11n
104
(K11n
104
)
1
Arc Sequences
5 1 8 9 2 11 1 10 4 5 7
Solving Sequence
1,5
2
3,7
8 11 6 10 9 4
c
1
c
2
c
7
c
11
c
6
c
10
c
8
c
4
c
3
, c
5
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= ha
4
4a
3
+ 4a
2
+ 1, u + 1, b 1i
I
u
2
= hu
8
2u
7
10u
6
+ 30u
5
+ 8u
4
14u
3
+ 2u
2
+ 2u 1, b u, u
6
2u
5
9u
4
+ 28u
3
u
2
+ 8a + 14u + 1i
I
u
3
= hu 1, a + 1, b + 1i
There are 3 irreducible components with 13 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
4a
3
+ 4a
2
+ 1, u + 1, b 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
7
=
a
1
a
8
=
a 1
1
a
11
=
a + 1
1
a
6
=
1
0
a
10
=
a + 1
a 2
a
9
=
a
3
+ 3a
2
2a
a
3
4a
2
+ 5a 1
a
4
=
a
2
+ 2a 1
a + 2
a
4
=
a
2
+ 2a 1
a + 2
(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 = 0.098684 0.455090I
b = 1.00000
5.75727 3.66386I 16.0000 + 4.0000I
u = 1.00000
a = 0.098684 + 0.455090I
b = 1.00000
5.75727 + 3.66386I 16.0000 4.0000I
u = 1.00000
a = 2.09868 0.45509I
b = 1.00000
5.75727 + 3.66386I 16.0000 4.0000I
u = 1.00000
a = 2.09868 + 0.45509I
b = 1.00000
5.75727 3.66386I 16.0000 + 4.0000I
3
II.
I
u
2
= hu
8
2u
7
+ · · · + 2u 1, b u, u
6
2u
5
9u
4
+ 28u
3
u
2
+ 8a + 14u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
7
=
1
8
u
6
+
1
4
u
5
+ ···
7
4
u
1
8
u
a
8
=
1
8
u
6
+
1
4
u
5
+ ···
11
4
u
1
8
u
a
11
=
1
8
u
7
1
4
u
6
+ ··· +
1
8
u + 1
u
2
a
6
=
u
u
3
+ u
a
10
=
1
8
u
7
1
4
u
6
+ ··· +
1
8
u + 1
1
8
u
7
+
1
4
u
6
+ ···
7
4
u
2
1
8
u
a
9
=
5
8
u
7
+
15
8
u
6
+ ···
65
8
u +
13
8
1
2
u
7
11
8
u
6
+ ··· + 4u
9
8
a
4
=
5
8
u
7
+
11
8
u
6
+ ···
9
8
u +
9
8
1
8
u
7
1
4
u
6
+ ··· +
7
4
u
2
+
1
8
u
a
4
=
5
8
u
7
+
11
8
u
6
+ ···
9
8
u +
9
8
1
8
u
7
1
4
u
6
+ ··· +
7
4
u
2
+
1
8
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 = 3.34406
a = 0.656809
b = 3.34406
15.7287 9.67086
u = 0.594812 0.065631I
a = 1.77892 + 0.41529I
b = 0.594812 0.065631I
4.20158 3.92770I 10.18918 + 5.00146I
u = 0.594812 + 0.065631I
a = 1.77892 0.41529I
b = 0.594812 + 0.065631I
4.20158 + 3.92770I 10.18918 5.00146I
u = 0.279091 0.329009I
a = 0.414734 + 0.712553I
b = 0.279091 0.329009I
0.535301 + 1.039085I 7.61110 6.36007I
u = 0.279091 + 0.329009I
a = 0.414734 0.712553I
b = 0.279091 + 0.329009I
0.535301 1.039085I 7.61110 + 6.36007I
u = 0.495898
a = 1.31523
b = 0.495898
1.19322 8.17952
u = 2.73980 1.24096I
a = 0.621831 + 0.351657I
b = 2.73980 1.24096I
11.45110 + 7.34942I 11.27453 2.75920I
u = 2.73980 + 1.24096I
a = 0.621831 0.351657I
b = 2.73980 + 1.24096I
11.45110 7.34942I 11.27453 + 2.75920I
5
III. I
u
3
= hu 1, a + 1, b + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
7
=
1
1
a
8
=
0
1
a
11
=
0
1
a
6
=
1
0
a
10
=
0
1
a
9
=
0
1
a
4
=
0
1
a
4
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
3.28987 12.0000
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u + 1)
5
(u
8
+ 2u
7
10u
6
30u
5
+ 8u
4
+ 14u
3
+ 2u
2
2u 1)
c
2
(u + 1)
5
(u
8
+ 24u
7
+ 236u
6
+ 1112u
5
+ 870u
4
+ 264u
3
+ 44u
2
+ 8u + 1)
c
3
, c
10
u(u
4
2u
2
+ 2)
(u
8
+ 4u
7
12u
6
70u
5
54u
4
+ 46u
3
+ 38u
2
10u + 10)
c
4
, c
9
(u)(u
4
+ 2u
2
+ 2)(u
8
+ 4u
7
+ ··· + 6u + 2)
c
5
(u 1)
4
(u + 1)(u
8
+ 2u
7
+ ··· 2u 1)
c
6
, c
7
(u 1)(u + 1)
4
(u
8
+ 2u
7
+ ··· 2u 1)
c
8
(u)(u
2
2u + 2)
2
(u
8
+ 4u
7
+ ··· + 4u + 4)
c
11
(u 1)
5
(u
8
+ 2u
7
10u
6
30u
5
+ 8u
4
+ 14u
3
+ 2u
2
2u 1)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
c
7
, c
11
(y 1)
5
(y
8
24y
7
+ 236y
6
1112y
5
+ 870y
4
264y
3
+ 44y
2
8y + 1)
c
2
(y 1)
5
(y
8
104y
7
+ ··· + 24y + 1)
c
3
, c
10
(y)(y
2
2y + 2)
2
(y
8
40y
7
+ ··· + 660y + 100)
c
4
, c
9
(y)(y
2
+ 2y + 2)
2
(y
8
+ 4y
7
+ ··· + 4y + 4)
c
8
(y)(y
2
+ 4)
2
(y
8
+ 32y
6
+ ··· 176y + 16)
9