10
139
(K10n
27
)
1
Arc Sequences
6 9 7 8 1 2 4 2 8 5
Solving Sequence
2,9 3,7
4 6 1 5 8 10
c
2
c
3
c
6
c
1
c
5
c
8
c
10
c
4
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
2
2, b 1, 2a + u + 2i
I
u
2
= hu
4
4u
3
+ 6u
2
2u 2, u
3
2u
2
+ 2a + 2u, u
3
+ 3u
2
+ b 2u 1i
I
v
1
= hv + 1, b + 1, ai
There are 3 irreducible components with 7 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
2
2, b 1, 2a + u + 2i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
2
u 1
1
a
3
=
1
2
u
1
a
7
=
0
u
a
4
=
1
2
u
u + 1
a
6
=
u
u
a
1
=
1
2
a
5
=
0
u
a
8
=
1
2
u
1
a
10
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 20
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.41421
a = 0.292893
b = 1.00000
8.22467 20.0000
u = 1.41421
a = 1.70711
b = 1.00000
8.22467 20.0000
3
II.
I
u
2
= hu
4
4u
3
+ 6u
2
2u 2, u
3
2u
2
+ 2a + 2u, u
3
+ 3u
2
+ b 2u 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
2
u
3
+ u
2
u
u
3
3u
2
+ 2u + 1
a
3
=
1
2
u
3
u
2
+ 1
2u
2
2u 1
a
7
=
0
u
a
4
=
1
2
u
3
u
2
+ 1
u
3
2u
2
+ u + 1
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
5
=
u
3
+ 2u
u
3
+ u
a
8
=
1
2
u
3
2u
2
+ u + 1
u
3
3u
2
+ 2u + 1
a
10
=
4u
3
9u
2
+ 2u + 3
4u
3
8u
2
+ 2u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u 16
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.395337
a = 0.582522
b = 0.321336
0.588647 16.7907
u = 1.46036 1.13932I
a = 0.660443 + 0.716885I
b = 1.15548 + 1.89385I
4.51885 + 4.85117I 13.07929 2.27864I
u = 1.46036 + 1.13932I
a = 0.660443 0.716885I
b = 1.15548 1.89385I
4.51885 4.85117I 13.07929 + 2.27864I
u = 1.47463
a = 0.903408
b = 0.632293
6.80412 13.0507
5
III. I
v
1
= hv + 1, b + 1, ai
(i) Arc colorings
a
2
=
1
0
a
9
=
0
1
a
3
=
1
1
a
7
=
1
0
a
4
=
0
1
a
6
=
1
0
a
1
=
1
0
a
5
=
1
0
a
8
=
1
1
a
10
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
3.28987 12.0000
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
c
10
u(u
2
2)(u
4
+ 4u
3
+ 6u
2
+ 2u 2)
c
2
(u + 1)
3
(u
4
2u
3
+ 4u
2
+ 2u 1)
c
3
, c
4
(u 1)(u + 1)
2
(u
4
+ 2u
3
+ 4u
2
2u 1)
c
7
(u + 1)
3
(u
4
+ 2u
3
+ 4u
2
2u 1)
c
8
(u 1)(u + 1)
2
(u
4
2u
3
+ 4u
2
+ 2u 1)
c
9
(u 1)
2
(u + 1)(u
4
+ 4u
3
+ 22u
2
12u + 1)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
c
10
y(y 2)
2
(y
4
4y
3
+ 16y
2
28y + 4)
c
2
, c
3
, c
4
c
7
, c
8
(y 1)
3
(y
4
+ 4y
3
+ 22y
2
12y + 1)
c
9
(y 1)
3
(y
4
+ 28y
3
+ 582y
2
100y + 1)
9