9
6
(K9a
23
)
1
Arc Sequences
7 4 8 2 9 1 3 5 6
Solving Sequence
4,8
3 2 5 9 7 1 6
c
3
c
2
c
4
c
8
c
7
c
1
c
6
c
5
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu + 1i
I
u
2
= hu
12
2u
10
u
9
+ 4u
8
+ u
7
3u
6
3u
5
+ 3u
4
+ u
3
u
2
2u + 1i
There are 2 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 + 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
1
a
3
=
1
1
a
2
=
0
1
a
5
=
1
1
a
9
=
1
0
a
7
=
1
0
a
1
=
1
1
a
6
=
0
1
a
6
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 18
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
4.93480 18.0000
3
II. I
u
2
= hu
12
2u
10
u
9
+ 4u
8
+ u
7
3u
6
3u
5
+ 3u
4
+ u
3
u
2
2u + 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
2
=
u
2
+ 1
u
2
a
5
=
u
4
u
2
+ 1
u
4
a
9
=
u
9
+ 2u
7
3u
5
+ 2u
3
u
u
9
+ u
7
u
5
+ u
a
7
=
u
u
3
+ u
a
1
=
u
6
+ u
4
2u
2
+ 1
u
8
2u
6
+ 2u
4
2u
2
a
6
=
u
11
+ 2u
9
4u
7
+ 4u
5
3u
3
+ 2u
u
11
+ u
10
+ u
9
u
8
3u
7
+ 3u
6
+ u
5
u
4
2u
3
+ 2u
2
a
6
=
u
11
+ 2u
9
4u
7
+ 4u
5
3u
3
+ 2u
u
11
+ u
10
+ u
9
u
8
3u
7
+ 3u
6
+ u
5
u
4
2u
3
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
11
8u
9
4u
8
+ 12u
7
+ 4u
6
8u
5
8u
4
+ 4u
3
+ 4u
2
4u 14
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.075027 0.655125I
10.34901 7.20360I 14.0875 + 4.7166I
u = 1.075027 + 0.655125I
10.34901 + 7.20360I 14.0875 4.7166I
u = 0.850204 0.630914I
1.76919 2.46907I 6.47747 + 3.95252I
u = 0.850204 + 0.630914I
1.76919 + 2.46907I 6.47747 3.95252I
u = 0.511432 0.812623I
8.67410 + 1.70959I 11.87181 0.16720I
u = 0.511432 + 0.812623I
8.67410 1.70959I 11.87181 + 0.16720I
u = 0.470358
0.660692 15.0692
u = 0.635020 0.640255I
0.207771 0.498503I 10.63137 + 1.38008I
u = 0.635020 + 0.640255I
0.207771 + 0.498503I 10.63137 1.38008I
u = 0.985497 0.634576I
1.23208 + 5.52285I 12.56374 6.48307I
u = 0.985497 + 0.634576I
1.23208 5.52285I 12.56374 + 6.48307I
u = 1.16193
14.5896 17.6671
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
c
8
, c
9
(u 1)(u
12
+ 2u
11
+ ··· + 4u + 1)
c
2
, c
4
(u + 1)(u
12
+ 4u
11
+ ··· + 6u + 1)
c
3
, c
7
(u + 1)(u
12
2u
10
+ ··· 2u + 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
c
8
, c
9
(y 1)(y
12
16y
11
+ ··· 6y + 1)
c
2
, c
4
(y 1)(y
12
+ 8y
11
+ ··· 14y + 1)
c
3
, c
7
(y 1)(y
12
4y
11
+ ··· 6y + 1)
7