 1.38403I 4.93593 0.98614I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.974615 + 0.396279I
a = 1.45063 + 0.42945I
b = 0.234307 + 0.335757I
1.87549 1.38403I 4.93593 + 0.98614I
u = 1.044582 0.476716I
a = 2.18798 + 1.05121I
b = 1.03318 + 1.14851I
1.46560 + 5.42569I 4.53155 7.12185I
u = 1.044582 + 0.476716I
a = 2.18798 1.05121I
b = 1.03318 1.14851I
1.46560 5.42569I 4.53155 + 7.12185I
u = 1.045009 0.319225I
a = 4.20142 1.16980I
b = 0.318153 0.099651I
3.74856 + 1.03244I 28.3012 + 20.0875I
u = 1.045009 + 0.319225I
a = 4.20142 + 1.16980I
b = 0.318153 + 0.099651I
3.74856 1.03244I 28.3012 20.0875I
u = 1.06725
a = 5.55769
b = 0.468968
3.80714 30.5601
u = 1.099043 0.390529I
a = 0.649498 1.116556I
b = 1.12138 + 1.16538I
5.73183 + 3.04738I 12.16397 4.67101I
u = 1.099043 + 0.390529I
a = 0.649498 + 1.116556I
b = 1.12138 1.16538I
5.73183 3.04738I 12.16397 + 4.67101I
u = 1.105682 0.199763I
a = 0.597739 + 0.551419I
b = 0.013877 + 0.420367I
2.36954 + 0.63816I 5.46759 + 1.52326I
u = 1.105682 + 0.199763I
a = 0.597739 0.551419I
b = 0.013877 0.420367I
2.36954 0.63816I 5.46759 1.52326I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.110383 0.566495I
a = 1.91144 0.85746I
b = 1.02208 1.07291I
5.89382 + 11.02702I 6.79585 7.24164I
u = 1.110383 + 0.566495I
a = 1.91144 + 0.85746I
b = 1.02208 + 1.07291I
5.89382 11.02702I 6.79585 + 7.24164I
u = 1.146836 0.310629I
a = 1.52883 1.82683I
b = 1.21335 0.97040I
5.91812 0.62603I 12.66614 + 3.47864I
u = 1.146836 + 0.310629I
a = 1.52883 + 1.82683I
b = 1.21335 + 0.97040I
5.91812 + 0.62603I 12.66614 3.47864I
u = 1.174904 0.635746I
a = 0.553389 0.009077I
b = 0.206090 + 0.501160I
4.70519 + 2.39134I 16.5054 10.8772I
u = 1.174904 + 0.635746I
a = 0.553389 + 0.009077I
b = 0.206090 0.501160I
4.70519 2.39134I 16.5054 + 10.8772I
u = 1.218446 0.212469I
a = 0.858080 0.269238I
b = 0.868853 1.090657I
2.09067 2.95787I 3.92137 + 7.14092I
u = 1.218446 + 0.212469I
a = 0.858080 + 0.269238I
b = 0.868853 + 1.090657I
2.09067 + 2.95787I 3.92137 7.14092I
u = 1.336129 0.262407I
a = 0.293592 + 0.128517I
b = 0.94575 + 1.05162I
6.58899 8.11200I 6.71308 + 6.24405I
u = 1.336129 + 0.262407I
a = 0.293592 0.128517I
b = 0.94575 1.05162I
6.58899 + 8.11200I 6.71308 6.24405I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.44028 0.24640I
a = 0.0866043 0.0611694I
b = 0.059583 0.522775I
5.52870 + 0.86398I 20.6612 3.3440I
u = 1.44028 + 0.24640I
a = 0.0866043 + 0.0611694I
b = 0.059583 + 0.522775I
5.52870 0.86398I 20.6612 + 3.3440I
12
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
82
+ 6u
81
+ ··· 8u 1)
c
2
(u + 1)
5
(u
82
+ 42u
81
+ ··· + 32u + 1)
c
3
u
5
(u
82
+ u
81
+ ··· 160u + 32)
c
4
(u + 1)
5
(u
82
+ 6u
81
+ ··· 8u 1)
c
5
(u
5
3u
4
+ ··· u + 1)(u
82
+ 6u
81
+ ··· 2u 1)
c
6
u
5
(u
82
+ u
81
+ ··· 160u + 32)
c
7
(u
5
+ u
4
+ ··· + u 1)(u
82
+ 2u
81
+ ··· 14u + 1)
c
8
(u
5
+ u
4
+ ··· + u + 1)(u
82
+ 2u
81
+ ··· + 2362u 484)
c
9
(u
5
u
4
+ ··· + u + 1)(u
82
+ 2u
81
+ ··· 20520u 1647)
c
10
(u
5
u
4
+ ··· + u + 1)(u
82
+ 2u
81
+ ··· 14u + 1)
c
11
(u
5
+ u
4
+ ··· + u + 1)(u
82
+ 14u
81
+ ··· + 2u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
82
42y
81
+ ··· 32y + 1)
c
2
(y 1)
5
(y
82
+ 2y
81
+ ··· 484y + 1)
c
3
y
5
(y
82
33y
81
+ ··· 19968y + 1024)
c
5
(y
5
y
4
+ ··· + 3y 1)(y
82
14y
81
+ ··· 6y + 1)
c
6
y
5
(y
82
33y
81
+ ··· 19968y + 1024)
c
7
(y
5
5y
4
+ ··· y 1)(y
82
58y
81
+ ··· + 14y + 1)
c
8
(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
(y
82
+ 90y
81
+ ··· + 10862436y + 234256)
c
9
(y
5
5y
4
+ 8y
3
3y
2
y 1)
(y
82
+ 50y
81
+ ··· 261288342y + 2712609)
c
10
(y 1)(y
5
5y
4
+ ··· y 1)(y
81
57y
80
+ ··· 15y 1)
c
11
(y
5
+ 3y
4
+ ··· y 1)(y
82
+ 6y
81
+ ··· + 14y + 1)
14