10
63
(K10a
51
)
1
Arc Sequences
5 10 9 6 2 8 1 4 3 7
Solving Sequence
1,5
2
6,7
8 4 10 3 9
c
1
c
5
c
7
c
4
c
10
c
2
c
9
c
3
, c
6
, c
8
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 1, a + 2, b + 1i
I
u
2
= ha
2
4a + 6, u + 1, b 1i
I
u
3
= hu
13
+ u
12
2u
11
3u
10
+ 4u
9
+ 6u
8
2u
7
7u
6
+ 2u
5
+ 6u
4
4u
2
+ u + 1, b u,
u
11
+ u
10
u
9
2u
8
+ 3u
7
+ 4u
6
+ u
5
3u
4
+ 3u
3
+ 3u
2
+ 2a + u 1i
I
u
4
= hu
18
+ u
17
+ ··· + 4u + 3, 831u
17
+ 26u
16
+ ··· + 1249b 103,
3742u
17
1171u
16
+ ··· + 3747a 5305i
There are 4 irreducible components with 34 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 + 2, b + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
6
=
1
0
a
7
=
2
1
a
8
=
1
1
a
4
=
1
1
a
10
=
1
1
a
3
=
1
1
a
9
=
1
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 = 2.00000
b = 1.00000
3.28987 12.0000
3
II. I
u
2
= ha
2
4a + 6, u + 1, b 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
6
=
1
0
a
7
=
a
1
a
8
=
a 1
1
a
4
=
1
1
a
10
=
a + 1
1
a
3
=
a + 5
a + 3
a
9
=
2a 3
a 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.00000
a = 2.00000 1.41421I
b = 1.00000
1.64493 12.0000
u = 1.00000
a = 2.00000 + 1.41421I
b = 1.00000
1.64493 12.0000
5
III. I
u
3
= hu
13
+ u
12
+ · · · + u + 1, b u, u
11
+ u
10
+ · · · + 2a 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
7
=
1
2
u
11
1
2
u
10
+ ···
1
2
u +
1
2
u
a
8
=
1
2
u
11
1
2
u
10
+ ···
3
2
u +
1
2
u
a
4
=
u
3
u
5
u
3
+ u
a
10
=
1
2
u
12
+
1
2
u
11
+ ···
1
2
u + 1
u
2
a
3
=
u
12
3
2
u
11
+ ··· +
5
2
u +
3
2
1
2
u
12
+
1
2
u
11
+ ··· +
1
2
u
2
1
2
u
a
9
=
u
11
u
10
+ u
9
+ 2u
8
2u
7
4u
6
u
5
+ 3u
4
u
3
3u
2
u + 1
1
2
u
11
1
2
u
10
+ ··· +
3
2
u +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes
= u
11
+ 3u
10
u
9
8u
8
+ u
7
+ 16u
6
+ 5u
5
13u
4
5u
3
+ 13u
2
+ 7u 13
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.168156 0.683587I
a = 1.92699 + 0.82808I
b = 1.168156 0.683587I
7.93590 11.51169I 7.17210 + 6.84034I
u = 1.168156 + 0.683587I
a = 1.92699 0.82808I
b = 1.168156 + 0.683587I
7.93590 + 11.51169I 7.17210 6.84034I
u = 0.982157 0.559210I
a = 1.39246 + 1.35905I
b = 0.982157 0.559210I
1.52198 4.38846I 11.77625 + 4.32757I
u = 0.982157 + 0.559210I
a = 1.39246 1.35905I
b = 0.982157 + 0.559210I
1.52198 + 4.38846I 11.77625 4.32757I
u = 0.622947 0.904317I
a = 0.700348 + 0.299710I
b = 0.622947 0.904317I
11.49217 0.51506I 3.16885 + 2.03529I
u = 0.622947 + 0.904317I
a = 0.700348 0.299710I
b = 0.622947 + 0.904317I
11.49217 + 0.51506I 3.16885 2.03529I
u = 0.421510
a = 0.603651
b = 0.421510
0.714503 13.6630
u = 0.644264 0.592137I
a = 0.437115 + 0.842660I
b = 0.644264 0.592137I
2.65637 + 1.35876I 4.47319 3.17078I
u = 0.644264 + 0.592137I
a = 0.437115 0.842660I
b = 0.644264 + 0.592137I
2.65637 1.35876I 4.47319 + 3.17078I
u = 0.750183 0.366139I
a = 0.02604 + 1.61200I
b = 0.750183 0.366139I
2.83101 + 1.40076I 6.04773 4.90140I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.750183 + 0.366139I
a = 0.02604 1.61200I
b = 0.750183 + 0.366139I
2.83101 1.40076I 6.04773 + 4.90140I
u = 1.089569 0.623417I
a = 1.75482 + 1.07931I
b = 1.089569 0.623417I
0.00714 + 8.67404I 9.53036 8.43648I
u = 1.089569 + 0.623417I
a = 1.75482 1.07931I
b = 1.089569 + 0.623417I
0.00714 8.67404I 9.53036 + 8.43648I
8
IV. I
u
4
= hu
18
+ u
17
+ · · · + 4u + 3, 831u
17
+ 26u
16
+ · · · + 1249b
103, 3742u
17
1171u
16
+ · · · + 3747a 5305i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
7
=
0.998666u
17
+ 0.312517u
16
+ ··· 0.0579130u + 1.41580
0.665332u
17
0.0208167u
16
+ ··· + 1.27542u + 0.0824660
a
8
=
1
3
u
17
+
1
3
u
16
+ ···
4
3
u +
4
3
0.665332u
17
0.0208167u
16
+ ··· + 1.27542u + 0.0824660
a
4
=
u
3
u
5
u
3
+ u
a
10
=
0.713638u
17
+ 0.733920u
16
+ ··· 4.17187u 0.830531
0.686149u
17
+ 0.0960769u
16
+ ··· 2.57886u 2.99600
a
3
=
0.617027u
17
0.692287u
16
+ ··· 0.378970u 2.33440
0.337870u
17
0.470777u
16
+ ··· 0.463571u 1.51962
a
9
=
0.864158u
17
+ 0.414198u
16
+ ··· 0.0955431u + 2.12837
0.305845u
17
0.0288231u
16
+ ··· + 0.0736589u 0.501201
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1792
1249
u
17
+
1976
1249
u
16
+ ···
1164
1249
u
10326
1249
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.174714 0.153689I
a = 1.78075 0.41253I
b = 0.943806 0.303030I
3.25448 + 1.10969I 11.44626 6.23947I
u = 1.174714 + 0.153689I
a = 1.78075 + 0.41253I
b = 0.943806 + 0.303030I
3.25448 1.10969I 11.44626 + 6.23947I
u = 1.054065 0.732497I
a = 0.202539 + 0.522210I
b = 0.437217 + 0.966793I
10.17134 5.50049I 4.51063 + 2.97298I
u = 1.054065 + 0.732497I
a = 0.202539 0.522210I
b = 0.437217 0.966793I
10.17134 + 5.50049I 4.51063 2.97298I
u = 0.857711 0.553032I
a = 2.16348 0.64494I
b = 1.339953 0.113954I
3.77376 2.21388I 7.75885 + 3.04598I
u = 0.857711 + 0.553032I
a = 2.16348 + 0.64494I
b = 1.339953 + 0.113954I
3.77376 + 2.21388I 7.75885 3.04598I
u = 0.640279 0.479450I
a = 0.360423 0.269890I
b = 0.640279 + 0.479450I
0.453072 10.3333
u = 0.640279 + 0.479450I
a = 0.360423 + 0.269890I
b = 0.640279 0.479450I
0.453072 10.3333
u = 0.437217 0.966793I
a = 0.665721 0.126228I
b = 1.054065 + 0.732497I
10.17134 + 5.50049I 4.51063 2.97298I
u = 0.437217 + 0.966793I
a = 0.665721 + 0.126228I
b = 1.054065 0.732497I
10.17134 5.50049I 4.51063 + 2.97298I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.444651 0.766223I
a = 0.369006 0.372522I
b = 0.935577 + 0.603792I
1.87293 3.41073I 6.11762 + 4.39642I
u = 0.444651 + 0.766223I
a = 0.369006 + 0.372522I
b = 0.935577 0.603792I
1.87293 + 3.41073I 6.11762 4.39642I
u = 0.935577 0.603792I
a = 0.309927 + 0.279242I
b = 0.444651 + 0.766223I
1.87293 + 3.41073I 6.11762 4.39642I
u = 0.935577 + 0.603792I
a = 0.309927 0.279242I
b = 0.444651 0.766223I
1.87293 3.41073I 6.11762 + 4.39642I
u = 0.943806 0.303030I
a = 2.13524 0.46209I
b = 1.174714 0.153689I
3.25448 + 1.10969I 11.44626 6.23947I
u = 0.943806 + 0.303030I
a = 2.13524 + 0.46209I
b = 1.174714 + 0.153689I
3.25448 1.10969I 11.44626 + 6.23947I
u = 1.339953 0.113954I
a = 1.59865 0.61604I
b = 0.857711 0.553032I
3.77376 2.21388I 7.75885 + 3.04598I
u = 1.339953 + 0.113954I
a = 1.59865 + 0.61604I
b = 0.857711 + 0.553032I
3.77376 + 2.21388I 7.75885 3.04598I
11
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u 1)(u + 1)
2
(u
13
+ u
12
+ ··· + u + 1)(u
18
+ u
17
+ ··· + 4u + 3)
c
2
, c
3
, c
8
c
9
u(u
2
+ 2)(u
9
u
8
+ 6u
7
5u
6
+ 11u
5
7u
4
+ 6u
3
2u
2
+ u 1)
2
(u
13
+ 3u
12
+ ··· 4u 2)
c
4
, c
6
(u 1)
3
(u
13
+ 5u
12
+ ··· + 9u + 1)(u
18
+ 9u
17
+ ··· + 40u + 9)
c
5
, c
10
(u 1)
2
(u + 1)(u
13
+ u
12
+ ··· + u + 1)(u
18
+ u
17
+ ··· + 4u + 3)
12
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
7
c
10
(y 1)
3
(y
13
5y
12
+ ··· + 9y 1)(y
18
9y
17
+ ··· 40y + 9)
c
2
, c
3
, c
8
c
9
y(y + 2)
2
(y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
6y
2
3y 1)
2
(y
13
+ 15y
12
+ ··· + 16y 4)
c
4
, c
6
(y 1)
3
(y
13
+ 11y
12
+ ··· + 25y 1)(y
18
y
17
+ ··· + 524y + 81)
13