11a
121
(K11a
121
)
1
Arc Sequences
6 1 10 8 2 5 3 11 4 9 7
Solving Sequence
1,6
2 3 5 7 8 4 11 9 10
c
1
c
2
c
5
c
6
c
7
c
4
c
11
c
8
c
10
c
3
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
11
+ 2u
9
+ 4u
7
+ 4u
5
u
4
+ 3u
3
u
2
+ 2u 1i
I
u
2
= hu
48
u
47
+ ··· + 2u + 1i
There are 2 irreducible components with 59 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
11
+ 2u
9
+ 4u
7
+ 4u
5
u
4
+ 3u
3
u
2
+ 2u 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
7
=
u
3
u
5
+ u
3
+ u
a
8
=
u
9
+ 2u
7
+ 3u
5
+ 2u
3
+ u
u
9
u
7
u
5
+ u
a
4
=
u
8
u
7
+ u
6
2u
5
+ 2u
4
2u
3
+ u
2
2u
u
10
2u
8
+ u
7
3u
6
+ u
5
3u
4
+ 2u
3
u
2
+ u
a
11
=
u
8
u
6
u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
9
=
u
10
+ u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ 2u
3
u
2
+ u
u
9
+ u
8
u
7
+ 2u
6
2u
5
+ 2u
4
u
3
+ 2u
2
a
10
=
u
10
+ 2u
8
+ u
7
+ 3u
6
+ u
5
+ 2u
4
+ u
3
+ u
2
+ u
u
9
2u
7
+ u
6
3u
5
+ u
4
3u
3
+ 2u
2
2u + 1
a
10
=
u
10
+ 2u
8
+ u
7
+ 3u
6
+ u
5
+ 2u
4
+ u
3
+ u
2
+ u
u
9
2u
7
+ u
6
3u
5
+ u
4
3u
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 = 0.817015 0.707633I
6.68078 3.13136I 8.76083 + 0.56604I
u = 0.817015 + 0.707633I
6.68078 + 3.13136I 8.76083 0.56604I
u = 0.617277 0.966546I
0.07920 + 7.68222I 0.97285 8.49443I
u = 0.617277 + 0.966546I
0.07920 7.68222I 0.97285 + 8.49443I
u = 0.111009 1.030814I
5.93919 + 3.55367I 6.35449 4.86751I
u = 0.111009 + 1.030814I
5.93919 3.55367I 6.35449 + 4.86751I
u = 0.444369
0.869046 11.7657
u = 0.594105 0.723647I
1.48764 1.96750I 4.49213 + 3.23948I
u = 0.594105 + 0.723647I
1.48764 + 1.96750I 4.49213 3.23948I
u = 0.729012 1.011347I
4.8176 14.7555I 5.24582 + 10.31160I
u = 0.729012 + 1.011347I
4.8176 + 14.7555I 5.24582 10.31160I
3
II. I
u
2
= hu
48
u
47
+ · · · + 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
7
=
u
3
u
5
+ u
3
+ u
a
8
=
u
9
+ 2u
7
+ 3u
5
+ 2u
3
+ u
u
9
u
7
u
5
+ u
a
4
=
u
21
4u
19
+ ··· 2u
3
u
u
21
+ 3u
19
+ 7u
17
+ 10u
15
+ 10u
13
+ 7u
11
+ u
9
2u
7
3u
5
u
3
+ u
a
11
=
u
8
u
6
u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
9
=
u
27
4u
25
+ ··· + 10u
5
+ 3u
3
u
29
+ 5u
27
+ ··· u
3
+ u
a
10
=
u
46
7u
44
+ ··· 4u
4
+ 1
u
47
+ 7u
45
+ ··· 2u 1
a
10
=
u
46
7u
44
+ ··· 4u
4
+ 1
u
47
+ 7u
45
+ ··· 2u 1
(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 = 0.798379 0.782289I
7.98533 10.0430
u = 0.798379 + 0.782289I
7.98533 10.0430
u = 0.779513 0.728650I
3.78730 1.05884I 9.33375 + 1.03697I
u = 0.779513 + 0.728650I
3.78730 + 1.05884I 9.33375 1.03697I
u = 0.748039 0.955471I
7.45278 + 5.81585I 8.97012 5.48927I
u = 0.748039 + 0.955471I
7.45278 5.81585I 8.97012 + 5.48927I
u = 0.730704 1.006044I
5.77023 + 8.94227I 7.03302 5.48937I
u = 0.730704 + 1.006044I
5.77023 8.94227I 7.03302 + 5.48937I
u = 0.718495 0.983429I
3.01107 + 6.72706I 7.45449 6.34172I
u = 0.718495 + 0.983429I
3.01107 6.72706I 7.45449 + 6.34172I
u = 0.619099 0.144052I
3.01107 + 6.72706I 7.45449 6.34172I
u = 0.619099 + 0.144052I
3.01107 6.72706I 7.45449 + 6.34172I
u = 0.559504 0.928720I
3.47428 + 2.08425I 3.81787 2.59078I
u = 0.559504 + 0.928720I
3.47428 2.08425I 3.81787 + 2.59078I
u = 0.520871 0.529700I
0.94545 3.01303I 3.90717 + 2.47987I
u = 0.520871 + 0.529700I
0.94545 + 3.01303I 3.90717 2.47987I
u = 0.516429 0.228211I
2.07060 + 1.71275I 0.95839 4.38827I
u = 0.516429 + 0.228211I
2.07060 1.71275I 0.95839 + 4.38827I
u = 0.447030 0.894068I
0.85406 3.28062I 1.88284 + 1.76353I
u = 0.447030 + 0.894068I
0.85406 + 3.28062I 1.88284 1.76353I
u = 0.156596 1.043695I
0.78809 + 9.12338I 0.18249 8.13527I
u = 0.156596 + 1.043695I
0.78809 9.12338I 0.18249 + 8.13527I
u = 0.037970 1.018318I
3.47428 2.08425I 3.81787 + 2.59078I
u = 0.037970 + 1.018318I
3.47428 + 2.08425I 3.81787 2.59078I
u = 0.103335 0.964930I
2.07060 1.71275I 0.95839 + 4.38827I
u = 0.103335 + 0.964930I
2.07060 + 1.71275I 0.95839 4.38827I
u = 0.161802 1.028075I
0.15229 3.45771I 1.61918 + 3.33537I
u = 0.161802 + 1.028075I
0.15229 + 3.45771I 1.61918 3.33537I
u = 0.357761 0.828361I
1.54659 2.07802I 3.74247 + 4.14356I
u = 0.357761 + 0.828361I
1.54659 + 2.07802I 3.74247 4.14356I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.605322 0.114770I
3.78730 1.05884I 9.33375 + 1.03697I
u = 0.605322 + 0.114770I
3.78730 + 1.05884I 9.33375 1.03697I
u = 0.644764 0.924836I
0.94545 3.01303I 3.90717 + 2.47987I
u = 0.644764 + 0.924836I
0.94545 + 3.01303I 3.90717 2.47987I
u = 0.684868 0.970999I
0.85406 3.28062I 1.88284 + 1.76353I
u = 0.684868 + 0.970999I
0.85406 + 3.28062I 1.88284 1.76353I
u = 0.709249 0.753994I
1.54659 2.07802I 3.74247 + 4.14356I
u = 0.709249 + 0.753994I
1.54659 + 2.07802I 3.74247 4.14356I
u = 0.712082 1.003136I
0.78809 9.12338I 0.18249 + 8.13527I
u = 0.712082 + 1.003136I
0.78809 + 9.12338I 0.18249 8.13527I
u = 0.750786 0.945298I
6.98909 8.15485
u = 0.750786 + 0.945298I
6.98909 8.15485
u = 0.786969 0.691947I
0.15229 + 3.45771I 1.61918 3.33537I
u = 0.786969 + 0.691947I
0.15229 3.45771I 1.61918 + 3.33537I
u = 0.795531 0.794799I
7.45278 5.81585I 8.97012 + 5.48927I
u = 0.795531 + 0.794799I
7.45278 + 5.81585I 8.97012 5.48927I
u = 0.820160 0.698926I
5.77023 + 8.94227I 7.03302 5.48937I
u = 0.820160 + 0.698926I
5.77023 8.94227I 7.03302 + 5.48937I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
5
c
9
(u
11
+ 2u
9
+ ··· + 2u + 1)(u
48
+ u
47
+ ··· 2u + 1)
c
2
, c
6
, c
8
(u
11
+ 4u
10
+ ··· + 2u 1)(u
48
+ 15u
47
+ ··· + 8u
3
+ 1)
c
4
, c
11
(u
11
+ 2u
9
2u
8
+ 10u
7
+ 12u
5
3u
4
+ 5u
3
u
2
1)
(u
48
+ 5u
47
+ ··· + 12u + 1)
c
7
(u
11
+ 7u
10
+ ··· + 28u + 8)
(7 18u + 8u
2
+ 22u
3
29u
4
+ 18u
5
27u
6
+ 9u
7
+ 40u
8
18u
9
38u
10
4u
11
+ 52u
12
+ 6u
13
32u
14
17u
15
+ 15u
16
+ 16u
17
+ 4u
18
21u
19
+ 3u
20
+ 7u
21
3u
23
+ u
24
)
2
c
10
(u + 1)(u
11
+ 4u
10
+ ··· + 2u 1)(u
47
+ 14u
46
+ ··· u + 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
5
c
9
(y
11
+ 4y
10
+ ··· + 2y 1)(y
48
+ 15y
47
+ ··· + 8y
3
+ 1)
c
2
, c
6
, c
8
(y
11
+ 8y
10
+ ··· + 22y 1)(y
48
+ 35y
47
+ ··· + 88y
2
+ 1)
c
4
, c
11
(y
11
+ 4y
10
+ ··· 2y 1)(y
48
5y
47
+ ··· 24y + 1)
c
7
(y
11
3y
10
+ ··· + 48y 64)
(49 212y + 450y
2
678y
3
+ 501y
4
+ 306y
5
1147y
6
+ 1387y
7
74y
8
2840y
9
+ 5468y
10
6400y
11
+ 5448y
12
3632y
13
+ 2020y
14
1169y
15
+ 1105y
16
1226y
17
+ 1156y
18
807y
19
+ 429y
20
167y
21
+ 48y
22
9y
23
+ y
24
)
2
c
10
(y
11
+ 8y
10
+ ··· + 22y 1)(y
48
+ 35y
47
+ ··· + 88y
2
+ 1)
8