11a
42
(K11a
42
)
1
Arc Sequences
4 1 7 2 10 8 3 6 11 5 9
Solving Sequence
5,10
6
2,11
4 1 3 9 8 7
c
5
c
10
c
4
c
1
c
2
c
9
c
8
c
7
c
3
, c
6
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
2
+ 3, u 1, b + 2a 1i
I
u
2
= hu
55
3u
54
+ ··· 5u + 1, u
54
+ 2u
53
+ ··· + b + 1, u
54
+ 2u
53
+ ··· + 2a 3i
There are 2 irreducible components with 57 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
+ 3, u 1, b + 2a 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
1
2
b +
1
2
b
a
6
=
1
2
b
1
2
1
2
b
1
2
a
2
=
1
0
a
11
=
1
2
b +
1
2
1
2
b
1
2
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
9
=
1
2
b
1
2
1
2
b
1
2
a
8
=
1
2
b
1
2
1
2
b
1
2
a
7
=
1
2
b
1
2
1
2
b
1
2
a
7
=
1
2
b
1
2
1
2
b
1
2
(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 = 0.500000 0.866025I
b = 1.73205I
1.64493 2.02988I 9.00000 + 3.46410I
u = 1.00000
a = 0.500000 + 0.866025I
b = 1.73205I
1.64493 + 2.02988I 9.00000 3.46410I
3
II.
I
u
2
= hu
55
3u
54
+· · ·−5u+1, u
54
+2u
53
+· · ·+b+1, u
54
+2u
53
+· · ·+2a3i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
2
u
54
u
53
+ ··· + u +
3
2
u
54
2u
53
+ ··· + 5u 1
a
6
=
9
2
u
54
10u
53
+ ··· + 25u
11
2
11
2
u
54
13u
53
+ ··· + 29u
15
2
a
2
=
1
0
a
11
=
1
2
u
54
u
53
+ ··· + u +
3
2
1
2
u
54
u
53
+ ··· + 3u
1
2
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
9
=
11
2
u
54
12u
53
+ ··· + 27u
11
2
3
2
u
54
3u
53
+ ··· + 6u
3
2
a
8
=
15
2
u
54
16u
53
+ ··· + 35u
15
2
11
2
u
54
11u
53
+ ··· + 22u
11
2
a
7
=
13
2
u
54
14u
53
+ ··· + 34u
15
2
11
2
u
54
13u
53
+ ··· + 29u
15
2
a
7
=
13
2
u
54
14u
53
+ ··· + 34u
15
2
11
2
u
54
13u
53
+ ··· + 29u
15
2
(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 = 1.261435 0.376282I
a = 0.697095 + 0.617164I
b = 0.16305 + 2.27697I
5.43130 + 5.17320I 6.90069 4.73531I
u = 1.261435 + 0.376282I
a = 0.697095 0.617164I
b = 0.16305 2.27697I
5.43130 5.17320I 6.90069 + 4.73531I
u = 1.231540 0.429492I
a = 0.274632 0.808651I
b = 0.09608 3.46895I
9.45702 1.46557I 10.98966 + 0.70313I
u = 1.231540 + 0.429492I
a = 0.274632 + 0.808651I
b = 0.09608 + 3.46895I
9.45702 + 1.46557I 10.98966 0.70313I
u = 1.228782 0.367943I
a = 0.429684 0.565269I
b = 0.568777 1.209213I
3.97885 0.10775I 4.88979 + 0.08750I
u = 1.228782 + 0.367943I
a = 0.429684 + 0.565269I
b = 0.568777 + 1.209213I
3.97885 + 0.10775I 4.88979 0.08750I
u = 1.203818 0.035992I
a = 0.542692 + 0.696990I
b = 0.16653 + 1.87259I
0.64263 + 2.45826I 0.00992 4.47694I
u = 1.203818 + 0.035992I
a = 0.542692 0.696990I
b = 0.16653 1.87259I
0.64263 2.45826I 0.00992 + 4.47694I
u = 1.193238 0.473729I
a = 0.177519 + 0.944542I
b = 0.64456 + 3.70015I
5.21292 8.08647I 6.73043 + 5.85393I
u = 1.193238 + 0.473729I
a = 0.177519 0.944542I
b = 0.64456 3.70015I
5.21292 + 8.08647I 6.73043 5.85393I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.176884 0.449966I
a = 0.721653 0.199367I
b = 1.352345 + 0.099845I
3.79702 2.69601I 4.74745 + 1.11286I
u = 1.176884 + 0.449966I
a = 0.721653 + 0.199367I
b = 1.352345 0.099845I
3.79702 + 2.69601I 4.74745 1.11286I
u = 1.051281 0.242889I
a = 0.075607 0.693528I
b = 0.82609 1.96180I
3.08639 0.93841I 10.69919 0.06174I
u = 1.051281 + 0.242889I
a = 0.075607 + 0.693528I
b = 0.82609 + 1.96180I
3.08639 + 0.93841I 10.69919 + 0.06174I
u = 0.861415
a = 0.246368
b = 0.533730
1.25349 8.32256
u = 0.826919 0.421607I
a = 0.195127 + 1.184787I
b = 1.83856 + 1.65528I
0.57728 4.35301I 2.97848 + 6.56505I
u = 0.826919 + 0.421607I
a = 0.195127 1.184787I
b = 1.83856 1.65528I
0.57728 + 4.35301I 2.97848 6.56505I
u = 0.733611 0.391605I
a = 1.152346 0.477462I
b = 0.136647 + 1.276548I
0.872499 + 0.772147I 1.84032 + 0.30762I
u = 0.733611 + 0.391605I
a = 1.152346 + 0.477462I
b = 0.136647 1.276548I
0.872499 0.772147I 1.84032 0.30762I
u = 0.051867 0.757851I
a = 1.40670 + 0.73471I
b = 0.988967 + 0.906739I
1.91291 + 3.57990I 3.62103 2.40810I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.051867 + 0.757851I
a = 1.40670 0.73471I
b = 0.988967 0.906739I
1.91291 3.57990I 3.62103 + 2.40810I
u = 0.025988 0.692151I
a = 0.756561 + 0.939481I
b = 0.181465 + 0.013937I
0.47649 1.47491I 0.96799 + 2.91910I
u = 0.025988 + 0.692151I
a = 0.756561 0.939481I
b = 0.181465 0.013937I
0.47649 + 1.47491I 0.96799 2.91910I
u = 0.053635 0.828871I
a = 1.330235 0.044660I
b = 1.253514 0.127614I
5.60698 2.95534I 7.15421 + 2.80412I
u = 0.053635 + 0.828871I
a = 1.330235 + 0.044660I
b = 1.253514 + 0.127614I
5.60698 + 2.95534I 7.15421 2.80412I
u = 0.059148 0.370271I
a = 1.97231 + 0.66266I
b = 0.080965 0.338748I
0.194248 1.398021I 2.04153 + 4.84692I
u = 0.059148 + 0.370271I
a = 1.97231 0.66266I
b = 0.080965 + 0.338748I
0.194248 + 1.398021I 2.04153 4.84692I
u = 0.134773 0.873581I
a = 1.279473 0.574810I
b = 1.055753 0.729357I
1.08802 9.45303I 2.37560 + 7.01645I
u = 0.134773 + 0.873581I
a = 1.279473 + 0.574810I
b = 1.055753 + 0.729357I
1.08802 + 9.45303I 2.37560 7.01645I
u = 0.147161 0.837390I
a = 0.525785 1.023071I
b = 0.112135 0.160240I
0.24633 3.95028I 0.20932 + 2.43300I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.147161 + 0.837390I
a = 0.525785 + 1.023071I
b = 0.112135 + 0.160240I
0.24633 + 3.95028I 0.20932 2.43300I
u = 0.517863 0.687312I
a = 1.343645 0.033063I
b = 0.443588 0.610860I
4.96946 3.90710I 2.99798 + 3.61065I
u = 0.517863 + 0.687312I
a = 1.343645 + 0.033063I
b = 0.443588 + 0.610860I
4.96946 + 3.90710I 2.99798 3.61065I
u = 0.579824 0.662419I
a = 0.135598 1.236030I
b = 0.583268 0.853194I
5.27340 + 1.63575I 3.78529 2.66903I
u = 0.579824 + 0.662419I
a = 0.135598 + 1.236030I
b = 0.583268 + 0.853194I
5.27340 1.63575I 3.78529 + 2.66903I
u = 0.763301 0.197825I
a = 0.861337 1.043039I
b = 0.18600 1.58276I
0.85220 1.53742I 0.475415 0.770388I
u = 0.763301 + 0.197825I
a = 0.861337 + 1.043039I
b = 0.18600 + 1.58276I
0.85220 + 1.53742I 0.475415 + 0.770388I
u = 0.769448 0.437822I
a = 0.492087 + 0.418648I
b = 0.143587 + 0.364706I
1.33757 + 1.88655I 2.60483 4.69845I
u = 0.769448 + 0.437822I
a = 0.492087 0.418648I
b = 0.143587 0.364706I
1.33757 1.88655I 2.60483 + 4.69845I
u = 0.916664 0.588628I
a = 0.959390 + 0.200687I
b = 0.531300 0.436341I
4.30983 + 3.18273I 1.66711 3.75336I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.916664 + 0.588628I
a = 0.959390 0.200687I
b = 0.531300 + 0.436341I
4.30983 3.18273I 1.66711 + 3.75336I
u = 0.941364 0.397946I
a = 0.160282 + 0.980063I
b = 0.12527 + 2.30115I
1.92448 + 4.39584I 6.07569 8.49319I
u = 0.941364 + 0.397946I
a = 0.160282 0.980063I
b = 0.12527 2.30115I
1.92448 4.39584I 6.07569 + 8.49319I
u = 0.963009 0.589107I
a = 0.101646 1.051558I
b = 0.81552 2.34440I
3.69457 + 8.78385I 0.00690 9.18113I
u = 0.963009 + 0.589107I
a = 0.101646 + 1.051558I
b = 0.81552 + 2.34440I
3.69457 8.78385I 0.00690 + 9.18113I
u = 1.177722 0.463008I
a = 0.429842 + 0.542675I
b = 0.605265 + 0.974978I
3.70073 + 5.77207I 4.07257 5.94241I
u = 1.177722 + 0.463008I
a = 0.429842 0.542675I
b = 0.605265 0.974978I
3.70073 5.77207I 4.07257 + 5.94241I
u = 1.194958 0.434256I
a = 0.738501 0.621216I
b = 0.30334 2.22969I
5.49641 + 0.62899I 7.13648 1.17442I
u = 1.194958 + 0.434256I
a = 0.738501 + 0.621216I
b = 0.30334 + 2.22969I
5.49641 0.62899I 7.13648 + 1.17442I
u = 1.204847 0.521092I
a = 0.744462 + 0.155027I
b = 1.226821 + 0.039906I
2.90431 + 8.91686I 3.42760 5.44869I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.204847 + 0.521092I
a = 0.744462 0.155027I
b = 1.226821 0.039906I
2.90431 8.91686I 3.42760 + 5.44869I
u = 1.219037 0.483533I
a = 0.286862 + 0.821520I
b = 0.12471 + 3.40599I
9.06604 + 7.69131I 10.03059 5.89654I
u = 1.219037 + 0.483533I
a = 0.286862 0.821520I
b = 0.12471 3.40599I
9.06604 7.69131I 10.03059 + 5.89654I
u = 1.221341 0.526305I
a = 0.155573 0.933221I
b = 0.36313 3.58809I
4.3508 + 14.5336I 5.51307 9.84314I
u = 1.221341 + 0.526305I
a = 0.155573 + 0.933221I
b = 0.36313 + 3.58809I
4.3508 14.5336I 5.51307 + 9.84314I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u
55
+ 3u
54
+ ··· 5u 1)
c
2
(u + 1)
2
(u
55
+ 31u
54
+ ··· 7u + 1)
c
3
, c
7
u
2
(u
55
+ u
54
+ ··· + 8u + 4)
c
4
(u + 1)
2
(u
55
+ 3u
54
+ ··· 5u 1)
c
5
(u
2
+ u + 1)(u
55
+ 2u
54
+ ··· + 2u + 1)
c
6
, c
8
u
2
(u
55
+ 15u
54
+ ··· 168u 16)
c
9
(u
2
u + 1)(u
55
+ 20u
54
+ ··· + 14u 1)
c
10
(u
2
u + 1)(u
55
+ 2u
54
+ ··· + 2u + 1)
c
11
(u
2
+ u + 1)(u
55
+ 20u
54
+ ··· + 14u 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
2
(y
55
31y
54
+ ··· 7y 1)
c
2
(y 1)
2
(y
55
11y
54
+ ··· + 101y 1)
c
3
, c
7
y
2
(y
55
+ 15y
54
+ ··· 168y 16)
c
5
, c
10
(y
2
+ y + 1)(y
55
+ 20y
54
+ ··· + 14y 1)
c
6
, c
8
y
2
(y
55
+ 47y
54
+ ··· + 3616y 256)
c
9
, c
11
+ 1.00000000000000(1y
2
+ 1y + 1.00000000000000)
(1.00y
55
+ 32.0y
54
+ ··· + 490.y 1.00)
12