11n
79
(K11n
79
)
1
Arc Sequences
4 1 8 2 11 10 4 1 7 6 9
Solving Sequence
5,11 2,6
4 1 3 10 7 8 9
c
5
c
4
c
1
c
2
c
10
c
6
c
7
c
9
c
3
, c
8
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
4
+ a
3
+ 3a
2
+ 2a + 1, b + 1, u 1i
I
u
2
= hu
11
5u
10
+ 2u
9
+ 21u
8
15u
7
43u
6
+ 30u
5
+ 30u
4
18u
3
+ 12u
2
+ 1, b u,
u
10
+ 4u
9
+ 2u
8
19u
7
4u
6
+ 39u
5
+ 9u
4
21u
3
3u
2
+ 8a 23u 15i
There are 2 irreducible components with 15 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
= ha
4
+ a
3
+ 3a
2
+ 2a + 1, b + 1, u 1i
(i) Arc colorings
a
5
=
0
1
a
11
=
a
1
a
2
=
1
0
a
6
=
a
2
a + 1
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
10
=
a
3
+ a
a
2
a 1
a
7
=
a
3
+ a
2
+ 2a + 1
a
3
+ a
2
+ 2a + 1
a
8
=
a
3
+ a
2
+ 2a + 1
a
3
+ a
2
+ 2a + 1
a
9
=
a
3
+ a
2
+ 2a + 1
0
a
9
=
a
3
+ a
2
+ 2a + 1
0
(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 = 0.395123 0.506844I
b = 1.00000
1.85594 1.41510I 4.47493 + 4.18840I
u = 1.00000
a = 0.395123 + 0.506844I
b = 1.00000
1.85594 + 1.41510I 4.47493 4.18840I
u = 1.00000
a = 0.10488 1.55249I
b = 1.00000
5.14581 3.16396I 2.02507 + 3.47609I
u = 1.00000
a = 0.10488 + 1.55249I
b = 1.00000
5.14581 + 3.16396I 2.02507 3.47609I
3
II. I
u
2
= hu
11
5u
10
+ · · · + 12u
2
+ 1, b u, u
10
+ 4u
9
+ · · · + 8a 15i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
8
u
10
1
2
u
9
+ ··· +
23
8
u +
15
8
u
a
2
=
1
0
a
6
=
3
8
u
10
+
7
4
u
9
+ ···
25
8
u +
5
8
1
8
u
10
+
1
2
u
9
+ ··· +
9
8
u +
1
8
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
10
=
1
2
u
10
5
2
u
9
+ ··· 13u
2
+
1
2
1
4
u
10
+
3
4
u
9
+ ···
5
2
u
2
+
3
4
u
a
7
=
1
4
u
9
+
1
4
u
8
+ ··· 3u +
5
4
1
4
u
10
+
1
4
u
9
+ ··· 3u
2
+
5
4
u
a
8
=
5
4
u
10
+
13
4
u
9
+ ···
11
4
u + 2
3
2
u
10
+
15
4
u
9
+ ··· +
3
2
u +
3
4
a
9
=
3
4
u
10
11
4
u
9
+ ···
7
4
u + 1
3
4
u
10
5
2
u
9
+ ··· +
5
4
u
1
4
a
9
=
3
4
u
10
11
4
u
9
+ ···
7
4
u + 1
3
4
u
10
5
2
u
9
+ ··· +
5
4
u
1
4
(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 = 1.28076 0.60275I
a = 0.461110 0.819358I
b = 1.28076 0.60275I
1.86679 1.71507I 2.68555 + 1.25777I
u = 1.28076 + 0.60275I
a = 0.461110 + 0.819358I
b = 1.28076 + 0.60275I
1.86679 + 1.71507I 2.68555 1.25777I
u = 1.22744
a = 0.329552
b = 1.22744
2.29341 5.45609
u = 0.069460 0.264957I
a = 1.69250 0.70148I
b = 0.069460 0.264957I
0.058810 0.998414I 1.10999 + 6.77459I
u = 0.069460 + 0.264957I
a = 1.69250 + 0.70148I
b = 0.069460 + 0.264957I
0.058810 + 0.998414I 1.10999 6.77459I
u = 0.286174 0.444607I
a = 2.30485 1.70194I
b = 0.286174 0.444607I
6.37335 2.43510I 2.89338 + 1.98880I
u = 0.286174 + 0.444607I
a = 2.30485 + 1.70194I
b = 0.286174 + 0.444607I
6.37335 + 2.43510I 2.89338 1.98880I
u = 2.04088 0.26755I
a = 0.238512 0.730828I
b = 2.04088 0.26755I
10.35591 + 6.75197I 2.99345 2.75276I
u = 2.04088 + 0.26755I
a = 0.238512 + 0.730828I
b = 2.04088 + 0.26755I
10.35591 6.75197I 2.99345 + 2.75276I
u = 2.13689 0.09549I
a = 0.390031 0.242224I
b = 2.13689 0.09549I
17.2404 + 2.6821I 5.87634 2.38377I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 2.13689 + 0.09549I
a = 0.390031 + 0.242224I
b = 2.13689 + 0.09549I
17.2404 2.6821I 5.87634 + 2.38377I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
11
+ 5u
10
+ ··· 12u
2
1)
c
2
(u + 1)
4
(u
11
+ 21u
10
+ ··· 24u + 1)
c
3
, c
7
u
4
(u
11
+ u
10
+ ··· + 8u 16)
c
4
(u + 1)
4
(u
11
+ 5u
10
+ ··· 12u
2
1)
c
5
, c
6
(u
4
u
3
+ 3u
2
2u + 1)(u
11
+ 2u
10
+ ··· + 2u + 1)
c
8
(u
4
u
3
+ u
2
+ 1)(u
11
+ 12u
9
+ ··· + u
2
1)
c
9
, c
10
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
11
+ 2u
10
+ ··· + 2u + 1)
c
11
(u
4
+ u
3
+ u
2
+ 1)(u
11
+ 12u
9
+ ··· + u
2
1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(y
11
21y
10
+ ··· 24y 1)
c
2
(y 1)
4
(y
11
73y
10
+ ··· + 168y 1)
c
3
, c
7
y
4
(y
11
+ 27y
10
+ ··· + 320y 256)
c
5
, c
6
, c
9
c
10
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
11
+ 12y
10
+ ··· + 2y 1)
c
8
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
11
+ 24y
10
+ ··· + 2y 1)
7