11n
64
(K11n
64
)
1
Arc Sequences
4 1 8 2 9 11 10 4 1 7 6
Solving Sequence
1,4
2
3,6
11 7 10 9 5 8
c
1
c
2
c
11
c
6
c
10
c
9
c
5
c
8
c
3
, c
4
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
4
b
3
+ 3b
2
2b + 1, a + 1, u 1i
I
u
2
= hu
14
5u
13
+ u
12
+ 26u
11
13u
10
80u
9
+ 46u
8
+ 124u
7
61u
6
94u
5
+ 50u
4
+ 6u
3
23u
2
+ 7u 1, a 1,
u
13
4u
12
3u
11
+ 23u
10
+ 10u
9
70u
8
24u
7
+ 100u
6
+ 39u
5
55u
4
5u
3
7u
2
+ 8b 22u + 1i
There are 2 irreducible components with 18 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
4
b
3
+ 3b
2
2b + 1, a + 1, u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
1
b
a
11
=
b + 1
b
2
a
7
=
b
2
+ b 1
b
3
+ b
a
10
=
b
3
+ b
2
2b + 1
b
3
b
2
+ 2b 1
a
9
=
0
b
3
b
2
+ 2b 1
a
5
=
1
0
a
8
=
0
b
3
b
2
+ 2b 1
a
8
=
0
b
3
b
2
+ 2b 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 = 1.00000
a = 1.00000
b = 0.10488 1.55249I
8.43568 3.16396I 9.97493 + 3.47609I
u = 1.00000
a = 1.00000
b = 0.10488 + 1.55249I
8.43568 + 3.16396I 9.97493 3.47609I
u = 1.00000
a = 1.00000
b = 0.395123 0.506844I
1.43393 1.41510I 7.52507 + 4.18840I
u = 1.00000
a = 1.00000
b = 0.395123 + 0.506844I
1.43393 + 1.41510I 7.52507 4.18840I
3
II. I
u
2
= hu
14
5u
13
+ · · · + 7u 1, a 1, u
13
4u
12
+ · · · + 8b + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
1
1
8
u
13
+
1
2
u
12
+ ··· +
11
4
u
1
8
a
11
=
1
8
u
13
+
1
2
u
12
+ ··· +
11
4
u +
7
8
1
8
u
13
+
3
4
u
12
+ ··· + 2u
3
8
a
7
=
1
4
u
13
+
5
4
u
12
+ ··· +
19
4
u +
1
2
u
13
+ 4u
12
+ ··· + 10u
3
2
a
10
=
5
4
u
13
+
21
4
u
12
+ ··· +
59
4
u 1
u
13
11
4
u
12
+ ··· +
5
4
u
3
4
a
9
=
1
4
u
13
+
5
2
u
12
+ ··· + 16u
7
4
u
13
11
4
u
12
+ ··· +
5
4
u
3
4
a
5
=
u
u
3
+ u
a
8
=
1
4
u
13
+
5
2
u
12
+ ··· + 16u
7
4
7
4
u
13
+
23
4
u
12
+ ··· +
41
4
u 2
a
8
=
1
4
u
13
+
5
2
u
12
+ ··· + 16u
7
4
7
4
u
13
+
23
4
u
12
+ ··· +
41
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 = 1.30923 0.72980I
a = 1.00000
b = 0.03110 + 1.66209I
12.09879 1.91262I 10.51406 + 1.13289I
u = 1.30923 + 0.72980I
a = 1.00000
b = 0.03110 1.66209I
12.09879 + 1.91262I 10.51406 1.13289I
u = 1.079731 0.462597I
a = 1.00000
b = 0.131850 + 0.795140I
3.41989 1.31906I 10.67824 + 1.83447I
u = 1.079731 + 0.462597I
a = 1.00000
b = 0.131850 0.795140I
3.41989 + 1.31906I 10.67824 1.83447I
u = 0.797754
a = 1.00000
b = 0.251089
1.17986 7.85993
u = 0.166912 0.164823I
a = 1.00000
b = 0.310969 0.512101I
0.044354 1.170564I 0.70219 + 5.58030I
u = 0.166912 + 0.164823I
a = 1.00000
b = 0.310969 + 0.512101I
0.044354 + 1.170564I 0.70219 5.58030I
u = 0.321066 0.413985I
a = 1.00000
b = 0.07909 1.57522I
7.25996 2.49887I 4.67922 + 1.75896I
u = 0.321066 + 0.413985I
a = 1.00000
b = 0.07909 + 1.57522I
7.25996 + 2.49887I 4.67922 1.75896I
u = 1.87363
a = 1.00000
b = 0.778815
11.8742 5.71438
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.90427 0.14543I
a = 1.00000
b = 0.550866 0.900632I
14.5979 + 4.4309I 8.77759 3.45380I
u = 1.90427 + 0.14543I
a = 1.00000
b = 0.550866 + 0.900632I
14.5979 4.4309I 8.77759 + 3.45380I
u = 1.95877 0.28461I
a = 1.00000
b = 0.16129 1.68407I
15.9841 + 7.2397I 10.36154 2.69654I
u = 1.95877 + 0.28461I
a = 1.00000
b = 0.16129 + 1.68407I
15.9841 7.2397I 10.36154 + 2.69654I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u + 1)
4
(u
14
+ 5u
13
+ ··· 7u 1)
c
2
(u + 1)
4
(u
14
+ 23u
13
+ ··· + 3u + 1)
c
3
, c
8
u
4
(u
14
+ u
13
+ ··· + 24u 16)
c
5
(u
4
u
3
+ u
2
+ 1)(u
14
+ 2u
13
+ ··· + 3u + 1)
c
6
, c
7
, c
10
c
11
(u
4
u
3
+ 3u
2
2u + 1)(u
14
+ 2u
13
+ ··· + 5u + 1)
c
9
(u
4
+ u
3
+ u
2
+ 1)(u
14
+ 6u
13
+ ··· + 117u + 19)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(y
14
23y
13
+ ··· 3y + 1)
c
2
(y 1)
4
(y
14
59y
13
+ ··· + 681y + 1)
c
3
, c
8
y
4
(y
14
27y
13
+ ··· + 960y + 256)
c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
14
30y
13
+ ··· 9y + 1)
c
6
, c
7
, c
10
c
11
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
14
+ 18y
13
+ ··· 9y + 1)
c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
14
18y
13
+ ··· 22277y + 361)
8