10
132
(K10n
13
)
1
Arc Sequences
5 10 7 6 2 4 10 7 3 8
Solving Sequence
1,5
2 6
4,8
10 3 7 9
c
1
c
5
c
4
c
10
c
2
c
7
c
8
c
3
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
3
u
2
+ 1, a + u, b + 1i
I
u
2
= hu
5
+ 2u
4
+ 2u
3
+ u + 1, a u, u
3
+ u
2
+ bi
There are 2 irreducible components with 8 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
3
u
2
+ 1, a + u, b + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
2
+ u + 1
a
4
=
u
2
+ 1
u
2
a
8
=
u
1
a
10
=
u + 1
1
a
3
=
1
u
2
a
7
=
1
0
a
9
=
u + 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
2
u + 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.754878
a = 0.754878
b = 1.00000
0.531480 1.61520
u = 0.877439 0.744862I
a = 0.877439 + 0.744862I
b = 1.00000
4.66906 2.82812I 0.69240 + 3.35914I
u = 0.877439 + 0.744862I
a = 0.877439 0.744862I
b = 1.00000
4.66906 + 2.82812I 0.69240 3.35914I
3
II. I
u
2
= hu
5
+ 2u
4
+ 2u
3
+ u + 1, a u, u
3
+ u
2
+ bi
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
4
=
u
3
2u
4
3u
3
1
a
8
=
u
u
3
u
2
a
10
=
u
4
+ u
3
+ 1
u
4
+ u
2
+ u
a
3
=
3u
3
u
2
+ 2
5u
4
+ 7u
3
3u
2
+ 3u + 4
a
7
=
2u
4
2u
3
1
2u
4
+ u
3
u
2
+ 1
a
9
=
11u
4
+ 8u
3
u
2
+ 6u + 4
7u
4
19u
3
+ 4u
2
+ 3u 7
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
4
+ 6u
3
+ 3u
2
6u + 7
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.10221 1.09532I
a = 1.10221 1.09532I
b = 2.64316 + 0.26340I
16.0529 + 4.0569I 0.27760 1.88627I
u = 1.10221 + 1.09532I
a = 1.10221 + 1.09532I
b = 2.64316 0.26340I
16.0529 4.0569I 0.27760 + 1.88627I
u = 0.668466
a = 0.668466
b = 0.148145
0.907840 11.5575
u = 0.436447 0.655029I
a = 0.436447 0.655029I
b = 0.717228 + 0.665045I
1.70245 1.37362I 0.55634 + 3.01933I
u = 0.436447 + 0.655029I
a = 0.436447 + 0.655029I
b = 0.717228 0.665045I
1.70245 + 1.37362I 0.55634 3.01933I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
3
u
2
+ 1)(u
5
+ 2u
4
+ 2u
3
+ u + 1)
c
2
, c
9
u
3
(u
5
+ u
4
+ 17u
3
4u
2
+ 20u 8)
c
3
, c
4
(u
3
+ u
2
+ 2u + 1)(u
5
+ 6u
3
+ u + 1)
c
5
(u
3
+ u
2
1)(u
5
+ 2u
4
+ 2u
3
+ u + 1)
c
6
(u
3
u
2
+ 2u 1)(u
5
+ 6u
3
+ u + 1)
c
7
(u 1)
3
(u
5
+ 4u
4
+ u
3
5u
2
+ 6u + 1)
c
8
(u + 1)
3
(u
5
+ 14u
4
+ 53u
3
+ 21u
2
+ 46u + 1)
c
10
(u + 1)
3
(u
5
+ 4u
4
+ u
3
5u
2
+ 6u + 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
3
y
2
+ 2y 1)(y
5
+ 6y
3
+ y 1)
c
2
, c
9
y
3
(y
5
+ 33y
4
+ 337y
3
+ 680y
2
+ 336y 64)
c
3
, c
4
, c
6
(y
3
+ 3y
2
+ 2y 1)(y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y 1)
c
7
, c
10
(y 1)
3
(y
5
14y
4
+ 53y
3
21y
2
+ 46y 1)
c
8
(y 1)
3
(y
5
90y
4
+ 2313y
3
+ 4407y
2
+ 2074y 1)
7