10
141
(K10n
25
)
1
Arc Sequences
6 5 10 8 1 2 9 5 1 4
Solving Sequence
1,5
6 2
3,8
9 4 7 10
c
5
c
1
c
2
c
8
c
4
c
7
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
2
2, 2a u, b + u + 1i
I
u
2
= ha
6
3a
5
+ 5a
3
7a
2
+ 4a 1, b + 1, 2a
4
4a
3
5a
2
+ 7a + u 4i
I
u
3
= hu
7
3u
6
+ 5u
4
4u
2
+ 2u 2, u
6
2u
5
2u
4
+ 3u
3
+ 2u
2
+ b u + 1,
u
6
+ u
5
+ 2u
4
+ u
3
2u
2
+ 2a 2u 2i
I
v
1
= hv + 1, b + 1, ai
There are 4 irreducible components with 16 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
2, 2a u, b + u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
6
=
u
u
a
2
=
1
2
a
3
=
1
0
a
8
=
1
2
u
u 1
a
9
=
1
2
u
1
a
4
=
1
2
u
1
a
7
=
0
u
a
10
=
1
2
u + 1
1
(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 = 1.41421
a = 0.707107
b = 0.414214
1.64493 4.00000
u = 1.41421
a = 0.707107
b = 2.41421
1.64493 4.00000
3
II. I
u
2
= ha
6
3a
5
+ 5a
3
7a
2
+ 4a 1, b + 1, 2a
4
4a
3
5a
2
+ 7a + u 4i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
2a
4
+ 4a
3
+ 5a
2
7a + 4
a
6
=
2a
4
+ 4a
3
+ 5a
2
7a + 4
2a
4
+ 4a
3
+ 5a
2
7a + 4
a
2
=
a
4
+ 2a
3
+ 3a
2
4a + 1
a
4
+ 2a
3
+ 3a
2
4a
a
3
=
a
4
+ 2a
3
+ 3a
2
4a + 1
3a
4
6a
3
8a
2
+ 11a 5
a
8
=
a
1
a
9
=
a
a
5
2a
4
3a
3
+ 4a
2
1
a
4
=
2a
5
+ 5a
4
+ 3a
3
10a
2
+ 8a 2
2a
5
6a
4
a
3
+ 12a
2
11a + 4
a
7
=
a
4
2a
3
3a
2
+ 4a 1
3a
4
6a
3
8a
2
+ 11a 5
a
10
=
a
5
+ 2a
4
+ 3a
3
4a
2
+ a + 1
a
5
2a
4
3a
3
+ 4a
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.445042
a = 1.57338
b = 1.00000
2.58490 2.00000
u = 1.80194
a = 0.500000 0.664847I
b = 1.00000
14.3344 2.00000
u = 1.80194
a = 0.500000 + 0.664847I
b = 1.00000
14.3344 2.00000
u = 1.24698
a = 0.500000 0.326949I
b = 1.00000
3.05488 2.00000
u = 1.24698
a = 0.500000 + 0.326949I
b = 1.00000
3.05488 2.00000
u = 0.445042
a = 2.57338
b = 1.00000
2.58490 2.00000
5
III. I
u
3
= hu
7
3u
6
+ 5u
4
4u
2
+ 2u 2, u
6
2u
5
2u
4
+ 3u
3
+ 2u
2
+ b
u + 1, u
6
+ u
5
+ 2u
4
+ u
3
2u
2
+ 2a 2u 2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
6
=
u
u
a
2
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
4
2u
2
a
8
=
1
2
u
6
1
2
u
5
u
4
1
2
u
3
+ u
2
+ u + 1
u
6
+ 2u
5
+ 2u
4
3u
3
2u
2
+ u 1
a
9
=
1
2
u
6
1
2
u
5
u
4
1
2
u
3
+ u
2
+ u + 1
u
6
u
5
2u
4
+ 2u
2
+ 1
a
4
=
1
2
u
6
+
1
2
u
5
+ u
4
1
2
u
3
1
u
6
u
5
3u
4
+ u
3
+ 3u
2
+ 1
a
7
=
u
3
+ 2u
u
3
+ u
a
10
=
1
2
u
6
+
1
2
u
5
+ u
4
1
2
u
3
u
2
+ u
u
6
u
5
2u
4
+ 2u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
5
+ 8u
3
6u
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.050174 0.492398I
a = 0.336313 + 0.965783I
b = 1.32157 + 0.92345I
3.39904 + 5.13113I 0.70211 5.71003I
u = 1.050174 + 0.492398I
a = 0.336313 0.965783I
b = 1.32157 0.92345I
3.39904 5.13113I 0.70211 + 5.71003I
u = 0.122110 0.584395I
a = 0.730892 0.908921I
b = 0.462714 0.668157I
0.192432 1.318886I 1.84900 + 4.97200I
u = 0.122110 + 0.584395I
a = 0.730892 + 0.908921I
b = 0.462714 + 0.668157I
0.192432 + 1.318886I 1.84900 4.97200I
u = 1.33623
a = 0.456914
b = 1.18859
3.10278 2.54953
u = 1.75995 0.15485I
a = 0.623036 0.617248I
b = 1.81001 0.80249I
13.3363 7.9365I 0.87212 + 4.07397I
u = 1.75995 + 0.15485I
a = 0.623036 + 0.617248I
b = 1.81001 + 0.80249I
13.3363 + 7.9365I 0.87212 4.07397I
7
IV. I
v
1
= hv + 1, b + 1, ai
(i) Arc colorings
a
1
=
1
0
a
5
=
1
0
a
6
=
1
0
a
2
=
1
0
a
3
=
1
0
a
8
=
0
1
a
9
=
1
1
a
4
=
1
1
a
7
=
1
0
a
10
=
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
3.28987 12.0000
9
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
u(u
2
2)(u
3
u
2
2u + 1)
2
(u
7
+ 3u
6
5u
4
+ 4u
2
+ 2u + 2)
c
2
u(u
2
2)(u
3
3u
2
4u 1)
2
(u
7
+ 9u
6
+ 30u
5
+ 45u
4
+ 46u
3
+ 32u
2
+ 22u + 14)
c
3
, c
8
(u 1)
2
(u + 1)(u
6
+ u
5
+ ··· + 2u 1)(u
7
+ u
6
+ ··· + u
2
1)
c
4
, c
10
(u 1)(u + 1)
2
(u
6
+ u
5
+ ··· + 2u 1)(u
7
+ u
6
+ ··· + u
2
1)
c
7
, c
9
(u 1)
3
(u
6
+ u
5
+ 4u
4
+ 10u
3
+ 8u
2
+ 4u + 1)
(u
7
+ u
6
+ 8u
5
+ 3u
4
+ 13u
3
+ 3u
2
+ 2u + 1)
10
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
y(y 2)
2
(y
3
5y
2
+ 6y 1)
2
(y
7
9y
6
+ 30y
5
45y
4
+ 28y
3
+ 4y
2
12y 4)
c
2
y(y 2)
2
(y
3
17y
2
+ 10y 1)
2
(y
7
21y
6
+ 182y
5
+ 203y
4
+ 304y
3
260y
2
412y 196)
c
3
, c
4
, c
8
c
10
(y 1)
3
(y
6
y
5
+ 4y
4
10y
3
+ 8y
2
4y + 1)
(y
7
y
6
+ 8y
5
3y
4
+ 13y
3
3y
2
+ 2y 1)
c
7
, c
9
(y 1)
3
(y
6
+ 7y
5
+ 12y
4
42y
3
8y
2
+ 1)
(y
7
+ 15y
6
+ 84y
5
+ 197y
4
+ 181y
3
+ 37y
2
2y 1)
11