10
74
(K10a
62
)
1
Arc Sequences
7 8 9 1 10 2 6 4 3 5
Solving Sequence
1,7 2,4
5 6 8 9 3 10
c
1
c
4
c
6
c
7
c
8
c
3
c
10
c
2
, c
5
, c
9
Representation Ideals
I =
6
\
i=1
I
u
i
I
u
1
= hu
2
+ 1, b u, a + u + 1i
I
u
2
= hu
3
u
2
+ 2u 1, a + u, b ui
I
u
3
= hu
8
+ 3u
6
2u
5
+ 2u
4
4u
3
+ u
2
u + 2, u
5
2u
3
+ u
2
+ b + 1,
u
7
+ 2u
6
+ 3u
5
+ 2u
4
+ 2u
3
2u
2
+ 2a + u 1i
I
u
4
= hu
10
+ u
9
+ 6u
8
+ 6u
7
+ 13u
6
+ 13u
5
+ 11u
4
+ 10u
3
+ 2u
2
+ 1,
u
9
+ 6u
7
+ 13u
5
+ 2u
4
+ 11u
3
+ 5u
2
+ 2a + u + 3,
u
9
6u
7
2u
6
13u
5
8u
4
11u
3
9u
2
+ 2b u 1i
I
u
5
= hu
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1, a + u, b ui
I
u
6
= hb
6
+ b
5
+ 2b
4
+ 2b
3
+ 2b
2
+ 2b + 1, b
5
+ 2b
3
+ b
2
+ 2b + u + 1, 2b
4
+ b
3
+ 2b
2
+ 2b + a + 1i
There are 6 irreducible components with 35 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
+ 1, b u, a + u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
u 1
u
a
2
=
u
1
a
4
=
0
u
a
5
=
u
u
a
6
=
u
0
a
8
=
1
u
a
9
=
1
u + 1
a
3
=
u
1
a
10
=
0
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 = 1.00000I
a = 1.00000 + 1.00000I
b = 1.00000I
1.64493 4.00000
u = 1.00000I
a = 1.00000 1.00000I
b = 1.00000I
1.64493 4.00000
3
II. I
u
2
= hu
3
u
2
+ 2u 1, a + u, b ui
(i) Arc colorings
a
1
=
1
0
a
7
=
u
u
a
2
=
u
2
+ 1
u
2
a
4
=
0
u
a
5
=
u
u
a
6
=
u
2
1
u
2
u + 1
a
8
=
1
u
2
+ 2u
a
9
=
1
2u
2
+ 2u
a
3
=
u
3u + 2
a
10
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 4u 10
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.215080 1.307141I
a = 0.215080 + 1.307141I
b = 0.215080 1.307141I
3.02413 + 2.82812I 2.49024 2.97945I
u = 0.215080 + 1.307141I
a = 0.215080 1.307141I
b = 0.215080 + 1.307141I
3.02413 2.82812I 2.49024 + 2.97945I
u = 0.569840
a = 0.569840
b = 0.569840
1.11345 9.01951
5
III. I
u
3
= hu
8
+ 3u
6
2u
5
+ 2u
4
4u
3
+ u
2
u + 2, u
5
2u
3
+ u
2
+ b +
1, u
7
+ 2u
6
+ 3u
5
+ 2u
4
+ 2u
3
2u
2
+ 2a + u 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
1
2
u
7
u
6
+ ···
1
2
u +
1
2
u
5
+ 2u
3
u
2
1
a
2
=
1
2
u
7
1
2
u
5
u
2
1
2
u
1
2
u
6
+ 2u
4
u
3
+ u
2
u + 1
a
4
=
0
u
a
5
=
u
u
a
6
=
u
3
2u
u
3
+ u
a
8
=
1
2
u
7
+
3
2
u
5
+ u
3
1
2
u +
1
2
u
6
2u
4
+ u
3
u
2
+ u 1
a
9
=
1
2
u
7
+
3
2
u
5
+ u
3
1
2
u +
1
2
1
a
3
=
1
2
u
7
+
5
2
u
5
+ ··· +
1
2
u
3
2
u
5
2u
3
+ u
2
+ 1
a
10
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
6
+ 4u
5
+ 8u
4
8u 2
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.511330 0.719091I
a = 0.151720 + 1.193721I
b = 0.547424 0.585652I
4.26996 1.39709I 0.22981 + 3.86736I
u = 0.511330 + 0.719091I
a = 0.151720 1.193721I
b = 0.547424 + 0.585652I
4.26996 + 1.39709I 0.22981 3.86736I
u = 0.309502 1.349503I
a = 1.46995 1.24354I
b = 0.547424 + 1.120873I
0.66484 7.64338I 5.77019 + 6.51087I
u = 0.309502 + 1.349503I
a = 1.46995 + 1.24354I
b = 0.547424 1.120873I
0.66484 + 7.64338I 5.77019 6.51087I
u = 0.036094 1.304743I
a = 0.787307 + 0.495756I
b = 0.547424 + 0.585652I
4.26996 + 1.39709I 0.22981 3.86736I
u = 0.036094 + 1.304743I
a = 0.787307 0.495756I
b = 0.547424 0.585652I
4.26996 1.39709I 0.22981 + 3.86736I
u = 0.856926 0.228629I
a = 0.21908 + 2.23818I
b = 0.547424 1.120873I
0.66484 + 7.64338I 5.77019 6.51087I
u = 0.856926 + 0.228629I
a = 0.21908 2.23818I
b = 0.547424 + 1.120873I
0.66484 7.64338I 5.77019 + 6.51087I
7
IV.
I
u
4
= hu
10
+u
9
+· · · +2u
2
+1, u
9
+6u
7
+· · · +2a + 3, u
9
6u
7
+· · · +2b 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
1
2
u
9
3u
7
+ ···
1
2
u
3
2
1
2
u
9
+ 3u
7
+ ··· +
1
2
u +
1
2
a
2
=
u
3
+ 2u
1
2
u
9
2u
7
+ ··· +
1
2
u +
1
2
a
4
=
0
u
a
5
=
u
u
a
6
=
u
3
2u
u
3
+ u
a
8
=
1
1
2
u
9
+ u
8
+ ··· +
1
2
u +
1
2
a
9
=
1
1
2
u
9
+ u
8
+ ··· +
1
2
u +
1
2
a
3
=
u
1
2
u
9
2u
7
+ ··· +
1
2
u +
1
2
a
10
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
8
+ 2u
7
+ 20u
6
+ 10u
5
+ 32u
4
+ 20u
3
+ 12u
2
+ 14u 8
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.748770 0.138462I
a = 0.48267 + 2.38022I
b = 0.439859 1.118372I
4.02991 3.81695I 11.33347 + 4.73761I
u = 0.748770 + 0.138462I
a = 0.48267 2.38022I
b = 0.439859 + 1.118372I
4.02991 + 3.81695I 11.33347 4.73761I
u = 0.28433 1.41260I
a = 0.009841 + 0.380107I
b = 0.910142 + 0.314063I
8.47865 6.45670I 1.02275 + 3.64794I
u = 0.28433 + 1.41260I
a = 0.009841 0.380107I
b = 0.910142 0.314063I
8.47865 + 6.45670I 1.02275 3.64794I
u = 0.05139 1.48296I
a = 0.651098 0.313606I
b = 0.782018 + 0.812236I
11.63696 2.88363I 2.09026 + 2.85464I
u = 0.05139 + 1.48296I
a = 0.651098 + 0.313606I
b = 0.782018 0.812236I
11.63696 + 2.88363I 2.09026 2.85464I
u = 0.229588 0.355227I
a = 1.027516 + 0.543852I
b = 0.177588 0.796469I
0.563291 + 1.057734I 7.69328 6.23330I
u = 0.229588 + 0.355227I
a = 1.027516 0.543852I
b = 0.177588 + 0.796469I
0.563291 1.057734I 7.69328 + 6.23330I
u = 0.35489 1.40814I
a = 1.18610 1.38404I
b = 0.609606 + 1.180280I
5.86173 + 12.00604I 2.08626 7.39232I
u = 0.35489 + 1.40814I
a = 1.18610 + 1.38404I
b = 0.609606 1.180280I
5.86173 12.00604I 2.08626 + 7.39232I
9
V. I
u
5
= hu
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1, a + u, b ui
(i) Arc colorings
a
1
=
1
0
a
7
=
u
u
a
2
=
u
2
+ 1
u
2
a
4
=
0
u
a
5
=
u
u
a
6
=
u
3
2u
u
3
+ u
a
8
=
u
5
2u
3
u
u
5
+ u
3
+ u
a
9
=
u
5
2u
3
u
1
a
3
=
2u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 3
u
5
2u
3
u
2
u 1
a
10
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
4u 6
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.713912 0.305839I
a = 0.713912 + 0.305839I
b = 0.713912 0.305839I
3.02413 2.82812I 2.49024 + 2.97945I
u = 0.713912 + 0.305839I
a = 0.713912 0.305839I
b = 0.713912 + 0.305839I
3.02413 + 2.82812I 2.49024 2.97945I
u = 0.284920 1.115141I
a = 0.284920 + 1.115141I
b = 0.284920 1.115141I
1.11345 9.01951
u = 0.284920 + 1.115141I
a = 0.284920 1.115141I
b = 0.284920 + 1.115141I
1.11345 9.01951
u = 0.498832 1.001303I
a = 0.498832 + 1.001303I
b = 0.498832 1.001303I
3.02413 2.82812I 2.49024 + 2.97945I
u = 0.498832 + 1.001303I
a = 0.498832 1.001303I
b = 0.498832 + 1.001303I
3.02413 + 2.82812I 2.49024 2.97945I
11
VI. I
u
6
=
hb
6
+b
5
+2b
4
+2b
3
+2b
2
+2b+1, b
5
+2b
3
+b
2
+2b+u+1, 2b
4
+b
3
+2b
2
+2b+a+1i
(i) Arc colorings
a
1
=
1
0
a
7
=
2b
4
b
3
2b
2
2b 1
b
a
2
=
2b
5
+ b
4
+ 2b
3
+ 2b
2
+ b + 1
b
2
a
4
=
0
b
5
2b
3
b
2
2b 1
a
5
=
b
5
+ 2b
3
+ b
2
+ 2b + 1
b
5
2b
3
b
2
2b 1
a
6
=
b
5
+ b
3
+ b
2
+ b + 1
b
3
+ b
a
8
=
b
4
b
2
b
b
5
+ b
3
+ b
a
9
=
b
4
b
2
b
1
a
3
=
b
5
+ b
3
+ b
2
b
5
2b
3
b
2
b 1
a
10
=
b
5
b
3
b
2
b 1
b
5
+ b
3
+ b
2
+ b + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b
3
4b 6
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 0.215080 + 1.307141I
a = 0.205862 0.549540I
b = 0.713912 0.305839I
3.02413 2.82812I 2.49024 + 2.97945I
u = 0.215080 1.307141I
a = 0.205862 + 0.549540I
b = 0.713912 + 0.305839I
3.02413 + 2.82812I 2.49024 2.97945I
u = 0.569840
a = 1.03980 + 2.79873I
b = 0.284920 1.115141I
1.11345 9.01951
u = 0.569840
a = 1.03980 2.79873I
b = 0.284920 + 1.115141I
1.11345 9.01951
u = 0.215080 + 1.307141I
a = 1.74566 + 0.73212I
b = 0.498832 1.001303I
3.02413 2.82812I 2.49024 + 2.97945I
u = 0.215080 1.307141I
a = 1.74566 0.73212I
b = 0.498832 + 1.001303I
3.02413 + 2.82812I 2.49024 2.97945I
13
VII. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
2
+ 1)(u
3
+ u
2
+ 2u + 1)(u
4
+ u
2
+ u + 1)
2
(u
6
u
5
+ 2u
4
2u
3
+ 2u
2
2u + 1)
2
(u
10
+ 2u
9
+ 4u
8
+ 4u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 5u
2
+ 3u + 2)
c
2
u
2
(u
3
+ u
2
1)
5
(u
4
3u
3
+ 4u
2
3u + 2)
2
(u
10
+ 2u
9
+ u
8
+ 4u
7
+ 10u
6
+ 2u
5
+ 27u
4
+ 66u
3
+ 32u
2
4u + 8)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(u
2
+ 1)(u
3
+ u
2
+ 2u + 1)
3
(u
6
u
5
+ 2u
4
2u
3
+ 2u
2
2u + 1)
(u
8
+ 3u
6
+ 2u
5
+ 2u
4
+ 4u
3
+ u
2
+ u + 2)
(u
10
u
9
+ 6u
8
6u
7
+ 13u
6
13u
5
+ 11u
4
10u
3
+ 2u
2
+ 1)
c
7
(u + 1)
2
(u
3
+ 3u
2
+ 2u 1)(u
4
+ 2u
3
+ 3u
2
+ u + 1)
2
(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
2
(u
10
+ 4u
9
+ 10u
8
+ 14u
7
+ 15u
6
+ 10u
5
+ 7u
4
+ 5u
3
+ 11u
2
+ 11u + 4)
14
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y + 1)
2
(y
3
+ 3y
2
+ 2y 1)(y
4
+ 2y
3
+ 3y
2
+ y + 1)
2
(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
2
(y
10
+ 4y
9
+ 10y
8
+ 14y
7
+ 15y
6
+ 10y
5
+ 7y
4
+ 5y
3
+ 11y
2
+ 11y + 4)
c
2
y
2
(y
3
y
2
+ 2y 1)
5
(y
4
y
3
+ 2y
2
+ 7y + 4)
2
(y
10
2y
9
+ ··· + 496y + 64)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 1)
2
(y
3
+ 3y
2
+ 2y 1)
3
(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
(y
8
+ 6y
7
+ 13y
6
+ 10y
5
2y
4
4y
3
+ y
2
+ 3y + 4)
(y
10
+ 11y
9
+ ··· + 4y + 1)
c
7
(y 1)
2
(y
3
5y
2
+ 10y 1)(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)
2
(1 + 8y
2
2y
3
+ 4y
4
y
5
+ y
6
)
2
(y
10
+ 4y
9
+ ··· 33y + 16)
15