10
133
(K10n
4
)
1
Arc Sequences
8 1 6 3 10 4 10 2 5 3
Solving Sequence
3,6
4
7,10
8 1 2 9 5
c
3
c
6
c
7
c
10
c
2
c
8
c
5
c
1
, c
4
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
3
+ b
2
+ 2b + 1, a, u 1i
I
u
2
= hu
12
+ 4u
11
+ 8u
10
+ 5u
9
5u
8
15u
7
9u
6
+ 8u
4
+ 2u
3
2u
2
4u 1,
u
11
+ 5u
10
+ 9u
9
+ 2u
8
15u
7
18u
6
+ u
5
+ 13u
4
+ 5u
3
u
2
+ 4b 7u + 1,
u
11
5u
10
11u
9
8u
8
+ 9u
7
+ 24u
6
+ 13u
5
7u
4
13u
3
3u
2
+ 2a + 5u + 5i
There are 2 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
= hb
3
+ b
2
+ 2b + 1, a, u 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
1
a
4
=
1
1
a
7
=
1
0
a
10
=
0
b
a
8
=
1
b
2
a
1
=
b
b
a
2
=
b
2
+ 1
b
2
a
9
=
0
b
a
5
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7b
2
5b 17
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0
b = 0.569840
2.75839 16.4238
u = 1.00000
a = 0
b = 0.215080 1.307141I
1.37919 2.82812I 4.28809 + 2.59975I
u = 1.00000
a = 0
b = 0.215080 + 1.307141I
1.37919 + 2.82812I 4.28809 2.59975I
3
II.
I
u
2
= hu
12
+ 4 u
11
+· · ·−4u1, u
11
+5u
10
+· · ·+4b+1, u
11
5u
10
+· · ·+2a+5i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
4
=
1
u
2
a
7
=
u
u
3
+ u
a
10
=
1
2
u
11
+
5
2
u
10
+ ···
5
2
u
5
2
1
4
u
11
5
4
u
10
+ ··· +
7
4
u
1
4
a
8
=
u
4
u
2
2u + 1
1
4
u
11
3
4
u
10
+ ··· +
3
4
u +
1
4
a
1
=
3
4
u
11
+
15
4
u
10
+ ···
17
4
u
9
4
1
4
u
11
5
4
u
10
+ ··· +
7
4
u
1
4
a
2
=
1
4
u
11
3
4
u
10
+ ··· +
3
4
u +
1
4
1
4
u
11
+
3
4
u
10
+ ··· +
1
4
u
1
4
a
9
=
3
2
u
11
9
2
u
10
+ ··· +
3
2
u +
3
2
3
4
u
11
+
7
4
u
10
+ ···
1
4
u +
3
4
a
5
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 2u
11
+
17
2
u
10
+ 16u
9
+
13
2
u
8
39
2
u
7
34u
6
9u
5
+
35
2
u
4
+ 19u
3
+
3
2
u
2
12u
19
2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.18067 1.13803I
a = 0.702429 + 1.111313I
b = 0.15451 1.86459I
14.0447 7.7983I 3.16952 + 4.22102I
u = 1.18067 + 1.13803I
a = 0.702429 1.111313I
b = 0.15451 + 1.86459I
14.0447 + 7.7983I 3.16952 4.22102I
u = 1.10559 1.21488I
a = 0.744589 1.118153I
b = 0.11602 + 1.80584I
14.3370 0.8045I 2.71291 0.16086I
u = 1.10559 + 1.21488I
a = 0.744589 + 1.118153I
b = 0.11602 1.80584I
14.3370 + 0.8045I 2.71291 + 0.16086I
u = 0.561933 0.696285I
a = 0.925264 + 0.846250I
b = 0.544421 1.250457I
2.66318 4.39533I 2.94428 + 5.22312I
u = 0.561933 + 0.696285I
a = 0.925264 0.846250I
b = 0.544421 + 1.250457I
2.66318 + 4.39533I 2.94428 5.22312I
u = 0.291129
a = 1.77307
b = 0.728189
1.41716 6.22072
u = 0.267707 0.884422I
a = 0.991606 0.968229I
b = 0.208639 + 1.095629I
3.72986 + 1.03019I 1.27943 1.44119I
u = 0.267707 + 0.884422I
a = 0.991606 + 0.968229I
b = 0.208639 1.095629I
3.72986 1.03019I 1.27943 + 1.44119I
u = 0.703419 0.354505I
a = 0.543453 0.851824I
b = 0.137910 + 0.436156I
0.87372 + 1.32529I 6.28742 4.78445I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.703419 + 0.354505I
a = 0.543453 + 0.851824I
b = 0.137910 0.436156I
0.87372 1.32529I 6.28742 + 4.78445I
u = 1.11609
a = 0.469158
b = 0.247448
2.23241 0.00782211
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
8
(u
3
+ u
2
1)
(u
12
+ 2u
11
+ u
10
2u
9
+ u
8
+ 6u
7
+ 4u
6
3u
5
+ 6u
3
+ 3u
2
u 1)
c
2
, c
10
(u
3
+ u
2
+ 2u + 1)(u
12
+ 2u
11
+ ··· + 7u + 1)
c
3
, c
6
(u 1)
3
(u
12
+ 4u
11
+ 8u
10
+ 5u
9
5u
8
15u
7
9u
6
+ 8u
4
+ 2u
3
2u
2
4u 1)
c
4
(u + 1)
3
(u
12
+ 14u
10
+ ··· + 12u + 1)
c
5
, c
9
u
3
(u
12
+ u
11
+ ··· + 36u + 8)
c
7
(u
3
+ u
2
+ 2u + 1)(u
12
+ 2u
11
+ ··· + 175u 49)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
(y
3
y
2
+ 2y 1)(y
12
2y
11
+ ··· 7y + 1)
c
2
, c
10
(y
3
+ 3y
2
+ 2y 1)(y
12
+ 18y
11
+ ··· 7y + 1)
c
3
, c
6
(y 1)
3
(y
12
+ 14y
10
+ ··· 12y + 1)
c
4
(y 1)
3
(y
12
+ 28y
11
+ ··· 136y + 1)
c
5
, c
9
y
3
(y
12
21y
11
+ ··· 464y + 64)
c
7
(y
3
+ 3y
2
+ 2y 1)(y
12
+ 54y
11
+ ··· 39739y + 2401)
8