11a
103
(K11a
103
)
1
Arc Sequences
6 1 7 10 2 4 3 11 5 9 8
Solving Sequence
2,5
6
1,10
4 7 3 9 11 8
c
5
c
1
c
4
c
6
c
3
c
9
c
10
c
8
c
2
, c
7
, c
11
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
6
2b
5
5b
4
+ 8b
3
+ 32b
2
66b + 37, 20b
5
33b
4
126b
3
+ 87b
2
+ 656b + 289a 917,
25b
5
20b
4
149b
3
+ 79b
2
+ 837b + 289u 819i
I
u
2
= hu
12
+ 4u
10
u
9
+ 6u
8
3u
7
+ 7u
6
3u
5
+ 7u
4
3u
3
+ 3u
2
2u + 1, u
11
4u
9
6u
7
6u
5
5u
3
+ a 2u,
u
11
+ u
10
3u
9
+ 3u
8
3u
7
+ 3u
6
3u
5
+ 3u
4
2u
3
+ 2u
2
+ bi
I
u
3
= hu
34
+ u
33
+ ··· + 8u + 1,
127209165398972u
33
+ 110135972200770u
32
+ ··· + 517397368676324b 165318083439065,
1.01393 × 10
16
u
33
+ 1.01360 × 10
16
u
32
+ ··· + 5.17397 × 10
15
a + 2.92536 × 10
16
i
There are 3 irreducible components with 52 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
6
2b
5
5b
4
+ 8b
3
+ 32b
2
66b + 37, 20b
5
33b
4
+ · · · + 289a
917, 25b
5
+ 289u + · · · + 837b 819i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
0.0865052b
5
+ 0.0692042b
4
+ ··· 2.89619b + 2.83391
a
6
=
0.0865052b
5
+ 0.0692042b
4
+ ··· 2.89619b + 2.83391
0.0865052b
5
+ 0.0692042b
4
+ ··· 2.89619b + 2.83391
a
1
=
0
1
a
10
=
0.0692042b
5
+ 0.114187b
4
+ ··· 2.26990b + 3.17301
b
a
4
=
0.0899654b
5
+ 0.0103806b
4
+ ··· + 2.39792b 0.519031
0.0899654b
5
+ 0.0103806b
4
+ ··· + 2.39792b 1.51903
a
7
=
0.0242215b
5
0.0899654b
4
+ ··· + 1.39446b 1.56055
0.0622837b
5
0.0207612b
4
+ ··· 1.50173b + 1.27336
a
3
=
1
1
a
9
=
0.0692042b
5
+ 0.114187b
4
+ ··· 2.26990b + 3.17301
0.0692042b
5
0.114187b
4
+ ··· + 3.26990b 3.17301
a
11
=
0.0415225b
5
0.0449827b
4
+ ··· + 2.02076b 0.221453
0.0484429b
5
0.0553633b
4
+ ··· 0.377163b 0.702422
a
8
=
0.110727b
5
0.159170b
4
+ ··· + 4.29066b 4.39446
0.0242215b
5
0.0899654b
4
+ ··· + 1.39446b 1.56055
a
8
=
0.110727b
5
0.159170b
4
+ ··· + 4.29066b 4.39446
0.0242215b
5
0.0899654b
4
+ ··· + 1.39446b 1.56055
(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 = 1.00000I
a = 0.754878
b = 2.07960 1.32472I
1.11345 7.01951
u = 1.00000I
a = 0.754878
b = 2.07960 + 1.32472I
1.11345 7.01951
u = 1.00000I
a = 0.877439 + 0.744862I
b = 0.977518 0.479777I
3.02413 2.82812I 0.49024 + 2.97945I
u = 1.00000I
a = 0.877439 0.744862I
b = 0.977518 + 0.479777I
3.02413 + 2.82812I 0.49024 2.97945I
u = 1.00000I
a = 0.877439 0.744862I
b = 2.10208 0.84494I
3.02413 + 2.82812I 0.49024 2.97945I
u = 1.00000I
a = 0.877439 + 0.744862I
b = 2.10208 + 0.84494I
3.02413 2.82812I 0.49024 + 2.97945I
3
II. I
u
2
= hu
12
+ 4u
10
+ · · · 2u + 1, u
11
4u
9
6u
7
6u
5
5u
3
+ a
2u, u
11
+ u
10
+ · · · + 2u
2
+ bi
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
10
=
u
11
+ 4u
9
+ 6u
7
+ 6u
5
+ 5u
3
+ 2u
u
11
u
10
+ 3u
9
3u
8
+ 3u
7
3u
6
+ 3u
5
3u
4
+ 2u
3
2u
2
a
4
=
u
2
+ 1
u
2
a
7
=
u
3
u
3
+ u
a
3
=
u
4
+ u
2
+ 1
u
4
a
9
=
u
11
+ 4u
9
+ 6u
7
+ 6u
5
+ 5u
3
+ 2u
u
11
+ 3u
9
+ 2u
7
+ u
5
+ u
3
u
a
11
=
u
8
+ 3u
6
+ 3u
4
+ 2u
2
+ 1
u
8
+ 2u
6
a
8
=
u
5
+ 2u
3
+ u
u
5
+ u
3
u
a
8
=
u
5
+ 2u
3
+ u
u
5
+ u
3
u
(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 = 0.737742 0.749761I
a = 1.185521 0.030411I
b = 1.48105 0.59444I
6.79074 + 3.16396I 1.82674 2.56480I
u = 0.737742 + 0.749761I
a = 1.185521 + 0.030411I
b = 1.48105 + 0.59444I
6.79074 3.16396I 1.82674 + 2.56480I
u = 0.337741 0.872538I
a = 0.853727 0.072741I
b = 1.46920 + 1.14722I
0.21101 + 1.41510I 1.82674 4.90874I
u = 0.337741 + 0.872538I
a = 0.853727 + 0.072741I
b = 1.46920 1.14722I
0.21101 1.41510I 1.82674 + 4.90874I
u = 0.117310 1.208577I
a = 0.562471 0.345689I
b = 2.40034 0.67347I
0.21101 1.41510I 1.82674 + 4.90874I
u = 0.117310 + 1.208577I
a = 0.562471 + 0.345689I
b = 2.40034 + 0.67347I
0.21101 + 1.41510I 1.82674 4.90874I
u = 0.02004 1.44158I
a = 0.623816 + 0.599557I
b = 2.00453 + 0.38432I
6.79074 3.16396I 1.82674 + 2.56480I
u = 0.02004 + 1.44158I
a = 0.623816 0.599557I
b = 2.00453 0.38432I
6.79074 + 3.16396I 1.82674 2.56480I
u = 0.455051 0.336038I
a = 0.25617 1.39382I
b = 0.163705 + 0.421062I
0.21101 + 1.41510I 1.82674 4.90874I
u = 0.455051 + 0.336038I
a = 0.25617 + 1.39382I
b = 0.163705 0.421062I
0.21101 1.41510I 1.82674 + 4.90874I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.757780 0.691817I
a = 0.014281 + 1.215620I
b = 0.571368 0.192544I
6.79074 + 3.16396I 1.82674 2.56480I
u = 0.757780 + 0.691817I
a = 0.014281 1.215620I
b = 0.571368 + 0.192544I
6.79074 3.16396I 1.82674 + 2.56480I
6
III.
I
u
3
= hu
34
+u
33
+· · ·+8u+1, 1.27×10
14
u
33
+1.10×10
14
u
32
+· · ·+5.17×10
14
b
1.65 × 10
14
, 1.01 × 10
16
u
33
+ 1.01 × 10
16
u
32
+ · · · + 5.17 × 10
15
a + 2.93 × 10
16
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
10
=
1.95968u
33
1.95905u
32
+ ··· 61.9142u 5.65399
0.245864u
33
0.212865u
32
+ ··· 4.85136u + 0.319519
a
4
=
1.41968u
33
+ 0.953207u
32
+ ··· + 24.2367u 3.33833
0.203926u
33
+ 0.0406624u
32
+ ··· 2.40839u 1.45002
a
7
=
2.17754u
33
1.94681u
32
+ ··· 71.5730u 11.0433
0.466475u
33
0.463665u
32
+ ··· 13.6958u 1.41968
a
3
=
u
4
+ u
2
+ 1
u
4
a
9
=
1.95968u
33
1.95905u
32
+ ··· 61.9142u 5.65399
0.0603903u
33
0.0125403u
32
+ ··· 2.89675u + 0.318885
a
11
=
1.25226u
33
0.850668u
32
+ ··· 19.8561u + 6.05213
0.0754751u
33
+ 0.174403u
32
+ ··· + 6.69274u + 2.20913
a
8
=
1.68221u
33
+ 1.35953u
32
+ ··· + 57.2089u + 9.39288
0.698669u
33
+ 0.577103u
32
+ ··· + 15.3212u + 1.62080
a
8
=
1.68221u
33
+ 1.35953u
32
+ ··· + 57.2089u + 9.39288
0.698669u
33
+ 0.577103u
32
+ ··· + 15.3212u + 1.62080
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.062641 0.522537I
a = 0.293628 1.060621I
b = 0.279197 + 0.816683I
14.3477 6.6660I 4.12754 + 3.29257I
u = 1.062641 + 0.522537I
a = 0.293628 + 1.060621I
b = 0.279197 0.816683I
14.3477 + 6.6660I 4.12754 3.29257I
u = 0.756680 1.182597I
a = 0.921688 + 0.122943I
b = 2.84089 + 0.05378I
12.2913 + 13.2218I 1.97446 7.47226I
u = 0.756680 + 1.182597I
a = 0.921688 0.122943I
b = 2.84089 0.05378I
12.2913 13.2218I 1.97446 + 7.47226I
u = 0.744389 0.572496I
a = 0.175509 + 1.165102I
b = 0.763425 0.947776I
5.18197 3.30193I 1.74557 + 3.15747I
u = 0.744389 + 0.572496I
a = 0.175509 1.165102I
b = 0.763425 + 0.947776I
5.18197 + 3.30193I 1.74557 3.15747I
u = 0.708232 0.962390I
a = 0.147425 1.018445I
b = 0.249723 0.484205I
6.14429 + 2.35773I 0.48410 2.53993I
u = 0.708232 + 0.962390I
a = 0.147425 + 1.018445I
b = 0.249723 + 0.484205I
6.14429 2.35773I 0.48410 + 2.53993I
u = 0.675576 1.053517I
a = 0.877316 + 0.147740I
b = 2.94208 0.47894I
3.77297 + 8.75654I 1.23706 8.00625I
u = 0.675576 + 1.053517I
a = 0.877316 0.147740I
b = 2.94208 + 0.47894I
3.77297 8.75654I 1.23706 + 8.00625I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.625868 0.866379I
a = 0.754520 0.618944I
b = 2.29822 + 1.02909I
1.54356 + 2.45179I 3.02847 2.63078I
u = 0.625868 + 0.866379I
a = 0.754520 + 0.618944I
b = 2.29822 1.02909I
1.54356 2.45179I 3.02847 + 2.63078I
u = 0.441745 0.826611I
a = 0.040512 + 0.635092I
b = 0.296423 + 0.239458I
0.14247 + 1.88117I 0.09008 3.89150I
u = 0.441745 + 0.826611I
a = 0.040512 0.635092I
b = 0.296423 0.239458I
0.14247 1.88117I 0.09008 + 3.89150I
u = 0.231881 0.577044I
a = 0.851478 0.184976I
b = 0.099543 + 0.504922I
0.165045 + 1.193182I 1.28022 6.36905I
u = 0.231881 + 0.577044I
a = 0.851478 + 0.184976I
b = 0.099543 0.504922I
0.165045 1.193182I 1.28022 + 6.36905I
u = 0.122630 0.166148I
a = 1.26902 + 5.31025I
b = 0.815723 + 0.522371I
4.64121 2.76844I 5.45074 + 3.04285I
u = 0.122630 + 0.166148I
a = 1.26902 5.31025I
b = 0.815723 0.522371I
4.64121 + 2.76844I 5.45074 3.04285I
u = 0.026982 1.057991I
a = 0.817861 0.736027I
b = 1.59877 0.17238I
1.40352 + 2.86614I 6.42514 2.90312I
u = 0.026982 + 1.057991I
a = 0.817861 + 0.736027I
b = 1.59877 + 0.17238I
1.40352 2.86614I 6.42514 + 2.90312I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.261633 0.992110I
a = 0.801344 + 0.344555I
b = 2.34355 0.20941I
3.37245 0.54787I 11.62744 + 0.56640I
u = 0.261633 + 0.992110I
a = 0.801344 0.344555I
b = 2.34355 + 0.20941I
3.37245 + 0.54787I 11.62744 0.56640I
u = 0.492701 0.994382I
a = 0.895018 0.043934I
b = 2.63329 0.29639I
1.90809 5.35995I 5.93507 + 8.80123I
u = 0.492701 + 0.994382I
a = 0.895018 + 0.043934I
b = 2.63329 + 0.29639I
1.90809 + 5.35995I 5.93507 8.80123I
u = 0.698424 1.002796I
a = 1.034759 0.114964I
b = 2.41238 + 0.48165I
5.84725 8.71325I 0.27553 + 7.33315I
u = 0.698424 + 1.002796I
a = 1.034759 + 0.114964I
b = 2.41238 0.48165I
5.84725 + 8.71325I 0.27553 7.33315I
u = 0.744862 0.976632I
a = 0.186674 0.629275I
b = 0.024226 0.537210I
6.18222 4.46313I 3.61159 + 3.03058I
u = 0.744862 + 0.976632I
a = 0.186674 + 0.629275I
b = 0.024226 + 0.537210I
6.18222 + 4.46313I 3.61159 3.03058I
u = 0.779313 1.162922I
a = 0.130006 + 0.876745I
b = 0.334798 + 0.669421I
12.8109 6.7247I 2.85757 + 2.78604I
u = 0.779313 + 1.162922I
a = 0.130006 0.876745I
b = 0.334798 0.669421I
12.8109 + 6.7247I 2.85757 2.78604I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.807854 0.740004I
a = 0.771405 + 0.169577I
b = 0.513966 + 0.364888I
6.89618 1.36512I 4.50793 + 2.51852I
u = 0.807854 + 0.740004I
a = 0.771405 0.169577I
b = 0.513966 0.364888I
6.89618 + 1.36512I 4.50793 2.51852I
u = 1.057876 0.566930I
a = 1.015336 + 0.282591I
b = 0.893688 0.181209I
14.6663 + 0.1016I 4.57906 + 1.48156I
u = 1.057876 + 0.566930I
a = 1.015336 0.282591I
b = 0.893688 + 0.181209I
14.6663 0.1016I 4.57906 1.48156I
11
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
2
+ 1)
3
(u
12
+ 4u
10
u
9
+ 6u
8
3u
7
+ 7u
6
3u
5
+ 7u
4
3u
3
+ 3u
2
2u + 1)
(u
34
+ u
33
+ ··· + 8u + 1)
c
2
(u + 1)
6
(u
12
+ 8u
11
+ ··· + 2u + 1)(u
34
+ 11u
33
+ ··· + 24u + 1)
c
3
, c
6
, c
7
(u
2
+ 1)
3
(u
12
+ 4u
10
u
9
+ 6u
8
3u
7
+ 7u
6
3u
5
+ 7u
4
3u
3
+ 3u
2
2u + 1)
(u
34
+ u
33
+ ··· + 10u + 1)
c
4
, c
9
(1 + u
2
+ u
3
+ u
4
)
3
(u
6
+ u
4
+ 2u
2
+ 1)(u
34
2u
33
+ ··· u + 2)
c
8
(u
3
u
2
+ 2u 1)
2
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
3
(u
34
+ 8u
33
+ ··· + 19u + 4)
c
10
, c
11
(u
3
+ u
2
+ 2u + 1)
2
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
3
(u
34
+ 8u
33
+ ··· + 19u + 4)
12
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y + 1)
6
(y
12
+ 8y
11
+ ··· + 2y + 1)(y
34
+ 11y
33
+ ··· + 24y + 1)
c
2
(y 1)
6
(y
12
8y
11
+ ··· + 18y + 1)(y
34
+ 31y
33
+ ··· + 916y + 1)
c
3
, c
6
, c
7
(y + 1)
6
(y
12
+ 8y
11
+ ··· + 2y + 1)(y
34
+ 39y
33
+ ··· + 56y + 1)
c
4
, c
9
(y
3
+ y
2
+ 2y + 1)
2
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
3
(y
34
+ 8y
33
+ ··· + 19y + 4)
c
8
, c
10
, c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
3
(y
34
+ 36y
33
+ ··· + 495y + 16)
13