10
76
(K10a
73
)
1
Arc Sequences
6 9 10 7 8 1 5 4 3 2
Solving Sequence
4,7
5
8,10
3 9 2 1 6
c
4
c
7
c
3
c
9
c
2
c
10
c
6
c
1
, c
5
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu 1, a, b + 1i
I
u
2
= hu
11
+ u
10
5u
9
4u
8
+ 9u
7
+ 4u
6
5u
5
+ 3u
4
3u
3
5u
2
+ 3u 1, b u,
u
10
u
9
+ 4u
8
+ 3u
7
5u
6
u
5
4u
3
+ 3u
2
+ a + 2ui
I
u
3
= hu
18
+ u
17
+ ··· 2u 1,
u
16
5u
14
+ 2u
13
+ 9u
12
8u
11
2u
10
+ 10u
9
13u
8
+ 2u
7
+ 12u
6
12u
5
+ 4u
4
+ 4u
3
6u
2
+ a + 3u 1,
u
17
6u
15
+ u
14
+ 14u
13
5u
12
13u
11
+ 9u
10
2u
9
5u
8
+ 11u
7
3u
6
3u
5
+ 3u
4
3u
3
+ u
2
+ b + u 1i
There are 3 irreducible components with 30 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
4
=
1
0
a
7
=
0
1
a
5
=
1
1
a
8
=
1
0
a
10
=
0
1
a
3
=
1
1
a
9
=
1
0
a
2
=
0
1
a
1
=
0
1
a
6
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
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
3.28987 12.0000
3
II. I
u
2
= hu
11
+ u
10
+ · · · + 3u 1, b u, u
10
u
9
+ · · · + a + 2ui
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
8
=
u
u
3
+ u
a
10
=
u
10
+ u
9
4u
8
3u
7
+ 5u
6
+ u
5
+ 4u
3
3u
2
2u
u
a
3
=
u
9
u
8
+ 4u
7
+ 3u
6
5u
5
u
4
3u
2
+ 3u
u
2
a
9
=
u
3
2u
u
3
+ u
a
2
=
u
9
u
8
+ 4u
7
+ 3u
6
5u
5
2u
4
u
2
+ 3u
u
4
2u
2
a
1
=
u
10
+ u
9
4u
8
4u
7
+ 5u
6
+ 5u
5
3u
2
3u
u
7
3u
5
+ 2u
3
+ u
a
6
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 2u
10
6u
9
+ 8u
8
+ 26u
7
14u
6
34u
5
+ 16u
4
4u
3
10u
2
+ 30u 14
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.42406 0.13076I
a = 1.140693 + 0.816975I
b = 1.42406 0.13076I
11.39952 4.33574I 15.3124 + 3.6840I
u = 1.42406 + 0.13076I
a = 1.140693 0.816975I
b = 1.42406 + 0.13076I
11.39952 + 4.33574I 15.3124 3.6840I
u = 1.360102 0.374662I
a = 0.38668 + 1.82768I
b = 1.360102 0.374662I
4.40916 11.51286I 10.44081 + 7.44023I
u = 1.360102 + 0.374662I
a = 0.38668 1.82768I
b = 1.360102 + 0.374662I
4.40916 + 11.51286I 10.44081 7.44023I
u = 0.062122 0.811051I
a = 0.07743 + 1.79273I
b = 0.062122 0.811051I
4.60381 + 2.87937I 1.58714 3.23335I
u = 0.062122 + 0.811051I
a = 0.07743 1.79273I
b = 0.062122 + 0.811051I
4.60381 2.87937I 1.58714 + 3.23335I
u = 0.264651 0.295634I
a = 0.685573 + 0.941556I
b = 0.264651 0.295634I
0.314917 + 0.927579I 5.88395 7.40073I
u = 0.264651 + 0.295634I
a = 0.685573 0.941556I
b = 0.264651 + 0.295634I
0.314917 0.927579I 5.88395 + 7.40073I
u = 1.296724 0.321683I
a = 0.67957 + 2.06108I
b = 1.296724 0.321683I
3.08453 + 5.20915I 9.44226 3.72118I
u = 1.296724 + 0.321683I
a = 0.67957 2.06108I
b = 1.296724 + 0.321683I
3.08453 5.20915I 9.44226 + 3.72118I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.32132
a = 2.22162
b = 1.32132
6.97991 12.6668
5
III.
I
u
3
= hu
18
+ u
17
+ · · · 2u 1, u
16
5u
14
+ · · · + a 1, u
17
6u
15
+ · · · + b 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
8
=
u
u
3
+ u
a
10
=
u
16
+ 5u
14
+ ··· 3u + 1
u
17
+ 6u
15
+ ··· u + 1
a
3
=
u
15
+ 6u
13
+ ··· 6u
2
+ 4u
u
17
5u
15
+ ··· u 2
a
9
=
u
3
2u
u
3
+ u
a
2
=
u
13
4u
11
+ ··· + 3u 1
2u
17
11u
15
+ ··· u 2
a
1
=
u
10
+ 3u
8
u
7
2u
6
+ 2u
5
3u
4
+ 3u
2
2u + 1
2u
17
+ 12u
15
+ ··· + u + 2
a
6
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
17
+ 24u
15
4u
14
56u
13
+ 20u
12
+ 48u
11
36u
10
+ 24u
9
+
16u
8
64u
7
+ 24u
6
+ 12u
5
20u
4
+ 24u
3
8u
2
2
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.311030 0.356898I
a = 0.571576 1.050599I
b = 0.138557 + 0.857281I
0.30826 7.08493I 6.42320 + 5.91335I
u = 1.311030 + 0.356898I
a = 0.571576 + 1.050599I
b = 0.138557 0.857281I
0.30826 + 7.08493I 6.42320 5.91335I
u = 1.308540 0.065670I
a = 0.226670 0.614277I
b = 0.535620 + 0.576021I
5.07330 2.09337I 12.51499 + 4.16283I
u = 1.308540 + 0.065670I
a = 0.226670 + 0.614277I
b = 0.535620 0.576021I
5.07330 + 2.09337I 12.51499 4.16283I
u = 1.253839 0.303492I
a = 0.358946 + 0.253812I
b = 1.112355 0.436175I
2.67293 2.45442I 9.67208 + 2.91298I
u = 1.253839 + 0.303492I
a = 0.358946 0.253812I
b = 1.112355 + 0.436175I
2.67293 + 2.45442I 9.67208 2.91298I
u = 0.285873
a = 2.38625
b = 1.11181
2.09142 3.34765
u = 0.035822 0.749326I
a = 1.14608 1.68926I
b = 1.209733 + 0.357771I
1.08148 1.33617I 4.71591 + 0.70175I
u = 0.035822 + 0.749326I
a = 1.14608 + 1.68926I
b = 1.209733 0.357771I
1.08148 + 1.33617I 4.71591 0.70175I
u = 0.138557 0.857281I
a = 1.12730 1.49368I
b = 1.311030 + 0.356898I
0.30826 + 7.08493I 6.42320 5.91335I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.138557 + 0.857281I
a = 1.12730 + 1.49368I
b = 1.311030 0.356898I
0.30826 7.08493I 6.42320 + 5.91335I
u = 0.535620 0.576021I
a = 0.442807 0.996705I
b = 1.308540 + 0.065670I
5.07330 + 2.09337I 12.51499 4.16283I
u = 0.535620 + 0.576021I
a = 0.442807 + 0.996705I
b = 1.308540 0.065670I
5.07330 2.09337I 12.51499 + 4.16283I
u = 1.11181
a = 0.613565
b = 0.285873
2.09142 3.34765
u = 1.112355 0.436175I
a = 0.474652 0.002042I
b = 1.253839 0.303492I
2.67293 2.45442I 9.67208 + 2.91298I
u = 1.112355 + 0.436175I
a = 0.474652 + 0.002042I
b = 1.253839 + 0.303492I
2.67293 + 2.45442I 9.67208 2.91298I
u = 1.209733 0.357771I
a = 0.724320 0.974134I
b = 0.035822 + 0.749326I
1.08148 + 1.33617I 4.71591 0.70175I
u = 1.209733 + 0.357771I
a = 0.724320 + 0.974134I
b = 0.035822 0.749326I
1.08148 1.33617I 4.71591 + 0.70175I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
u(u
9
u
8
+ 2u
7
u
6
+ 3u
5
u
4
+ 2u
3
+ u + 1)
2
(u
11
+ 3u
10
+ 6u
9
+ 7u
8
+ 7u
7
+ 3u
6
2u
5
8u
4
7u
3
5u
2
2u 2)
c
2
, c
3
, c
9
(u + 1)
(u
11
+ u
10
5u
9
4u
8
+ 9u
7
+ 4u
6
5u
5
+ 3u
4
3u
3
5u
2
+ 3u 1)
(u
18
+ u
17
+ ··· 2u 1)
c
4
, c
5
, c
7
(u 1)
(u
11
+ u
10
5u
9
4u
8
+ 9u
7
+ 4u
6
5u
5
+ 3u
4
3u
3
5u
2
+ 3u 1)
(u
18
+ u
17
+ ··· 2u 1)
c
8
, c
10
u(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u 1)
2
(u
11
+ 3u
10
+ ··· 16u 4)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
y(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y 1)
2
(y
11
+ 3y
10
+ ··· 16y 4)
c
2
, c
3
, c
4
c
5
, c
7
, c
9
(y 1)(y
11
11y
10
+ ··· y 1)(y
18
13y
17
+ ··· 12y + 1)
c
8
, c
10
y(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
15y
4
+ 22y
2
+ 13y 1)
2
(y
11
+ 7y
10
+ ··· + 24y 16)
10