10
1
(K10a
75
)
1
Arc Sequences
6 10 9 8 7 1 5 4 3 2
Solving Sequence
1,6
2 7 5 8 4 10 3 9
c
1
c
6
c
5
c
7
c
4
c
10
c
2
c
9
c
3
, c
8
Representation Ideals
I = I
u
1
I
u
1
= hu
8
+ u
7
+ u
6
+ 3u
4
+ 2u
3
+ 2u
2
+ 1i
There are 1 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
8
+ u
7
+ u
6
+ 3u
4
+ 2u
3
+ 2u
2
+ 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
u
u
a
5
=
u
3
u
3
+ u
a
8
=
u
5
+ u
u
5
+ u
3
+ u
a
4
=
u
7
+ 2u
3
u
7
+ u
5
+ 2u
3
+ u
a
10
=
u
2
+ 1
u
4
a
3
=
u
4
+ u
2
+ 1
u
6
u
2
a
9
=
u
6
+ u
4
+ 2u
2
+ 1
u
7
+ u
6
+ u
4
+ 2u
3
+ 2u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
6
4u
5
4u
4
12u
2
8u 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.961906 0.970163I
14.9579 + 3.5262I 1.98168 2.14300I
u = 0.961906 + 0.970163I
14.9579 3.5262I 1.98168 + 2.14300I
u = 0.623553 0.732535I
3.50819 + 2.28803I 1.30973 4.26686I
u = 0.623553 + 0.732535I
3.50819 2.28803I 1.30973 + 4.26686I
u = 0.242210 0.575229I
0.267684 0.921357I 5.17544 + 7.34493I
u = 0.242210 + 0.575229I
0.267684 + 0.921357I 5.17544 7.34493I
u = 0.843248 0.880399I
11.71742 3.09309I 1.88403 + 2.68898I
u = 0.843248 + 0.880399I
11.71742 + 3.09309I 1.88403 2.68898I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
8
+ u
7
+ u
6
+ 3u
4
+ 2u
3
+ 2u
2
+ 1)
c
2
, c
3
, c
4
c
5
, c
7
, c
8
c
9
, c
10
(u
8
+ u
7
+ 7u
6
+ 6u
5
+ 15u
4
+ 10u
3
+ 10u
2
+ 4u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
8
+ y
7
+ 7y
6
+ 6y
5
+ 15y
4
+ 10y
3
+ 10y
2
+ 4y + 1)
c
2
, c
3
, c
4
c
5
, c
7
, c
8
c
9
, c
10
(y
8
+ 13y
7
+ 67y
6
+ 174y
5
+ 239y
4
+ 166y
3
+ 50y
2
+ 4y + 1)
5