11a
333
(K11a
333
)
1
Arc Sequences
8 7 1 11 10 3 2 4 6 5 9
Solving Sequence
1,8
2 7 3 4 9 6 10 5 11
c
1
c
7
c
2
c
3
c
8
c
6
c
9
c
5
c
11
c
4
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
8
+ 5u
6
+ 7u
4
+ u
3
+ 2u
2
+ 2u + 1i
I
u
2
= hu
24
+ u
23
+ ··· u
3
+ 1i
There are 2 irreducible components with 32 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
8
+ 5u
6
+ 7u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
2u
2
a
4
=
u
4
+ 3u
2
+ 1
u
4
2u
2
a
9
=
u
7
4u
5
+ u
4
4u
3
+ 2u
2
u
4
2u
2
a
6
=
u
3
2u
u
5
+ 3u
3
+ u
a
10
=
u
7
4u
5
4u
3
u
6
+ 2u
4
u
2
a
5
=
u
6
+ 3u
4
+ 2u
2
+ 1
u
7
+ 3u
5
+ 2u
3
+ u
a
11
=
u
5
+ 2u
3
u
u
6
+ 3u
4
+ u
3
+ 2u
2
+ 2u + 1
a
11
=
u
5
+ 2u
3
u
u
6
+ 3u
4
+ u
3
+ 2u
2
+ 2u + 1
(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 = 0.404853 0.285137I
0.853870 + 0.627235I 8.80552 5.03557I
u = 0.404853 + 0.285137I
0.853870 0.627235I 8.80552 + 5.03557I
u = 0.14255 1.61382I
17.4667 + 10.2751I 4.16626 5.30618I
u = 0.14255 + 1.61382I
17.4667 10.2751I 4.16626 + 5.30618I
u = 0.08626 1.49661I
11.28299 3.64910I 1.10964 + 3.07905I
u = 0.08626 + 1.49661I
11.28299 + 3.64910I 1.10964 3.07905I
u = 0.461135 0.691908I
1.71296 5.69915I 0.47037 + 9.01967I
u = 0.461135 + 0.691908I
1.71296 + 5.69915I 0.47037 9.01967I
3
II. I
u
2
= hu
24
+ u
23
+ · · · u
3
+ 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
2u
2
a
4
=
u
4
+ 3u
2
+ 1
u
4
2u
2
a
9
=
u
9
6u
7
11u
5
6u
3
u
u
9
+ 5u
7
+ 7u
5
+ 2u
3
+ u
a
6
=
u
3
2u
u
5
+ 3u
3
+ u
a
10
=
u
17
10u
15
39u
13
74u
11
71u
9
38u
7
18u
5
4u
3
u
u
19
+ 11u
17
+ 48u
15
+ 105u
13
+ 121u
11
+ 75u
9
+ 30u
7
+ 8u
5
+ u
3
+ u
a
5
=
2u
22
+ 26u
20
+ ··· u + 2
2u
23
+ 26u
21
+ ··· u
2
+ 2u
a
11
=
u
18
+ 11u
16
+ 48u
14
+ 105u
12
+ 121u
10
+ 75u
8
+ 30u
6
+ 8u
4
+ u
2
+ 1
u
18
10u
16
39u
14
74u
12
71u
10
38u
8
18u
6
4u
4
u
2
a
11
=
u
18
+ 11u
16
+ 48u
14
+ 105u
12
+ 121u
10
+ 75u
8
+ 30u
6
+ 8u
4
+ u
2
+ 1
u
18
10u
16
39u
14
74u
12
71u
10
38u
8
18u
6
4u
4
u
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 = 0.591891 0.137722I
7.79349 4.22631I 1.67942 + 2.13120I
u = 0.591891 + 0.137722I
7.79349 + 4.22631I 1.67942 2.13120I
u = 0.489583 0.725679I
9.51380 + 7.90456I 1.91927 6.92574I
u = 0.489583 + 0.725679I
9.51380 7.90456I 1.91927 + 6.92574I
u = 0.409437 0.638189I
0.18849 + 2.30634I 4.56865 4.07548I
u = 0.409437 + 0.638189I
0.18849 2.30634I 4.56865 + 4.07548I
u = 0.288696 0.833188I
10.86471 1.36952I 4.42656 0.88523I
u = 0.288696 + 0.833188I
10.86471 + 1.36952I 4.42656 + 0.88523I
u = 0.11519 1.59101I
7.79349 + 4.22631I 1.67942 2.13120I
u = 0.11519 + 1.59101I
7.79349 4.22631I 1.67942 + 2.13120I
u = 0.07716 1.63217I
19.3156 5.87212
u = 0.07716 + 1.63217I
19.3156 5.87212
u = 0.02617 1.49212I
4.94432 + 1.72225I 2.81956 4.07903I
u = 0.02617 + 1.49212I
4.94432 1.72225I 2.81956 + 4.07903I
u = 0.08387 1.60577I
10.86471 1.36952I 4.42656 0.88523I
u = 0.08387 + 1.60577I
10.86471 + 1.36952I 4.42656 + 0.88523I
u = 0.13255 1.60291I
9.51380 7.90456I 1.91927 + 6.92574I
u = 0.13255 + 1.60291I
9.51380 + 7.90456I 1.91927 6.92574I
u = 0.271534 0.725672I
2.90121 3.57147
u = 0.271534 + 0.725672I
2.90121 3.57147
u = 0.493302 0.448019I
4.94432 1.72225I 2.81956 + 4.07903I
u = 0.493302 + 0.448019I
4.94432 + 1.72225I 2.81956 4.07903I
u = 0.516875 0.160721I
0.18849 + 2.30634I 4.56865 4.07548I
u = 0.516875 + 0.160721I
0.18849 2.30634I 4.56865 + 4.07548I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(u
8
+ 5u
6
+ ··· + 2u + 1)(u
24
+ u
23
+ ··· u
3
+ 1)
c
3
, c
11
(u
8
+ 2u
7
+ 3u
6
+ 5u
4
+ 5u
3
+ 6u
2
+ 2u + 1)
(u
24
+ 7u
23
+ ··· + 94u + 17)
c
8
(u
8
+ 5u
7
+ 11u
6
+ 16u
5
+ 22u
4
+ 27u
3
+ 23u
2
+ 12u + 4)
(3 2u 7u
2
+ 8u
3
+ 2u
4
4u
5
+ 2u
6
4u
7
+ u
8
+ 4u
9
u
10
2u
11
+ u
12
)
2
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(y
8
+ 10y
7
+ 39y
6
+ 74y
5
+ 71y
4
+ 37y
3
+ 14y
2
+ 1)
(y
24
+ 27y
23
+ ··· 2y
2
+ 1)
c
3
, c
11
(y
8
+ 2y
7
+ 19y
6
+ 22y
5
+ 55y
4
+ 41y
3
+ 26y
2
+ 8y + 1)
(y
24
9y
23
+ ··· 2376y + 289)
c
8
(y
8
3y
7
+ 5y
6
+ 4y
5
+ 14y
4
13y
3
+ 57y
2
+ 40y + 16)
(9 46y + 93y
2
96y
3
+ 30y
4
+ 52y
5
76y
6
+ 34y
7
+ 17y
8
30y
9
+ 19y
10
6y
11
+ y
12
)
2
7