10
61
(K10a
123
)
1
Arc Sequences
5 9 8 1 2 10 4 3 6 7
Solving Sequence
2,9 3,6
10 7 5 1 4 8
c
2
c
9
c
6
c
5
c
1
c
4
c
8
c
3
, c
7
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 1, b, a + 1i
I
u
2
= hb
2
+ 2, a + 1, u + 1i
I
u
3
= hu
10
u
9
4u
8
+ 2u
7
+ 6u
6
+ 2u
5
7u
4
3u
3
+ 8u
2
2u 3,
u
9
8u
8
+ u
7
+ 12u
6
+ 7u
5
+ 4u
4
18u
3
+ 4u
2
+ 17b 3u + 2,
7u
9
+ 22u
8
+ 10u
7
50u
6
15u
5
+ 40u
4
+ 58u
3
96u
2
+ 51a 47u + 122i
I
u
4
= hu
9
u
8
6u
7
+ 5u
6
+ 12u
5
6u
4
8u
3
u
2
+ u 1, a 1, u
7
+ u
6
+ 5u
5
4u
4
7u
3
+ 4u
2
+ 2b + u 1i
There are 4 irreducible components with 22 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 1, b, a + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
0
a
3
=
1
0
a
6
=
0
1
a
10
=
1
1
a
7
=
1
0
a
5
=
1
1
a
1
=
0
1
a
4
=
1
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
3.28987 12.0000
3
II. I
u
2
= hb
2
+ 2, a + 1, u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
b
a
3
=
b + 1
2
a
6
=
0
1
a
10
=
1
b 1
a
7
=
1
b
a
5
=
1
1
a
1
=
0
1
a
4
=
1
0
a
8
=
b + 1
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.41421I
8.22467 12.0000
u = 1.00000
a = 1.00000
b = 1.41421I
8.22467 12.0000
5
III.
I
u
3
= hu
10
u
9
+· · ·2u3, u
9
8u
8
+· · ·+17b+2, 7u
9
+22u
8
+· · ·+51a+122i
(i) Arc colorings
a
2
=
1
0
a
9
=
0.137255u
9
0.431373u
8
+ ··· + 0.921569u 2.39216
0.0588235u
9
+ 0.470588u
8
+ ··· + 0.176471u 0.117647
a
3
=
0.0784314u
9
+ 0.0392157u
8
+ ··· + 1.09804u + 0.490196
0.235294u
9
0.117647u
8
+ ··· 0.294118u 0.470588
a
6
=
0
u
a
10
=
0.137255u
9
0.431373u
8
+ ··· + 0.921569u 2.39216
1
a
7
=
0.627451u
9
+ 0.686275u
8
+ ··· 3.78431u + 1.07843
0.294118u
9
+ 0.352941u
8
+ ··· 1.11765u + 0.411765
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
u
3
+ 2u
u
3
+ u
a
8
=
0.274510u
9
0.862745u
8
+ ··· + 1.84314u 1.78431
0.470588u
9
0.235294u
8
+ ··· 0.588235u 0.941176
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
8
17
u
9
+
64
17
u
8
8
17
u
7
164
17
u
6
56
17
u
5
+
104
17
u
4
+
212
17
u
3
100
17
u
2
44
17
u +
18
17
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.349550 0.050168I
a = 0.455069 0.453019I
b = 0.233677 + 0.885557I
5.10967 2.21397I 8.88568 + 4.22289I
u = 1.349550 + 0.050168I
a = 0.455069 + 0.453019I
b = 0.233677 0.885557I
5.10967 + 2.21397I 8.88568 4.22289I
u = 0.660273 1.014194I
a = 0.795823 + 1.055371I
b = 0.05818 + 1.69128I
14.2482 3.3317I 9.91874 + 2.36228I
u = 0.660273 + 1.014194I
a = 0.795823 1.055371I
b = 0.05818 1.69128I
14.2482 + 3.3317I 9.91874 2.36228I
u = 0.506729
a = 2.27326
b = 0.416284
2.40769 0.391165
u = 0.591412 0.634202I
a = 1.10369 1.09872I
b = 0.233677 0.885557I
5.10967 + 2.21397I 8.88568 4.22289I
u = 0.591412 + 0.634202I
a = 1.10369 + 1.09872I
b = 0.233677 + 0.885557I
5.10967 2.21397I 8.88568 + 4.22289I
u = 1.15193
a = 0.439896
b = 0.416284
2.40769 0.391165
u = 1.59581 0.11029I
a = 0.455500 + 0.604056I
b = 0.05818 1.69128I
14.2482 + 3.3317I 9.91874 2.36228I
u = 1.59581 + 0.11029I
a = 0.455500 0.604056I
b = 0.05818 + 1.69128I
14.2482 3.3317I 9.91874 + 2.36228I
7
IV. I
u
4
= hu
9
u
8
+ · · · + u 1, a 1, u
7
+ u
6
+ · · · + 2b 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
1
2
u
7
1
2
u
6
+ ···
1
2
u +
1
2
a
3
=
1
2
u
7
+
1
2
u
6
+ ··· +
1
2
u +
1
2
1
2
u
8
+ 3u
6
+ ··· u +
1
2
a
6
=
0
u
a
10
=
1
1
2
u
7
1
2
u
6
+ ···
1
2
u +
1
2
a
7
=
u
1
2
u
8
+
1
2
u
7
+ ··· +
1
2
u
2
+
1
2
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
u
3
+ 2u
u
3
+ u
a
8
=
1
2
u
8
1
2
u
7
+ ···
1
2
u
2
+
3
2
u
1
2
u
8
1
2
u
7
+ ···
1
2
u
2
1
2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
8
u
7
+ 9u
6
+ 6u
5
25u
4
14u
3
+ 21u
2
+ 13u 6
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.59750 0.17287I
a = 1.00000
b = 0.638951 0.973621I
12.55800 5.12744I 10.43762 + 3.71423I
u = 1.59750 + 0.17287I
a = 1.00000
b = 0.638951 + 0.973621I
12.55800 + 5.12744I 10.43762 3.71423I
u = 0.581336 0.407332I
a = 1.00000
b = 0.00790 1.51466I
6.71646 1.46233I 6.34609 + 4.72292I
u = 0.581336 + 0.407332I
a = 1.00000
b = 0.00790 + 1.51466I
6.71646 + 1.46233I 6.34609 4.72292I
u = 0.234603 0.339731I
a = 1.00000
b = 0.215940 + 0.436674I
0.116751 + 0.880893I 2.67139 7.91481I
u = 0.234603 + 0.339731I
a = 1.00000
b = 0.215940 0.436674I
0.116751 0.880893I 2.67139 + 7.91481I
u = 1.56954
a = 1.00000
b = 0.903187
9.61991 8.30451
u = 1.65947 0.34544I
a = 1.00000
b = 0.18562 + 1.72176I
17.6214 + 8.4586I 11.39264 3.44703I
u = 1.65947 + 0.34544I
a = 1.00000
b = 0.18562 1.72176I
17.6214 8.4586I 11.39264 + 3.44703I
9
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u 1)(u + 1)
2
(u
9
+ u
8
+ ··· + u + 1)
(u
10
+ u
9
4u
8
2u
7
+ 6u
6
2u
5
7u
4
+ 3u
3
+ 8u
2
+ 2u 3)
c
2
, c
3
, c
7
c
8
u(u
2
+ 2)(u
5
u
4
+ 4u
3
3u
2
+ 3u 1)
2
(u
9
+ 3u
8
+ 10u
7
+ 19u
6
+ 31u
5
+ 37u
4
+ 34u
3
+ 22u
2
+ 8u + 2)
c
4
, c
5
, c
9
c
10
(u 1)
2
(u + 1)(u
9
+ u
8
+ ··· + u + 1)
(u
10
+ u
9
4u
8
2u
7
+ 6u
6
2u
5
7u
4
+ 3u
3
+ 8u
2
+ 2u 3)
10
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
5
c
6
, c
9
, c
10
(y 1)
3
(y
9
13y
8
+ 70y
7
197y
6
+ 300y
5
232y
4
+ 86y
3
29y
2
y 1)
(y
10
9y
9
+ ··· 52y + 9)
c
2
, c
3
, c
7
c
8
y(y + 2)
2
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y 1)
2
(y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 119y
5
+ 51y
4
52y
3
88y
2
24y 4)
11