9
48
(K9n
6
)
1
Arc Sequences
5 7 9 2 4 3 6 4 3
Solving Sequence
3,9 4,7
2 6 5 1 8
c
3
c
2
c
6
c
5
c
1
c
8
c
4
, c
7
, c
9
Representation Ideals
I =
5
\
i=1
I
u
i
I
u
1
= hu
2
+ 1, b + u, a 2u + 1i
I
u
2
= hu
3
u
2
+ u + 1, b u, u
2
+ ai
I
u
3
= hu
4
+ u
3
+ u
2
+ 1, u
2
+ b + u, u
3
2u
2
+ a 2ui
I
u
4
= hu
4
+ u
3
+ u
2
+ 1, b u, u
3
+ a 1i
I
u
5
= hu
4
2u
3
+ 3u
2
3u + 2, u
3
+ 2a u + 1, u
3
+ u
2
+ b u + 1i
There are 5 irreducible components with 17 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
+ 1, b + u, a 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
2u 1
u
a
4
=
u 1
1
a
7
=
0
u
a
2
=
1
1
a
6
=
u
u
a
5
=
2u 1
u + 1
a
1
=
u 1
u
a
8
=
u
0
a
8
=
u
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000I
a = 1.00000 2.00000I
b = 1.00000I
4.93480 4.00000
u = 1.00000I
a = 1.00000 + 2.00000I
b = 1.00000I
4.93480 4.00000
3
II. I
u
2
= hu
3
u
2
+ u + 1, b u, u
2
+ ai
(i) Arc colorings
a
3
=
1
0
a
9
=
u
2
u
a
4
=
u
2
u
u
2
a
7
=
0
u
a
2
=
1
u
2
a
6
=
u
u
a
5
=
u
2
u 1
a
1
=
u
2
u
u
a
8
=
u
2
+ u + 1
u
2
1
a
8
=
u
2
+ u + 1
u
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u + 8
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.543689
a = 0.295598
b = 0.543689
0.875992 11.2621
u = 0.771845 1.115143I
a = 0.64780 + 1.72143I
b = 0.771845 1.115143I
2.02941 9.53188I 3.36893 + 6.69086I
u = 0.771845 + 1.115143I
a = 0.64780 1.72143I
b = 0.771845 + 1.115143I
2.02941 + 9.53188I 3.36893 6.69086I
5
III. I
u
3
= hu
4
+ u
3
+ u
2
+ 1, u
2
+ b + u, u
3
2u
2
+ a 2ui
(i) Arc colorings
a
3
=
1
0
a
9
=
u
3
+ 2u
2
+ 2u
u
2
u
a
4
=
u
3
u 1
u
3
+ 1
a
7
=
0
u
a
2
=
1
u
2
a
6
=
u
u
a
5
=
u
3
1
1
a
1
=
u
3
u
2
u
u
2
+ u
a
8
=
u
3
u
3
+ u
a
8
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
4u + 2
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.851808 0.911292I
a = 0.409261 + 0.055548I
b = 0.956685 0.641200I
3.50087 + 3.16396I 5.82674 2.56480I
u = 0.851808 + 0.911292I
a = 0.409261 0.055548I
b = 0.956685 + 0.641200I
3.50087 3.16396I 5.82674 + 2.56480I
u = 0.351808 0.720342I
a = 0.59074 2.34806I
b = 0.043315 + 1.227186I
3.50087 1.41510I 2.17326 + 4.90874I
u = 0.351808 + 0.720342I
a = 0.59074 + 2.34806I
b = 0.043315 1.227186I
3.50087 + 1.41510I 2.17326 4.90874I
7
IV. I
u
4
= hu
4
+ u
3
+ u
2
+ 1, b u, u
3
+ a 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
u
3
+ 1
u
a
4
=
u
3
u
2
u
u
2
a
7
=
0
u
a
2
=
1
u
2
a
6
=
u
u
a
5
=
u
3
2u
2
2u
1
a
1
=
u
3
u 1
u
a
8
=
u
3
u
3
+ u
a
8
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
4u + 2
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.851808 0.911292I
a = 0.504108 + 1.226852I
b = 0.851808 0.911292I
3.50087 + 3.16396I 5.82674 2.56480I
u = 0.851808 + 0.911292I
a = 0.504108 1.226852I
b = 0.851808 + 0.911292I
3.50087 3.16396I 5.82674 + 2.56480I
u = 0.351808 0.720342I
a = 1.50411 0.10631I
b = 0.351808 0.720342I
3.50087 1.41510I 2.17326 + 4.90874I
u = 0.351808 + 0.720342I
a = 1.50411 + 0.10631I
b = 0.351808 + 0.720342I
3.50087 + 1.41510I 2.17326 4.90874I
9
V. I
u
5
= hu
4
2u
3
+ 3u
2
3u + 2, u
3
+ 2a u + 1, u
3
+ u
2
+ b u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
1
2
u
3
+
1
2
u
1
2
u
3
u
2
+ u 1
a
4
=
1
2
u
3
+ u
2
3
2
u +
3
2
u
3
u
2
+ 2u 1
a
7
=
0
u
a
2
=
1
u
2
a
6
=
u
u
a
5
=
1
2
u
3
1
2
u +
1
2
u
3
2u
2
+ 3u 3
a
1
=
1
2
u
3
+ u
2
3
2
u +
3
2
u
3
+ u
2
u + 1
a
8
=
u
3
u
3
+ u
a
8
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 2
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.043315 1.227186I
a = 0.38053 1.53420I
b = 0.351808 + 0.720342I
3.50087 + 1.41510I 2.17326 4.90874I
u = 0.043315 + 1.227186I
a = 0.38053 + 1.53420I
b = 0.351808 0.720342I
3.50087 1.41510I 2.17326 + 4.90874I
u = 0.956685 0.641200I
a = 0.130534 + 0.427872I
b = 0.851808 0.911292I
3.50087 + 3.16396I 5.82674 2.56480I
u = 0.956685 + 0.641200I
a = 0.130534 0.427872I
b = 0.851808 + 0.911292I
3.50087 3.16396I 5.82674 + 2.56480I
11
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
3
c
4
, c
6
, c
8
c
9
(u
2
+ 1)(u
3
+ u
2
+ u 1)(1 + u
2
u
3
+ u
4
)
2
(u
4
+ 2u
3
+ ··· + 3u + 2)
c
5
, c
7
(u + 1)
2
(u
3
+ u
2
+ 3u 1)(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
(u
4
+ 2u
3
+ u
2
+ 3u + 4)
12
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
3
c
4
, c
6
, c
8
c
9
(y + 1)
2
(y
3
+ y
2
+ 3y 1)(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
(y
4
+ 2y
3
+ y
2
+ 3y + 4)
c
5
, c
7
(y 1)
2
(y
3
+ 5y
2
+ 11y 1)(y
4
2y
3
3y
2
y + 16)
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
13