11a
240
(K11a
240
)
1
Arc Sequences
7 1 10 11 8 9 2 6 3 4 5
Solving Sequence
1,7
2
5,8
11 4 10 3 9 6
c
1
c
7
c
11
c
4
c
10
c
3
c
9
c
6
c
2
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u 1, b, a + u + 1i
I
u
2
= hu
32
2u
31
+ ··· 4u + 1, u
30
u
29
+ ··· + a + u, u
31
20u
29
+ ··· + b 1i
There are 2 irreducible components with 34 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
2
+ u 1, b, a + u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
u 1
0
a
2
=
1
0
a
5
=
0
u
a
8
=
u 1
0
a
11
=
1
u 1
a
4
=
u
u + 1
a
10
=
u
u
a
3
=
1
0
a
9
=
0
u
a
6
=
u 1
u
a
6
=
u 1
u
(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 = 1.61803
a = 0.618034
b = 0
10.5276 15.0000
u = 0.618034
a = 1.61803
b = 0
2.63189 15.0000
3
II.
I
u
2
= hu
32
2u
31
+· · · 4u+1, u
30
u
29
+· · · +a+u, u
31
20u
29
+· · · +b1i
(i) Arc colorings
a
1
=
1
0
a
7
=
u
30
+ u
29
+ ··· + u
2
u
u
31
+ 20u
29
+ ··· 2u + 1
a
2
=
u
5
4u
3
+ 3u
u
5
3u
3
+ u
a
5
=
0
u
a
8
=
u
30
+ u
29
+ ··· + 2u 1
u
31
20u
29
+ ··· + 5u 1
a
11
=
1
u
2
a
4
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
2u
2
a
3
=
u
3
+ 2u
u
5
3u
3
+ u
a
9
=
u
4
3u
2
+ 1
u
6
+ 4u
4
3u
2
a
6
=
u
30
+ u
29
+ ··· 6u
2
+ u
u
16
+ 10u
14
+ ··· 4u
2
+ 2u
a
6
=
u
30
+ u
29
+ ··· 6u
2
+ u
u
16
+ 10u
14
+ ··· 4u
2
+ 2u
(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 = 1.70563 0.05938I
a = 0.290510 0.632941I
b = 0.439191 1.152011I
14.2827 4.0537I 18.8548 + 2.1790I
u = 1.70563 + 0.05938I
a = 0.290510 + 0.632941I
b = 0.439191 + 1.152011I
14.2827 + 4.0537I 18.8548 2.1790I
u = 1.65686 0.03953I
a = 0.212700 + 0.393808I
b = 0.361390 + 0.762089I
8.99808 1.32195I 12.70571 + 0.82116I
u = 1.65686 + 0.03953I
a = 0.212700 0.393808I
b = 0.361390 0.762089I
8.99808 + 1.32195I 12.70571 0.82116I
u = 1.53142
a = 0.385233
b = 0.905611
11.6098 22.5451
u = 1.21444
a = 0.0978048
b = 1.23178
11.2682 22.0993
u = 0.994935 0.377573I
a = 0.29892 1.68518I
b = 1.187476 0.631599I
7.38074 8.76774I 18.7396 + 7.0546I
u = 0.994935 + 0.377573I
a = 0.29892 + 1.68518I
b = 1.187476 + 0.631599I
7.38074 + 8.76774I 18.7396 7.0546I
u = 0.932935 0.300495I
a = 0.72249 + 1.59561I
b = 1.007085 + 0.529996I
2.01989 4.86523I 15.1954 + 6.8122I
u = 0.932935 + 0.300495I
a = 0.72249 1.59561I
b = 1.007085 0.529996I
2.01989 + 4.86523I 15.1954 6.8122I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.910087 0.140122I
a = 1.33113 0.94875I
b = 0.879985 0.249008I
3.85845 0.52237I 20.0729 + 1.6653I
u = 0.910087 + 0.140122I
a = 1.33113 + 0.94875I
b = 0.879985 + 0.249008I
3.85845 + 0.52237I 20.0729 1.6653I
u = 0.130280 0.363295I
a = 0.39628 2.70995I
b = 0.351495 0.640635I
1.55433 0.80952I 9.54426 + 1.40879I
u = 0.130280 + 0.363295I
a = 0.39628 + 2.70995I
b = 0.351495 + 0.640635I
1.55433 + 0.80952I 9.54426 1.40879I
u = 0.109732 0.502858I
a = 0.82718 + 1.88465I
b = 0.811978 + 0.521589I
1.17199 + 2.12258I 8.07273 5.17972I
u = 0.109732 + 0.502858I
a = 0.82718 1.88465I
b = 0.811978 0.521589I
1.17199 2.12258I 8.07273 + 5.17972I
u = 0.187060 0.621355I
a = 1.29407 1.72242I
b = 1.099542 0.526509I
3.74023 + 5.38912I 14.5723 5.7053I
u = 0.187060 + 0.621355I
a = 1.29407 + 1.72242I
b = 1.099542 + 0.526509I
3.74023 5.38912I 14.5723 + 5.7053I
u = 0.322365
a = 0.344260
b = 0.474707
0.607216 16.6585
u = 0.588921 0.485955I
a = 1.216466 + 0.243812I
b = 1.054603 0.362834I
4.95979 1.69559I 17.9188 0.0178I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.588921 + 0.485955I
a = 1.216466 0.243812I
b = 1.054603 + 0.362834I
4.95979 + 1.69559I 17.9188 + 0.0178I
u = 0.718238 0.225952I
a = 0.544839 + 0.269319I
b = 0.498635 + 0.548258I
0.558920 + 0.474938I 11.62959 1.27773I
u = 0.718238 + 0.225952I
a = 0.544839 0.269319I
b = 0.498635 0.548258I
0.558920 0.474938I 11.62959 + 1.27773I
u = 0.946587 0.231196I
a = 0.590079 0.867745I
b = 0.393556 0.988578I
4.86072 + 2.90543I 17.9793 3.5680I
u = 0.946587 + 0.231196I
a = 0.590079 + 0.867745I
b = 0.393556 + 0.988578I
4.86072 2.90543I 17.9793 + 3.5680I
u = 1.69979 0.04032I
a = 1.091759 0.677006I
b = 1.093523 0.307765I
13.16725 + 1.26120I 19.5964 0.3072I
u = 1.69979 + 0.04032I
a = 1.091759 + 0.677006I
b = 1.093523 + 0.307765I
13.16725 1.26120I 19.5964 + 0.3072I
u = 1.70039 0.07638I
a = 0.808887 + 1.071757I
b = 1.132711 + 0.560553I
11.32619 + 6.33717I 16.5996 4.8794I
u = 1.70039 + 0.07638I
a = 0.808887 1.071757I
b = 1.132711 0.560553I
11.32619 6.33717I 16.5996 + 4.8794I
u = 1.71578 0.10104I
a = 0.532207 1.164915I
b = 1.25510 0.70710I
16.9357 + 10.6981I 19.7116 5.5874I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.71578 + 0.10104I
a = 0.532207 + 1.164915I
b = 1.25510 + 0.70710I
16.9357 10.6981I 19.7116 + 5.5874I
u = 1.75198
a = 0.513306
b = 1.45783
17.6150 22.3108
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
u
2
(u
32
+ u
31
+ ··· 12u 4)
c
2
u
2
(u
32
+ 15u
31
+ ··· + 152u + 16)
c
3
, c
4
, c
9
c
10
, c
11
(u
2
u 1)(u
32
+ 2u
31
+ ··· + 4u + 1)
c
5
, c
8
(u 1)
2
(u
32
+ 3u
31
+ ··· + 5u 1)
c
6
(u + 1)
2
(u
32
+ 3u
31
+ ··· + 5u 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
y
2
(y
32
15y
31
+ ··· 152y + 16)
c
2
y
2
(y
32
+ y
31
+ ··· 2848y + 256)
c
3
, c
4
, c
9
c
10
, c
11
(y
2
3y + 1)(y
32
42y
31
+ ··· 4y + 1)
c
5
, c
6
, c
8
(y 1)
2
(y
32
29y
31
+ ··· 7y + 1)
9