9
49
(K9n
8
)
1
Arc Sequences
8 6 8 2 9 2 5 3 5
Solving Sequence
2,8 1,5
4 3 7 6 9
c
1
c
4
c
3
c
7
c
6
c
9
c
2
, c
5
, c
8
Representation Ideals
I =
5
\
i=1
I
u
i
I
u
1
= hu
3
u
2
1, a + 1, b + ui
I
u
2
= hu
3
+ 3u
2
+ 2u 1, a + 1, b ui
I
u
3
= hu
4
+ 3u
3
+ 5u
2
+ 6u + 4, u
3
2u
2
+ 2a 2u 3, u
3
3u
2
+ 2b 3u 4i
I
u
4
= ha
4
a
3
+ 5a
2
8a + 4, a
3
3a
2
+ 8b + 3a 14, a
3
3a
2
+ 3a + 8u 14i
I
u
5
= hb
4
+ 3b
3
+ 5b
2
+ 6b + 4, a + 1, b
3
3b
2
3b + 2u 4i
There are 5 irreducible components with 18 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
3
u
2
1, a + 1, b + ui
(i) Arc colorings
a
2
=
1
0
a
8
=
1
u
a
1
=
u + 1
u
2
a
5
=
0
u
a
4
=
u
u
a
3
=
u
2
+ 2u
u 1
a
7
=
1
u
2
u
a
6
=
u
2
u 1
u
2
u
a
9
=
u + 1
u
2
+ 1
a
9
=
u + 1
u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
3
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.232786 0.792552I
a = 1.00000
b = 0.232786 + 0.792552I
5.50124 + 1.58317I 1.27815 1.10697I
u = 0.232786 + 0.792552I
a = 1.00000
b = 0.232786 0.792552I
5.50124 1.58317I 1.27815 + 1.10697I
u = 1.46557
a = 1.00000
b = 1.46557
4.42273 9.44370
3
II. I
u
2
= hu
3
+ 3u
2
+ 2u 1, a + 1, b ui
(i) Arc colorings
a
2
=
1
0
a
8
=
1
u
a
1
=
u + 1
u
2
a
5
=
0
u
a
4
=
u
u
a
3
=
u
2
+ 2u
2u
2
+ 3u 1
a
7
=
1
u
2
+ u
a
6
=
u
2
+ u 1
u
2
+ u
a
9
=
u + 1
u
2
+ 2u 1
a
9
=
u + 1
u
2
+ 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
15
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.66236 0.56228I
a = 1.00000
b = 1.66236 0.56228I
1.30745 9.42707I 7.65816 + 5.60826I
u = 1.66236 + 0.56228I
a = 1.00000
b = 1.66236 + 0.56228I
1.30745 + 9.42707I 7.65816 5.60826I
u = 0.324718
a = 1.00000
b = 0.324718
0.674976 14.6837
5
III.
I
u
3
= hu
4
+3u
3
+5u
2
+6u +4, u
3
2u
2
+2a 2u3, u
3
3u
2
+2b 3u4i
(i) Arc colorings
a
2
=
1
0
a
8
=
1
2
u
3
+ u
2
+ u +
3
2
1
2
u
3
+
3
2
u
2
+
3
2
u + 2
a
1
=
3
4
u
3
7
4
u
2
9
4
u 2
1
2
u
3
3
2
u
2
3
2
u 3
a
5
=
0
u
a
4
=
u
u
a
3
=
1
4
u
3
1
4
u
2
+
1
4
u
1
2
u
3
1
2
u
2
1
2
u 1
a
7
=
1
2
u
3
+ u
2
+ u +
3
2
1
2
u
3
+
1
2
u
2
+
1
2
u
a
6
=
u
3
+
3
2
u
2
+
3
2
u +
3
2
1
2
u
3
+
1
2
u
2
+
1
2
u
a
9
=
3
4
u
3
7
4
u
2
9
4
u 2
1
2
u
3
3
2
u
2
3
2
u 1
a
9
=
3
4
u
3
7
4
u
2
9
4
u 2
1
2
u
3
3
2
u
2
3
2
u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
3
2u
2
2u 10
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.30902 0.53523I
a = 1.059017 0.433013I
b = 1.61803
3.94784 2.02988I 8.00000 + 3.46410I
u = 1.30902 + 0.53523I
a = 1.059017 + 0.433013I
b = 1.61803
3.94784 + 2.02988I 8.00000 3.46410I
u = 0.19098 1.40126I
a = 0.059017 + 0.433013I
b = 0.618034
3.94784 + 2.02988I 8.00000 3.46410I
u = 0.19098 + 1.40126I
a = 0.059017 0.433013I
b = 0.618034
3.94784 2.02988I 8.00000 + 3.46410I
7
IV.
I
u
4
= ha
4
a
3
+ 5a
2
8a + 4, a
3
3a
2
+ 8b + 3a 14, a
3
3a
2
+ 3a + 8u 14i
(i) Arc colorings
a
2
=
1
0
a
8
=
a
1
8
a
3
+
3
8
a
2
3
8
a +
7
4
a
1
=
1
4
a
3
1
4
a
2
3
4
a +
1
2
1
8
a
3
3
8
a
2
+
3
8
a
11
4
a
5
=
0
1
8
a
3
+
3
8
a
2
3
8
a +
7
4
a
4
=
1
8
a
3
+
3
8
a
2
3
8
a +
7
4
1
8
a
3
+
3
8
a
2
3
8
a +
7
4
a
3
=
1
8
a
3
3
8
a
2
+
3
8
a +
1
4
3
8
a
3
1
8
a
2
+
17
8
a
9
4
a
7
=
a
3
8
a
3
+
1
8
a
2
17
8
a +
5
4
a
6
=
3
8
a
3
+
1
8
a
2
9
8
a +
5
4
3
8
a
3
+
1
8
a
2
17
8
a +
5
4
a
9
=
1
4
a
3
1
4
a
2
3
4
a +
1
2
3
4
a
3
1
4
a
2
+
13
4
a
9
2
a
9
=
1
4
a
3
1
4
a
2
3
4
a +
1
2
3
4
a
3
1
4
a
2
+
13
4
a
9
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =
3
2
a
3
1
2
a
2
+
17
2
a 15
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.618034
a = 0.30902 2.26728I
b = 0.618034
3.94784 + 2.02988I 8.00000 3.46410I
u = 0.618034
a = 0.30902 + 2.26728I
b = 0.618034
3.94784 2.02988I 8.00000 + 3.46410I
u = 1.61803
a = 0.809017 0.330792I
b = 1.61803
3.94784 + 2.02988I 8.00000 3.46410I
u = 1.61803
a = 0.809017 + 0.330792I
b = 1.61803
3.94784 2.02988I 8.00000 + 3.46410I
9
V. I
u
5
= hb
4
+ 3b
3
+ 5b
2
+ 6b + 4, a + 1, b
3
3b
2
3b + 2u 4i
(i) Arc colorings
a
2
=
1
0
a
8
=
1
b
a
1
=
b + 1
b
2
a
5
=
0
1
2
b
3
+
3
2
b
2
+
3
2
b + 2
a
4
=
1
2
b
3
+
3
2
b
2
+
3
2
b + 2
1
2
b
3
+
3
2
b
2
+
3
2
b + 2
a
3
=
b
3
+ 2b
2
+ 2b + 2
3
2
b
3
+
7
2
b
2
+
9
2
b + 4
a
7
=
1
1
2
b
3
+
3
2
b
2
+
5
2
b + 3
a
6
=
1
2
b
3
+
3
2
b
2
+
5
2
b + 2
1
2
b
3
+
3
2
b
2
+
5
2
b + 3
a
9
=
b + 1
1
2
b
3
3
2
b
2
3
2
b 1
a
9
=
b + 1
1
2
b
3
3
2
b
2
3
2
b 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2b
3
2b
2
2b 10
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.61803
a = 1.00000
b = 1.30902 0.53523I
3.94784 2.02988I 8.00000 + 3.46410I
u = 1.61803
a = 1.00000
b = 1.30902 + 0.53523I
3.94784 + 2.02988I 8.00000 3.46410I
u = 0.618034
a = 1.00000
b = 0.19098 1.40126I
3.94784 + 2.02988I 8.00000 3.46410I
u = 0.618034
a = 1.00000
b = 0.19098 + 1.40126I
3.94784 2.02988I 8.00000 + 3.46410I
11
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
, c
7
(u
2
u 1)
4
(u
3
+ u
2
+ 1)(u
3
+ 3u
2
+ 2u 1)(u
4
+ 3u
3
+ ··· + 6u + 4)
c
2
, c
5
, c
8
(u
2
u + 1)
4
(u
3
+ u 1)(u
3
+ 2u
2
+ 3u + 1)(u
4
+ 3u
3
+ ··· + 6u + 4)
c
3
, c
6
, c
9
(u
2
u + 1)
4
(u
3
+ u + 1)(u
3
+ 2u
2
+ 3u + 1)(u
4
+ 3u
3
+ ··· + 6u + 4)
12
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
7
(y
2
3y + 1)
4
(y
3
5y
2
+ 10y 1)(y
3
y
2
2y 1)
(y
4
+ y
3
3y
2
+ 4y + 16)
c
2
, c
3
, c
5
c
6
, c
8
, c
9
(y
2
+ y + 1)
4
(y
3
+ 2y
2
+ y 1)(y
3
+ 2y
2
+ 5y 1)
(y
4
+ y
3
3y
2
+ 4y + 16)
13