11a
365
(K11a
365
)
1
Arc Sequences
7 8 10 9 1 11 2 3 4 5 6
Solving Sequence
4,10
3 9 5 11 8 2 7 1 6
c
3
c
9
c
4
c
10
c
8
c
2
c
7
c
1
c
6
c
5
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
9
+ 4u
7
u
6
+ 5u
5
3u
4
2u
2
3u + 1i
I
u
2
= hu
16
+ u
15
+ 6u
14
+ 6u
13
+ 15u
12
+ 15u
11
+ 17u
10
+ 17u
9
+ 4u
8
+ 4u
7
8u
6
8u
5
4u
4
4u
3
+ 2u
2
+ 2u + 1i
There are 2 irreducible components with 25 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
9
+ 4u
7
u
6
+ 5u
5
3u
4
2u
2
3u + 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
3
=
1
u
2
a
9
=
u
u
a
5
=
u
2
+ 1
u
2
a
11
=
u
5
2u
3
u
u
5
u
3
+ u
a
8
=
u
3
+ 2u
u
5
u
3
+ u
a
2
=
u
6
3u
4
2u
2
+ 1
u
8
+ 2u
6
2u
2
a
7
=
u
6
3u
4
2u
2
+ 1
u
8
+ u
7
+ 2u
6
+ 2u
5
u
4
3u
2
2u + 1
a
1
=
u
3
+ 2u
u
8
u
7
+ 3u
6
4u
5
+ 3u
4
3u
3
+ 3u 1
a
6
=
u
4
u
2
+ 1
u
8
+ u
7
+ 3u
6
+ 2u
5
4u
2
2u + 1
a
6
=
u
4
u
2
+ 1
u
8
+ u
7
+ 3u
6
+ 2u
5
4u
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 = 0.704803
4.75227 19.3449
u = 0.445037 1.304011I
6.66561 9.83268I 11.48734 + 5.80501I
u = 0.445037 + 1.304011I
6.66561 + 9.83268I 11.48734 5.80501I
u = 0.092398 1.291150I
7.68628 2.63224I 3.26146 + 3.89078I
u = 0.092398 + 1.291150I
7.68628 + 2.63224I 3.26146 3.89078I
u = 0.278445
0.461193 21.4244
u = 0.285490 1.280784I
3.22608 + 7.14899I 8.72219 6.90579I
u = 0.285490 + 1.280784I
3.22608 7.14899I 8.72219 + 6.90579I
u = 0.930248
14.7946 18.2887
3
II. I
u
2
= hu
16
+ u
15
+ 6u
14
+ 6u
13
+ 15u
12
+ 15u
11
+ 17u
10
+ 17u
9
+ 4u
8
+
4u
7
8u
6
8u
5
4u
4
4u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
3
=
1
u
2
a
9
=
u
u
a
5
=
u
2
+ 1
u
2
a
11
=
u
5
2u
3
u
u
5
u
3
+ u
a
8
=
u
3
+ 2u
u
5
u
3
+ u
a
2
=
u
6
3u
4
2u
2
+ 1
u
8
+ 2u
6
2u
2
a
7
=
u
9
4u
7
5u
5
+ 3u
u
11
+ 3u
9
+ 2u
7
3u
5
3u
3
+ u
a
1
=
u
12
+ 5u
10
+ 9u
8
+ 4u
6
6u
4
5u
2
+ 1
u
14
4u
12
5u
10
+ 2u
8
+ 8u
6
+ 2u
4
3u
2
a
6
=
2u
15
+ 12u
13
+ 29u
11
+ 28u
9
6u
7
30u
5
+ u
4
11u
3
+ 3u
2
+ 6u + 3
u
13
+ 5u
11
+ 9u
9
+ 4u
7
6u
5
+ u
4
5u
3
+ 2u
2
+ u
a
6
=
2u
15
+ 12u
13
+ 29u
11
+ 28u
9
6u
7
30u
5
+ u
4
11u
3
+ 3u
2
+ 6u + 3
u
13
+ 5u
11
+ 9u
9
+ 4u
7
6u
5
+ u
4
5u
3
+ 2u
2
+ u
(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 = 0.926940 0.018527I
10.78258 4.93524I 14.9844 + 2.9942I
u = 0.926940 + 0.018527I
10.78258 + 4.93524I 14.9844 2.9942I
u = 0.457337 1.275724I
6.88602 11.8221
u = 0.457337 + 1.275724I
6.88602 11.8221
u = 0.329483 0.355718I
2.79859 1.27532I 9.18053 + 5.08518I
u = 0.329483 + 0.355718I
2.79859 + 1.27532I 9.18053 5.08518I
u = 0.300887 1.216991I
1.05533 3.63283I 14.4224 + 4.5180I
u = 0.300887 + 1.216991I
1.05533 + 3.63283I 14.4224 4.5180I
u = 0.076587 1.175000I
2.79859 + 1.27532I 9.18053 5.08518I
u = 0.076587 + 1.175000I
2.79859 1.27532I 9.18053 + 5.08518I
u = 0.289289 1.118510I
1.93558 11.0032
u = 0.289289 + 1.118510I
1.93558 11.0032
u = 0.453425 1.291549I
10.78258 + 4.93524I 14.9844 2.9942I
u = 0.453425 + 1.291549I
10.78258 4.93524I 14.9844 + 2.9942I
u = 0.695347 0.104492I
1.05533 + 3.63283I 14.4224 4.5180I
u = 0.695347 + 0.104492I
1.05533 3.63283I 14.4224 + 4.5180I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
7
c
8
, c
10
(u
8
u
7
5u
6
+ 4u
5
+ 7u
4
4u
3
2u
2
+ 2u 1)
2
(u
9
+ 3u
8
u
7
8u
6
u
5
+ 8u
4
+ 6u
3
7u 2)
c
3
, c
4
, c
5
c
6
, c
9
, c
11
(u
9
+ 4u
7
+ ··· 3u + 1)(u
16
+ u
15
+ ··· + 2u + 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
7
c
8
, c
10
(y
8
11y
7
+ 47y
6
98y
5
+ 103y
4
50y
3
+ 6y
2
+ 1)
2
(y
9
11y
8
+ 47y
7
98y
6
+ 103y
5
50y
4
+ 18y
3
52y
2
+ 49y 4)
c
3
, c
4
, c
5
c
6
, c
9
, c
11
(y
9
+ 8y
8
+ 26y
7
+ 39y
6
+ 13y
5
37y
4
40y
3
+ 2y
2
+ 13y 1)
(y
16
+ 11y
15
+ ··· + 12y
2
+ 1)
7