11a
39
(K11a
39
)
1
Arc Sequences
4 1 7 2 10 9 3 11 5 6 8
Solving Sequence
5,9
10 6 7
2,11
4 1 3 8
c
9
c
5
c
6
c
10
c
4
c
1
c
3
c
8
c
2
, c
7
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
5
+ b
3
6b
2
2b 11, u 1, 9b
4
+ 14b
3
11b
2
8b + 89a 19i
I
u
2
= hu
55
+ 6u
54
+ ··· 8u 1, 169u
54
+ 721u
53
+ ··· + 16b + 55, 227u
54
1299u
53
+ ··· + 16a + 439i
There are 2 irreducible components with 60 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
3
6b
2
2b 11, u 1, 9b
4
+ 14b
3
11b
2
8b + 89a 19i
(i) Arc colorings
a
5
=
0
1
a
9
=
0.101124b
4
0.157303b
3
+ ··· + 0.0898876b + 0.213483
b
a
10
=
0.101124b
4
0.157303b
3
+ ··· + 0.0898876b + 0.213483
0.101124b
4
+ 0.157303b
3
+ ··· + 0.910112b 0.213483
a
6
=
0.179775b
4
+ 0.168539b
3
+ ··· + 0.617978b + 0.842697
0.0224719b
4
0.146067b
3
+ ··· 0.202247b + 1.26966
a
7
=
0.101124b
4
+ 0.157303b
3
+ ··· + 0.910112b 0.213483
0.101124b
4
+ 0.157303b
3
+ ··· + 0.910112b 0.213483
a
2
=
1
0
a
11
=
0.134831b
4
0.123596b
3
+ ··· + 0.213483b + 1.38202
0.134831b
4
0.123596b
3
+ ··· + 0.213483b + 0.382022
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
8
=
0.101124b
4
+ 0.157303b
3
+ ··· + 0.910112b 0.213483
0.101124b
4
+ 0.157303b
3
+ ··· + 0.910112b 0.213483
a
8
=
0.101124b
4
+ 0.157303b
3
+ ··· + 0.910112b 0.213483
0.101124b
4
+ 0.157303b
3
+ ··· + 0.910112b 0.213483
(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 = 1.00000
a = 1.41878 0.21917I
b = 0.96309 1.41932I
4.22763 + 4.40083I 2.94226 4.18967I
u = 1.00000
a = 1.41878 + 0.21917I
b = 0.96309 + 1.41932I
4.22763 4.40083I 2.94226 + 4.18967I
u = 1.00000
a = 0.309916 0.549911I
b = 0.029195 1.372286I
1.31583 1.53058I 6.94263 + 4.09764I
u = 1.00000
a = 0.309916 + 0.549911I
b = 0.029195 + 1.372286I
1.31583 + 1.53058I 6.94263 4.09764I
u = 1.00000
a = 1.21774
b = 1.98456
0.756147 2.23022
3
II. I
u
2
= hu
55
+ 6u
54
+ · · · 8u 1, 169u
54
+ 721u
53
+ · · · + 16b +
55, 227u
54
1299u
53
+ · · · + 16a + 439i
(i) Arc colorings
a
5
=
0
u
a
9
=
14.1875u
54
+ 81.1875u
53
+ ··· 184.313u 27.4375
10.5625u
54
45.0625u
53
+ ··· 2.81250u 3.43750
a
10
=
14.1875u
54
+ 81.1875u
53
+ ··· 184.313u 27.4375
7.75000u
54
34.7500u
53
+ ··· + 14.5000u + 0.500000
a
6
=
14.2500u
54
+ 74.1250u
53
+ ··· 114.250u 16.3750
0.687500u
54
+ 3.56250u
53
+ ··· 5.56250u 0.812500
a
7
=
27
4
u
54
+ 30u
53
+ ··· + 13u +
21
4
27
4
u
54
+
145
4
u
53
+ ··· 59u
15
2
a
2
=
1
0
a
11
=
0.0625000u
54
+ 0.312500u
53
+ ··· 1.43750u + 1.93750
0.0625000u
54
+ 0.312500u
53
+ ··· 1.43750u 0.0625000
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
8
=
12.5000u
54
63.5000u
53
+ ··· + 51.2500u + 4.75000
15.5000u
54
77.7500u
53
+ ··· + 86.2500u + 10.5000
a
8
=
12.5000u
54
63.5000u
53
+ ··· + 51.2500u + 4.75000
15.5000u
54
77.7500u
53
+ ··· + 86.2500u + 10.5000
(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 = 1.150217 0.701224I
a = 0.829297 0.731413I
b = 1.39547 3.07119I
10.8274 14.0334I 2.74555 + 8.17005I
u = 1.150217 + 0.701224I
a = 0.829297 + 0.731413I
b = 1.39547 + 3.07119I
10.8274 + 14.0334I 2.74555 8.17005I
u = 1.131317 0.693924I
a = 0.014711 + 0.608451I
b = 0.06424 + 2.13487I
4.79235 10.53538I 0.79776 + 8.38025I
u = 1.131317 + 0.693924I
a = 0.014711 0.608451I
b = 0.06424 2.13487I
4.79235 + 10.53538I 0.79776 8.38025I
u = 1.103897 0.704264I
a = 0.589381 0.150141I
b = 0.408464 0.642739I
5.32499 6.24906I 0.59828 + 2.50741I
u = 1.103897 + 0.704264I
a = 0.589381 + 0.150141I
b = 0.408464 + 0.642739I
5.32499 + 6.24906I 0.59828 2.50741I
u = 1.099520 0.737359I
a = 1.012705 0.508057I
b = 0.061727 0.736233I
11.90408 3.49537I 4.18205 + 2.47888I
u = 1.099520 + 0.737359I
a = 1.012705 + 0.508057I
b = 0.061727 + 0.736233I
11.90408 + 3.49537I 4.18205 2.47888I
u = 1.053485 0.579410I
a = 0.843542 + 0.696553I
b = 2.31858 + 1.88725I
3.09735 8.51931I 0.39740 + 8.43145I
u = 1.053485 + 0.579410I
a = 0.843542 0.696553I
b = 2.31858 1.88725I
3.09735 + 8.51931I 0.39740 8.43145I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.999297 0.553816I
a = 0.200871 0.559124I
b = 0.43598 2.02723I
1.25442 5.19790I 5.46768 + 6.87562I
u = 0.999297 + 0.553816I
a = 0.200871 + 0.559124I
b = 0.43598 + 2.02723I
1.25442 + 5.19790I 5.46768 6.87562I
u = 0.912992 0.503784I
a = 0.967509 0.387449I
b = 1.47075 + 0.76322I
1.55049 1.99776I 0.73726 + 3.12660I
u = 0.912992 + 0.503784I
a = 0.967509 + 0.387449I
b = 1.47075 0.76322I
1.55049 + 1.99776I 0.73726 3.12660I
u = 0.838321 0.728876I
a = 0.955766 + 0.863790I
b = 0.231629 + 1.087459I
7.99304 2.76322I 5.66546 + 3.09828I
u = 0.838321 + 0.728876I
a = 0.955766 0.863790I
b = 0.231629 1.087459I
7.99304 + 2.76322I 5.66546 3.09828I
u = 0.825967 0.561444I
a = 0.445929 0.518379I
b = 0.771674 0.277334I
1.63950 2.24198I 1.57550 + 4.32492I
u = 0.825967 + 0.561444I
a = 0.445929 + 0.518379I
b = 0.771674 + 0.277334I
1.63950 + 2.24198I 1.57550 4.32492I
u = 0.682905 0.426557I
a = 0.654114 + 0.522448I
b = 0.511742 + 1.111086I
0.093506 + 0.957018I 1.47250 1.22419I
u = 0.682905 + 0.426557I
a = 0.654114 0.522448I
b = 0.511742 1.111086I
0.093506 0.957018I 1.47250 + 1.22419I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.636512 0.224540I
a = 2.06481 + 0.36689I
b = 0.732573 0.996569I
5.07190 + 4.05462I 5.67447 1.52659I
u = 0.636512 + 0.224540I
a = 2.06481 0.36689I
b = 0.732573 + 0.996569I
5.07190 4.05462I 5.67447 + 1.52659I
u = 0.576460 0.947356I
a = 0.537887 1.241392I
b = 0.34583 1.53044I
13.50704 2.65879I 6.05466 + 2.20857I
u = 0.576460 + 0.947356I
a = 0.537887 + 1.241392I
b = 0.34583 + 1.53044I
13.50704 + 2.65879I 6.05466 2.20857I
u = 0.541319 0.917635I
a = 0.248916 + 0.703169I
b = 0.327913 + 0.624460I
7.04434 + 0.29231I 2.77654 + 2.24461I
u = 0.541319 + 0.917635I
a = 0.248916 0.703169I
b = 0.327913 0.624460I
7.04434 0.29231I 2.77654 2.24461I
u = 0.496725 0.933713I
a = 0.757117 + 0.057430I
b = 0.730130 0.012101I
6.73070 + 4.57214I 1.95904 4.03979I
u = 0.496725 + 0.933713I
a = 0.757117 0.057430I
b = 0.730130 + 0.012101I
6.73070 4.57214I 1.95904 + 4.03979I
u = 0.483891 0.966514I
a = 0.815178 1.115773I
b = 1.303463 0.351285I
12.8723 + 7.9548I 5.34871 3.91540I
u = 0.483891 + 0.966514I
a = 0.815178 + 1.115773I
b = 1.303463 + 0.351285I
12.8723 7.9548I 5.34871 + 3.91540I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.325642 0.591557I
a = 1.21633 + 1.60536I
b = 0.092195 0.420374I
4.93016 + 3.86526I 3.70043 4.17528I
u = 0.325642 + 0.591557I
a = 1.21633 1.60536I
b = 0.092195 + 0.420374I
4.93016 3.86526I 3.70043 + 4.17528I
u = 0.182425 0.252129I
a = 1.72274 + 0.37286I
b = 0.077803 + 0.469162I
0.008201 + 1.198198I 0.25143 5.38204I
u = 0.182425 + 0.252129I
a = 1.72274 0.37286I
b = 0.077803 0.469162I
0.008201 1.198198I 0.25143 + 5.38204I
u = 0.351487 0.343411I
a = 1.95115 1.78026I
b = 1.021942 0.415456I
2.99828 0.14721I 1.47136 1.22747I
u = 0.351487 + 0.343411I
a = 1.95115 + 1.78026I
b = 1.021942 + 0.415456I
2.99828 + 0.14721I 1.47136 + 1.22747I
u = 0.786253 0.638767I
a = 1.25492 + 0.74913I
b = 0.87236 + 1.12865I
7.88540 2.59431I 2.77604 + 0.35254I
u = 0.786253 + 0.638767I
a = 1.25492 0.74913I
b = 0.87236 1.12865I
7.88540 + 2.59431I 2.77604 0.35254I
u = 0.813432 0.582817I
a = 0.712525 + 0.208166I
b = 0.114434 0.093188I
1.70221 + 0.32794I 0.771677 + 0.468923I
u = 0.813432 + 0.582817I
a = 0.712525 0.208166I
b = 0.114434 + 0.093188I
1.70221 0.32794I 0.771677 0.468923I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.855618
a = 0.254366
b = 0.346879
1.24884 7.99829
u = 0.881787 0.589772I
a = 0.012451 0.718606I
b = 0.46087 1.48148I
1.48241 + 4.33310I 1.64326 6.16138I
u = 0.881787 + 0.589772I
a = 0.012451 + 0.718606I
b = 0.46087 + 1.48148I
1.48241 4.33310I 1.64326 + 6.16138I
u = 0.903696 0.627740I
a = 0.967990 + 0.934692I
b = 0.32828 + 3.01721I
7.52118 + 7.54900I 1.95510 5.67894I
u = 0.903696 + 0.627740I
a = 0.967990 0.934692I
b = 0.32828 3.01721I
7.52118 7.54900I 1.95510 + 5.67894I
u = 1.022629 0.390914I
a = 1.047547 0.555883I
b = 1.68232 2.68934I
1.08244 + 3.51008I 2.94382 4.50441I
u = 1.022629 + 0.390914I
a = 1.047547 + 0.555883I
b = 1.68232 + 2.68934I
1.08244 3.51008I 2.94382 + 4.50441I
u = 1.047031 0.274962I
a = 0.084328 + 0.489880I
b = 0.38490 + 1.90238I
3.02612 + 0.98819I 10.81757 0.62676I
u = 1.047031 + 0.274962I
a = 0.084328 0.489880I
b = 0.38490 1.90238I
3.02612 0.98819I 10.81757 + 0.62676I
u = 1.162780 0.205267I
a = 1.072523 + 0.089182I
b = 2.41449 0.38325I
0.59955 1.29811I 2.15451 + 4.77022I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.162780 + 0.205267I
a = 1.072523 0.089182I
b = 2.41449 + 0.38325I
0.59955 + 1.29811I 2.15451 4.77022I
u = 1.302134 0.031330I
a = 0.329965 0.455290I
b = 0.16727 1.71576I
0.07504 1.93289I 0.05787 + 3.96687I
u = 1.302134 + 0.031330I
a = 0.329965 + 0.455290I
b = 0.16727 + 1.71576I
0.07504 + 1.93289I 0.05787 3.96687I
u = 1.341852 0.047444I
a = 1.090634 + 0.129231I
b = 0.99274 + 1.63051I
6.09964 4.98197I 3.02870 + 3.56815I
u = 1.341852 + 0.047444I
a = 1.090634 0.129231I
b = 0.99274 1.63051I
6.09964 + 4.98197I 3.02870 3.56815I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
55
+ 6u
54
+ ··· 8u 1)
c
2
(u + 1)
5
(u
55
+ 24u
54
+ ··· + 8u + 1)
c
3
, c
7
u
5
(u
55
+ u
54
+ ··· + 64u + 32)
c
4
(u + 1)
5
(u
55
+ 6u
54
+ ··· 8u 1)
c
5
(u
5
u
4
+ ··· + u + 1)(u
55
+ 2u
54
+ ··· + 2u 1)
c
6
(u
5
+ 3u
4
+ ··· u 1)(u
55
+ 6u
54
+ ··· + 302u + 77)
c
8
(u
5
+ u
4
+ ··· + u + 1)(u
55
+ 8u
54
+ ··· + 94u + 7)
c
9
, c
10
(u
5
+ u
4
+ ··· + u 1)(u
55
+ 2u
54
+ ··· + 2u 1)
c
11
(u
5
u
4
+ ··· + u 1)(u
55
+ 8u
54
+ ··· + 94u + 7)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
55
24y
54
+ ··· + 8y 1)
c
2
(y 1)
5
(y
55
+ 20y
54
+ ··· 220y 1)
c
3
, c
7
y
5
(y
55
+ 33y
54
+ ··· 14848y 1024)
c
5
, c
9
(y
5
5y
4
+ ··· y 1)(y
55
52y
54
+ ··· + 14y 1)
c
6
1.000000000000
(1y
5
y
4
+ ··· + 3.00000000000y 1.000000000000)
(1.00y
55
20.0y
54
+ ··· + 1.46 × 10
5
y 5.93 × 10
3
)
c
8
, c
11
(y
5
+ 3y
4
+ ··· y 1)(y
55
+ 48y
54
+ ··· + 4314y 49)
c
10
1.00000000000
(1y
5
5.0000000000y
4
+ ··· y 1.00000000000)
(1y
55
52.000000000y
54
+ ··· + 14.0000000000y 1.00000000000)
12