9
37
(K9a
18
)
1
Arc Sequences
6 8 1 9 2 5 3 7 4
Solving Sequence
2,8 3,5
6 7 9 1 4
c
2
c
5
c
6
c
8
c
1
c
4
c
3
, c
7
, 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
4
u
3
+ 2u
2
2u + 1, u
3
+ b u + 1, u
3
+ u
2
+ a 2u + 2i
I
u
3
= hu
6
+ u
4
+ 2u
3
+ u
2
+ u + 2, u
3
+ b 1, u
5
u
3
2u
2
+ 2a u 1i
I
u
4
= hu
8
u
7
+ 2u
6
2u
5
+ 4u
4
3u
3
+ 2u
2
+ 1, b u, u
7
+ 2u
3
+ u
2
+ 2a 3u + 1i
I
u
5
= hu
2
+ u + 1, a + 2, b ui
I
u
6
= hb
4
b
3
+ 2b
2
2b + 1, b
3
b + a, b
3
b + u + 1i
There are 6 irreducible components with 26 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
2
=
1
0
a
8
=
u + 1
u
a
3
=
u
1
a
5
=
0
u
a
6
=
u
u
a
7
=
u
0
a
9
=
1
u
a
1
=
0
1
a
4
=
u
u + 1
a
4
=
u
u + 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
4
u
3
+ 2u
2
2u + 1, u
3
+ b u + 1, u
3
+ u
2
+ a 2u + 2i
(i) Arc colorings
a
2
=
1
0
a
8
=
u
3
u
2
+ 2u 2
u
3
+ u 1
a
3
=
u
3
+ 2u
u
3
+ u
a
5
=
0
u
a
6
=
u
u
a
7
=
u
u
3
+ u
a
9
=
1
0
a
1
=
u
2
+ 1
u
2
a
4
=
u
u
a
4
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 4u 2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.121744 1.306622I
a = 0.070696 0.758745I
b = 0.500000 + 0.866025I
2.02988I 3.46410I
u = 0.121744 + 1.306622I
a = 0.070696 + 0.758745I
b = 0.500000 0.866025I
2.02988I 3.46410I
u = 0.621744 0.440597I
a = 1.070696 0.758745I
b = 0.500000 0.866025I
2.02988I 3.46410I
u = 0.621744 + 0.440597I
a = 1.070696 + 0.758745I
b = 0.500000 + 0.866025I
2.02988I 3.46410I
5
III.
I
u
3
= hu
6
+ u
4
+ 2u
3
+ u
2
+ u + 2, u
3
+ b 1, u
5
u
3
2u
2
+ 2a u 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
1
2
u
5
+
1
2
u
3
+ u
2
+
1
2
u +
1
2
u
3
+ 1
a
3
=
1
2
u
5
1
2
u
3
1
2
u +
1
2
u
4
+ u
2
+ u + 1
a
5
=
0
u
a
6
=
u
u
a
7
=
u
u
3
+ u
a
9
=
1
2
u
5
1
2
u
3
1
2
u +
1
2
u
5
+ u
4
+ u + 1
a
1
=
u
2
+ 1
u
2
a
4
=
1
2
u
5
+
3
2
u
3
+ u
2
+
1
2
u +
3
2
u
5
+ u
4
+ u
3
+ 2u
2
+ u + 1
a
4
=
1
2
u
5
+
3
2
u
3
+ u
2
+
1
2
u +
3
2
u
5
+ u
4
+ u
3
+ 2u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
4u
3
8u 6
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.931903 0.428993I
a = 0.885437 0.407603I
b = 0.705204 1.038718I
6.15087 5.13794I 3.31793 + 3.20902I
u = 0.931903 + 0.428993I
a = 0.885437 + 0.407603I
b = 0.705204 + 1.038718I
6.15087 + 5.13794I 3.31793 3.20902I
u = 0.226699 1.074326I
a = 0.188043 0.891136I
b = 0.226699 + 1.074326I
4.07707 8.63587
u = 0.226699 + 1.074326I
a = 0.188043 + 0.891136I
b = 0.226699 1.074326I
4.07707 8.63587
u = 0.705204 1.038718I
a = 0.447394 0.658981I
b = 0.931903 0.428993I
6.15087 5.13794I 3.31793 + 3.20902I
u = 0.705204 + 1.038718I
a = 0.447394 + 0.658981I
b = 0.931903 + 0.428993I
6.15087 + 5.13794I 3.31793 3.20902I
7
IV.
I
u
4
= hu
8
u
7
+2u
6
2u
5
+4u
4
3u
3
+2u
2
+1, bu, u
7
+2u
3
+u
2
+2a3u+1i
(i) Arc colorings
a
2
=
1
0
a
8
=
1
2
u
7
u
3
1
2
u
2
+
3
2
u
1
2
u
a
3
=
1
2
u
7
u
6
+ ··· +
1
2
u +
1
2
u
2
a
5
=
0
u
a
6
=
u
u
a
7
=
u
u
3
+ u
a
9
=
1
2
u
7
1
2
u
2
+
3
2
u
1
2
u
5
+ u
3
+ u
a
1
=
u
2
+ 1
u
2
a
4
=
1
2
u
7
+ u
5
+ ··· +
1
2
u +
1
2
1
2
u
7
+ u
5
+ ··· +
1
2
u
1
2
a
4
=
1
2
u
7
+ u
5
+ ··· +
1
2
u +
1
2
1
2
u
7
+ u
5
+ ··· +
1
2
u
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
6
+ 2u
5
4u
4
+ 6u
3
12u
2
+ 6u
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.666851 1.155534I
a = 1.55320 + 0.75511I
b = 0.666851 1.155534I
3.94193 + 10.98945I 0.47099 7.14773I
u = 0.666851 + 1.155534I
a = 1.55320 0.75511I
b = 0.666851 + 1.155534I
3.94193 10.98945I 0.47099 + 7.14773I
u = 0.273948 0.520074I
a = 1.011913 0.934421I
b = 0.273948 0.520074I
0.221012 + 1.276802I 1.83102 5.88514I
u = 0.273948 + 0.520074I
a = 1.011913 + 0.934421I
b = 0.273948 + 0.520074I
0.221012 1.276802I 1.83102 + 5.88514I
u = 0.578102 1.055334I
a = 1.84091 + 0.61494I
b = 0.578102 1.055334I
1.73404 6.79402I 3.11839 + 7.09473I
u = 0.578102 + 1.055334I
a = 1.84091 0.61494I
b = 0.578102 + 1.055334I
1.73404 + 6.79402I 3.11839 7.09473I
u = 0.862697 0.615401I
a = 1.224206 + 0.050581I
b = 0.862697 0.615401I
7.44069 0.66722I 4.81639 + 2.10627I
u = 0.862697 + 0.615401I
a = 1.224206 0.050581I
b = 0.862697 + 0.615401I
7.44069 + 0.66722I 4.81639 2.10627I
9
V. I
u
5
= hu
2
+ u + 1, a + 2, b ui
(i) Arc colorings
a
2
=
1
0
a
8
=
2
u
a
3
=
2u + 1
u + 1
a
5
=
0
u
a
6
=
u
u
a
7
=
u
u + 1
a
9
=
1
0
a
1
=
u
u 1
a
4
=
u
u
a
4
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 2
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 2.00000
b = 0.500000 0.866025I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 2.00000
b = 0.500000 + 0.866025I
2.02988I 3.46410I
11
VI. I
u
6
= hb
4
b
3
+ 2b
2
2b + 1, b
3
b + a, b
3
b + u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
b
3
+ b
b
a
3
=
b
3
+ b
2
2b + 2
b
2
a
5
=
0
b
3
+ b 1
a
6
=
b
3
+ b 1
b
3
+ b 1
a
7
=
b
3
+ b 1
b
3
+ b
a
9
=
b
3
b
2
+ 2b 1
2b 1
a
1
=
b
3
b + 1
b
3
b
a
4
=
b + 1
b
3
2b
2
+ 2b 1
a
4
=
b + 1
b
3
2b
2
+ 2b 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b
3
+ 4b 2
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.121744 1.306622I
2.02988I 3.46410I
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0.121744 + 1.306622I
2.02988I 3.46410I
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0.621744 0.440597I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.621744 + 0.440597I
2.02988I 3.46410I
13
VII. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
5
c
7
(u
2
+ 1)(u
2
+ u + 1)
3
(u
4
u
3
+ ··· 2u + 1)(u
6
+ u
4
+ ··· + u + 2)
(u
8
u
7
+ 2u
6
2u
5
+ 4u
4
3u
3
+ 2u
2
+ 1)
c
3
, c
4
, c
9
(u
2
+ 1)(u
2
+ u + 1)(u
3
+ 2u + 1)
2
(u
4
u
3
+ 2u
2
2u + 1)
2
(u
8
+ 2u
7
+ 6u
6
+ 8u
5
+ 10u
4
+ 9u
3
+ 5u
2
+ 3u + 2)
c
6
, c
8
(u + 1)
2
(u
2
+ u + 1)
3
(u
4
+ 3u
3
+ 2u
2
+ 1)
(u
6
+ 2u
5
+ 3u
4
+ 2u
3
+ u
2
+ 3u + 4)
(u
8
+ 3u
7
+ 8u
6
+ 10u
5
+ 14u
4
+ 11u
3
+ 12u
2
+ 4u + 1)
14
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
5
c
7
(y + 1)
2
(y
2
+ y + 1)
3
(y
4
+ 3y
3
+ 2y
2
+ 1)
(y
6
+ 2y
5
+ 3y
4
+ 2y
3
+ y
2
+ 3y + 4)
(y
8
+ 3y
7
+ 8y
6
+ 10y
5
+ 14y
4
+ 11y
3
+ 12y
2
+ 4y + 1)
c
3
, c
4
, c
9
(y + 1)
2
(y
2
+ y + 1)(y
3
+ 4y
2
+ 4y 1)
2
(y
4
+ 3y
3
+ 2y
2
+ 1)
2
(y
8
+ 8y
7
+ 24y
6
+ 30y
5
+ 8y
4
5y
3
+ 11y
2
+ 11y + 4)
c
6
, c
8
(y 1)
2
(y
2
+ y + 1)
3
(y
4
5y
3
+ 6y
2
+ 4y + 1)
(y
6
+ 2y
5
+ 3y
4
2y
3
+ 13y
2
y + 16)
(y
8
+ 7y
7
+ 32y
6
+ 82y
5
+ 146y
4
+ 151y
3
+ 84y
2
+ 8y + 1)
15