8
10
(K8a
3
)
1
Arc Sequences
7 5 6 8 3 1 4 2
Solving Sequence
2,5
3
6,7
1 8 4
c
2
c
5
c
1
c
8
c
4
c
3
, c
6
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu + 1, a, b + 1i
I
u
2
= hu
3
u 1, a, b ui
I
u
3
= hu
11
2u
10
4u
9
+ 8u
8
+ 6u
7
8u
6
7u
5
2u
4
+ 7u
3
+ 3u
2
u + 1,
u
10
u
9
4u
8
+ 3u
7
+ 5u
6
u
5
2u
4
2u
3
+ u
2
+ b 1,
u
10
+ u
9
+ 4u
8
3u
7
6u
6
+ u
5
+ 5u
4
+ 4u
3
3u
2
+ a 4ui
There are 3 irreducible components with 15 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 + 1, a, b + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
6
=
1
0
a
7
=
0
1
a
1
=
1
1
a
8
=
0
1
a
4
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0
b = 1.00000
0 0
3
II. I
u
2
= hu
3
u 1, a, b ui
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
6
=
u
1
a
7
=
0
u
a
1
=
1
u
2
a
8
=
u
2
+ 1
u
2
a
4
=
u
2
+ 1
u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.662359 0.562280I
a = 0
b = 0.662359 0.562280I
1.64493 6.00000
u = 0.662359 + 0.562280I
a = 0
b = 0.662359 + 0.562280I
1.64493 6.00000
u = 1.32472
a = 0
b = 1.32472
1.64493 6.00000
5
III.
I
u
3
= hu
11
2u
10
+ · · · u + 1, u
10
u
9
+ · · · + b 1, u
10
+ u
9
+ · · · + a 4ui
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
6
=
u
u
3
+ u
a
7
=
u
10
u
9
4u
8
+ 3u
7
+ 6u
6
u
5
5u
4
4u
3
+ 3u
2
+ 4u
u
10
+ u
9
+ 4u
8
3u
7
5u
6
+ u
5
+ 2u
4
+ 2u
3
u
2
+ 1
a
1
=
u
9
4u
7
2u
6
+ 5u
5
+ 6u
4
4u
2
3u
u
10
+ 5u
8
+ u
7
7u
6
5u
5
+ 6u
3
+ 4u
2
a
8
=
u
10
+ u
9
+ 5u
8
3u
7
9u
6
+ 6u
4
+ 6u
3
3u
u
10
+ 5u
8
+ u
7
7u
6
5u
5
+ 6u
3
+ 4u
2
a
4
=
u
2
+ 1
u
4
2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
9
+ 4u
8
+ 10u
7
16u
6
18u
5
+ 12u
4
+ 16u
3
+ 12u
2
6u
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.379210 0.103381I
a = 0.60860 + 1.75074I
b = 0.850023 0.614930I
3.20561 + 2.41892I 4.92816 2.88947I
u = 1.379210 + 0.103381I
a = 0.60860 1.75074I
b = 0.850023 + 0.614930I
3.20561 2.41892I 4.92816 + 2.88947I
u = 0.780044
a = 0.492097
b = 0.347303
1.12618 9.42944
u = 0.399448 0.789847I
a = 0.913169 0.807761I
b = 0.978643 + 0.595733I
0.67123 + 4.69742I 2.91876 5.88322I
u = 0.399448 + 0.789847I
a = 0.913169 + 0.807761I
b = 0.978643 0.595733I
0.67123 4.69742I 2.91876 + 5.88322I
u = 0.172742 0.362556I
a = 0.68149 2.03593I
b = 0.952018 + 0.226513I
1.73094 0.74196I 3.53927 + 1.11909I
u = 0.172742 + 0.362556I
a = 0.68149 + 2.03593I
b = 0.952018 0.226513I
1.73094 + 0.74196I 3.53927 1.11909I
u = 1.48612 0.29515I
a = 0.01716 + 1.63307I
b = 1.126055 0.711355I
6.76952 8.65115I 5.78570 + 5.57892I
u = 1.48612 + 0.29515I
a = 0.01716 1.63307I
b = 1.126055 + 0.711355I
6.76952 + 8.65115I 5.78570 5.57892I
u = 1.50982 0.17565I
a = 0.603484 1.060568I
b = 0.523691 + 0.948055I
8.61577 2.58451I 8.19194 + 1.01660I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.50982 + 0.17565I
a = 0.603484 + 1.060568I
b = 0.523691 0.948055I
8.61577 + 2.58451I 8.19194 1.01660I
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u + 1)(u
3
u 1)
(u
11
+ 2u
10
4u
8
2u
7
+ 4u
6
+ 5u
5
2u
4
5u
3
u
2
+ 3u + 1)
c
2
, c
3
, c
5
(u 1)(u
3
u + 1)
(u
11
+ 2u
10
4u
9
8u
8
+ 6u
7
+ 8u
6
7u
5
+ 2u
4
+ 7u
3
3u
2
u 1)
c
4
, c
7
u(u + 1)
3
(u
11
2u
10
u
9
+ 3u
8
+ u
7
2u
6
+ 4u
5
11u
4
+ 9u
3
u
2
2u + 2)
c
8
(u + 1)(u
3
+ 2u
2
+ u + 1)(u
11
+ 4u
10
+ ··· + 11u + 1)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y 1)(y
3
2y
2
+ y 1)(y
11
4y
10
+ ··· + 11y 1)
c
2
, c
3
, c
5
(y 1)(y
3
2y
2
+ y 1)(y
11
12y
10
+ ··· 5y 1)
c
4
, c
7
(y)(y 1)
3
(y
11
6y
10
+ ··· + 8y 4)
c
8
(y 1)(y
3
2y
2
3y 1)(y
11
+ 8y
10
+ ··· + 67y 1)
9