10
130
(K10n
20
)
1
Arc Sequences
4 8 5 2 10 9 5 2 6 7
Solving Sequence
5,10 2,6
4 1 9 7 8 3
c
5
c
4
c
1
c
9
c
6
c
8
c
2
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
3
a
2
+ 2a 1, u 1, b a + 1i
I
u
2
= hu
11
4u
10
u
9
+ 17u
8
+ u
7
40u
6
+ 3u
5
+ 37u
4
3u
3
9u
2
+ 7u 1,
u
10
+ 3u
9
+ 4u
8
13u
7
14u
6
+ 26u
5
+ 23u
4
14u
3
11u
2
+ 4a 6u 9,
u
10
+ 3u
9
+ 4u
8
13u
7
14u
6
+ 26u
5
+ 23u
4
18u
3
11u
2
+ 4b + 2u 1i
There are 2 irreducible components with 14 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
= ha
3
a
2
+ 2a 1, u 1, b a + 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
a
a 1
a
2
=
1
0
a
6
=
a
2
a
2
a + 1
a
4
=
1
1
a
1
=
0
1
a
9
=
a
2
a + 1
0
a
7
=
a
2
a + 1
a
2
a + 1
a
8
=
a
2
a + 1
0
a
3
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3a
2
4a + 4
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.215080 1.307141I
b = 0.78492 1.30714I
4.66906 2.82812I 1.84740 + 3.54173I
u = 1.00000
a = 0.215080 + 1.307141I
b = 0.78492 + 1.30714I
4.66906 + 2.82812I 1.84740 3.54173I
u = 1.00000
a = 0.569840
b = 0.430160
0.531480 2.69479
3
II.
I
u
2
= hu
11
4u
10
+· · ·+7u1, u
10
+3u
9
+· · ·+4a9, u
10
+3u
9
+· · ·+4b1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
4
u
10
3
4
u
9
+ ··· +
3
2
u +
9
4
1
4
u
10
3
4
u
9
+ ···
1
2
u +
1
4
a
2
=
1
0
a
6
=
1
2
u
10
2u
9
+ ···
3
2
u + 1
1
4
u
10
+
1
4
u
9
+ ··· +
1
4
u
2
+
1
4
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
9
=
7
4
u
10
23
4
u
9
+ ··· 14u +
21
4
3
2
u
10
7
2
u
9
+ ··· 7u +
3
2
a
7
=
1
4
u
10
9
4
u
9
+ ··· 7u +
15
4
5
4
u
10
+
9
4
u
9
+ ··· + 2u +
1
4
a
8
=
1
4
u
10
9
4
u
9
+ ··· 7u +
15
4
3
2
u
10
7
2
u
9
+ ··· 7u +
3
2
a
3
=
u
u
3
u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 3u
10
21
2
u
9
6u
8
+ 43u
7
+
29
2
u
6
191
2
u
5
17
2
u
4
+ 80u
3
2u
2
39
2
u +
31
2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.26769 0.68760I
a = 0.459995 0.932512I
b = 0.16375 2.54722I
8.01785 1.82060I 2.54374 + 1.21714I
u = 1.26769 + 0.68760I
a = 0.459995 + 0.932512I
b = 0.16375 + 2.54722I
8.01785 + 1.82060I 2.54374 1.21714I
u = 0.911055 0.299346I
a = 0.110424 0.459350I
b = 0.578747 0.579225I
1.69473 0.83621I 2.12521 + 2.51411I
u = 0.911055 + 0.299346I
a = 0.110424 + 0.459350I
b = 0.578747 + 0.579225I
1.69473 + 0.83621I 2.12521 2.51411I
u = 0.182568
a = 2.62925
b = 0.270203
0.824865 12.3321
u = 0.283200 0.366521I
a = 2.49507 1.49752I
b = 0.162752 0.803423I
3.43504 2.25109I 3.70368 + 2.34373I
u = 0.283200 + 0.366521I
a = 2.49507 + 1.49752I
b = 0.162752 + 0.803423I
3.43504 + 2.25109I 3.70368 2.34373I
u = 1.86528 0.08844I
a = 0.422196 0.322260I
b = 0.293316 1.067853I
12.35854 + 2.70718I 0.47291 2.44627I
u = 1.86528 + 0.08844I
a = 0.422196 + 0.322260I
b = 0.293316 + 1.067853I
12.35854 2.70718I 0.47291 + 2.44627I
u = 1.93898 0.26128I
a = 0.037929 0.811387I
b = 0.97683 3.21797I
19.3195 + 6.7782I 2.17368 2.81310I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.93898 + 0.26128I
a = 0.037929 + 0.811387I
b = 0.97683 + 3.21797I
19.3195 6.7782I 2.17368 + 2.81310I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
3
(u
11
+ 4u
10
u
9
17u
8
+ u
7
+ 40u
6
+ 3u
5
37u
4
3u
3
+ 9u
2
+ 7u + 1)
c
2
, c
8
u
3
(u
11
+ u
10
+ ··· 4u + 8)
c
3
(u 1)
3
(u
11
+ 18u
10
+ ··· + 31u + 1)
c
4
(u + 1)
3
(u
11
+ 4u
10
u
9
17u
8
+ u
7
+ 40u
6
+ 3u
5
37u
4
3u
3
+ 9u
2
+ 7u + 1)
c
5
, c
6
(u
3
+ u
2
+ 2u + 1)(u
11
+ 2u
10
+ ··· 2u 1)
c
7
(u
3
+ u
2
1)(u
11
+ 12u
9
+ 36u
7
+ 2u
6
+ 2u
5
+ 13u
4
+ 13u
3
+ u
2
+ 1)
c
9
(u
3
u
2
+ 2u 1)(u
11
+ 2u
10
+ ··· 2u 1)
c
10
(u
3
+ u
2
1)(u
11
+ 2u
10
+ ··· 6u + 9)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
3
(y
11
18y
10
+ ··· + 31y 1)
c
2
, c
8
y
3
(y
11
+ 21y
10
+ ··· + 336y 64)
c
3
(y 1)
3
(y
11
46y
10
+ ··· + 863y 1)
c
5
, c
6
, c
9
(y
3
+ 3y
2
+ 2y 1)(y
11
+ 12y
10
+ ··· 2y 1)
c
7
(y
3
y
2
+ 2y 1)(y
11
+ 24y
10
+ ··· 2y 1)
c
10
(y
3
y
2
+ 2y 1)(y
11
+ 12y
10
+ ··· 594y 81)
7