10
142
(K10n
30
)
1
Arc Sequences
8 6 9 8 2 3 1 4 5 2
Solving Sequence
2,5
6
3,8
1 4 9 7 10
c
5
c
2
c
1
c
4
c
8
c
7
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 1, a, b + 1i
I
u
2
= ha
2
+ 2, u + 1, b 1i
I
u
3
= hu
4
+ u
3
2u
2
+ 3, u
3
+ b + 2u 1, 2u
3
u
2
+ 3a 4u + 3i
I
u
4
= hu
6
+ u
5
5u
4
4u
3
+ 5u
2
u 1, b u, u
4
u
3
+ 4u
2
+ 2a + 5u 1i
There are 4 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
= hu 1, a, b + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
6
=
1
1
a
3
=
0
1
a
8
=
0
1
a
1
=
1
1
a
4
=
0
1
a
9
=
0
1
a
7
=
1
0
a
10
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0
b = 1.00000
3.28987 12.0000
3
II. I
u
2
= ha
2
+ 2, u + 1, b 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
6
=
1
1
a
3
=
0
1
a
8
=
a
1
a
1
=
a + 1
1
a
4
=
2
a 1
a
9
=
a
a 1
a
7
=
1
0
a
10
=
a
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.41421I
b = 1.00000
1.64493 12.0000
u = 1.00000
a = 1.41421I
b = 1.00000
1.64493 12.0000
5
III. I
u
3
= hu
4
+ u
3
2u
2
+ 3, u
3
+ b + 2u 1, 2u
3
u
2
+ 3a 4u + 3i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
3
=
u
2
+ 1
u
2
a
8
=
2
3
u
3
+
1
3
u
2
+
4
3
u 1
u
3
2u + 1
a
1
=
2
3
u
3
1
3
u
2
1
3
u + 3
u
3
+ u
2
+ u 4
a
4
=
1
3
u
3
+
2
3
u
2
+
2
3
u 1
u
3
u
2
u + 2
a
9
=
1
3
u
3
+
2
3
u
2
+
2
3
u 1
1
a
7
=
u
3
2u
u
3
+ u
a
10
=
1
3
u
3
+
2
3
u
2
+
2
3
u 1
u
3
+ u
2
+ u 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
4u 6
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.45326 0.16311I
a = 0.273719 + 0.626639I
b = 0.953264 0.702911I
3.28987 + 2.02988I 12.00000 3.46410I
u = 1.45326 + 0.16311I
a = 0.273719 0.626639I
b = 0.953264 + 0.702911I
3.28987 2.02988I 12.00000 + 3.46410I
u = 0.953264 0.702911I
a = 0.773719 0.337964I
b = 1.45326 0.16311I
3.28987 + 2.02988I 12.00000 3.46410I
u = 0.953264 + 0.702911I
a = 0.773719 + 0.337964I
b = 1.45326 + 0.16311I
3.28987 2.02988I 12.00000 + 3.46410I
7
IV.
I
u
4
= hu
6
+ u
5
5u
4
4u
3
+ 5u
2
u 1, b u, u
4
u
3
+ 4u
2
+ 2a + 5u 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
3
=
u
2
+ 1
u
2
a
8
=
1
2
u
4
+
1
2
u
3
2u
2
5
2
u +
1
2
u
a
1
=
1
2
u
5
1
2
u
4
+ ···
1
2
u + 1
u
2
a
4
=
1
2
u
5
u
4
+ ··· 2u
3
2
1
2
u
4
1
2
u
3
2u
2
+
3
2
u +
1
2
a
9
=
1
2
u
5
1
2
u
4
+ ···
1
2
u + 1
1
2
u
5
+
1
2
u
4
2u
3
3
2
u
2
+
1
2
u
a
7
=
u
3
2u
u
3
+ u
a
10
=
1
2
u
5
1
2
u
4
+ ···
1
2
u + 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
3u
3
+ 4u
2
+ 11u 13
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.85126 0.30576I
a = 0.464021 + 0.716665I
b = 1.85126 0.30576I
13.0621 6.7708I 12.38492 + 2.96218I
u = 1.85126 + 0.30576I
a = 0.464021 0.716665I
b = 1.85126 + 0.30576I
13.0621 + 6.7708I 12.38492 2.96218I
u = 0.338910
a = 1.10469
b = 0.338910
0.610583 16.1650
u = 0.526900 0.379519I
a = 1.19618 + 1.56453I
b = 0.526900 0.379519I
3.26038 + 1.42716I 6.28345 4.88332I
u = 0.526900 + 0.379519I
a = 1.19618 1.56453I
b = 0.526900 + 0.379519I
3.26038 1.42716I 6.28345 + 4.88332I
u = 1.98762
a = 0.640373
b = 1.98762
17.6195 14.4983
9
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u + 1)(u
4
+ u
3
2u
2
+ 3)(u
6
+ u
5
+ ··· u 1)
c
2
(u 1)(u + 1)
2
(u
4
+ u
3
2u
2
+ 3)(u
6
+ u
5
+ ··· u 1)
c
3
, c
4
, c
8
u(u
2
+ 2)(u
2
u + 1)
2
(u
6
+ 3u
5
+ 7u
4
+ 10u
3
+ 10u
2
+ 8u + 2)
c
5
, c
6
(u 1)
3
(u
4
+ u
3
2u
2
+ 3)(u
6
+ u
5
5u
4
4u
3
+ 5u
2
u 1)
c
7
(u + 1)
3
(u
4
+ u
3
2u
2
+ 3)(u
6
+ u
5
5u
4
4u
3
+ 5u
2
u 1)
c
9
u(u
2
+ 2)(u
2
u + 1)
2
(u
6
+ 3u
5
11u
4
32u
3
2u
2
16u + 10)
c
10
(u 1)
2
(u + 1)(u
4
+ 5u
3
+ 10u
2
+ 12u + 9)
(u
6
+ 11u
5
+ 43u
4
+ 66u
3
+ 27u
2
+ 11u + 1)
10
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
5
c
6
, c
7
(y 1)
3
(y
4
5y
3
+ 10y
2
12y + 9)
(y
6
11y
5
+ 43y
4
66y
3
+ 27y
2
11y + 1)
c
3
, c
4
, c
8
y(y + 2)
2
(y
2
+ y + 1)
2
(y
6
+ 5y
5
+ 9y
4
4y
3
32y
2
24y + 4)
c
9
y(y + 2)
2
(y
2
+ y + 1)
2
(y
6
31y
5
+ 309y
4
864y
3
1240y
2
296y + 100)
c
10
(y 1)
3
(y
4
5y
3
2y
2
+ 36y + 81)
(y
6
35y
5
+ 451y
4
2274y
3
637y
2
67y + 1)
11