11n
88
(K11n
88
)
1
Arc Sequences
6 1 8 9 2 3 11 4 5 1 8
Solving Sequence
3,8
4 9
1,5
2 11 7 6 10
c
3
c
8
c
4
c
2
c
11
c
7
c
6
c
10
c
1
, c
5
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
2
u + 1, a, b + 1i
I
u
2
= ha
4
+ 2a
2
+ 4, b 1, a
2
+ 2ui
I
u
3
= hu
11
+ 10u
10
+ 43u
9
+ 90u
8
+ 33u
7
281u
6
696u
5
776u
4
422u
3
72u
2
+ 24u 1,
569915u
10
5649897u
9
+ ··· + 4181332b 8449886,
9934266u
10
99754175u
9
+ ··· + 4181332a 174758259i
There are 3 irreducible components with 17 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
u + 1, a, b + 1i
(i) Arc colorings
a
3
=
0
u
a
8
=
0
1
a
4
=
0
u
a
9
=
0
1
a
1
=
1
0
a
5
=
0
u
a
2
=
u
u
a
11
=
1
1
a
7
=
1
0
a
6
=
1
u 1
a
10
=
0
1
a
10
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0
b = 1.00000
1.64493 + 2.02988I 12.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0
b = 1.00000
1.64493 2.02988I 12.00000 + 3.46410I
3
II. I
u
2
= ha
4
+ 2a
2
+ 4, b 1, a
2
+ 2ui
(i) Arc colorings
a
3
=
0
1
2
a
2
a
8
=
a
1
a
4
=
a
2
2
1
2
a
3
1
2
a
2
a
9
=
a
1
2
a
3
a 1
a
1
=
1
0
a
5
=
0
1
2
a
2
a
2
=
1
2
a
2
1
2
a
2
a
11
=
a + 1
1
a
7
=
1
0
a
6
=
1
1
2
a
2
+ 1
a
10
=
a
1
a
10
=
a
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.70711 1.22474I
b = 1.00000
6.57974 + 2.02988I 14.0000 3.4641I
u = 0.500000 + 0.866025I
a = 0.70711 + 1.22474I
b = 1.00000
6.57974 2.02988I 14.0000 + 3.4641I
u = 0.500000 + 0.866025I
a = 0.70711 1.22474I
b = 1.00000
6.57974 2.02988I 14.0000 + 3.4641I
u = 0.500000 0.866025I
a = 0.70711 + 1.22474I
b = 1.00000
6.57974 + 2.02988I 14.0000 3.4641I
5
III.
I
u
3
= hu
11
+ 10u
10
+ · · · + 24u 1, 5.70 × 10
5
u
10
5.65 × 10
6
u
9
+ · · · + 4.18 ×
10
6
b8.45×10
6
, 9.93×10
6
u
10
9.98×10
7
u
9
+· · ·+4.18×10
6
a1.75×10
8
i
(i) Arc colorings
a
3
=
0
u
a
8
=
2.37586u
10
+ 23.8570u
9
+ ··· 237.195u + 41.7949
0.136300u
10
+ 1.35122u
9
+ ··· 13.8266u + 2.02086
a
4
=
2.72957u
10
27.4206u
9
+ ··· + 261.097u 53.0344
0.129860u
10
1.26983u
9
+ ··· + 10.1196u 2.14629
a
9
=
1.58645u
10
15.9784u
9
+ ··· + 148.869u 32.6294
0.00266685u
10
0.0194938u
9
+ ··· + 3.13583u 1.09971
a
1
=
1
0
a
5
=
0.138695u
10
1.35638u
9
+ ··· + 8.05243u 5.18841
0.0525199u
10
0.445799u
9
+ ··· 1.89934u 0.135010
a
2
=
u
u
a
11
=
2.14629u
10
21.5928u
9
+ ··· + 203.990u 41.3914
0.0389885u
10
0.425795u
9
+ ··· + 7.57946u 1.71898
a
7
=
0.135010u
10
1.40262u
9
+ ··· + 6.96657u 5.13958
0.0305644u
10
+ 0.229369u
9
+ ··· 1.85974u 0.138695
a
6
=
0.135010u
10
1.40262u
9
+ ··· + 6.96657u 5.13958
0.0488352u
10
0.492036u
9
+ ··· 2.98521u 0.0861747
a
10
=
2.18528u
10
22.0186u
9
+ ··· + 211.569u 43.1104
0.0389885u
10
0.425795u
9
+ ··· + 7.57946u 1.71898
a
10
=
2.18528u
10
22.0186u
9
+ ··· + 211.569u 43.1104
0.0389885u
10
0.425795u
9
+ ··· + 7.57946u 1.71898
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 2.51273 0.34743I
a = 0.717456 + 0.144601I
b = 1.97688 0.71300I
12.58098 1.38651I 13.48038 + 0.69811I
u = 2.51273 + 0.34743I
a = 0.717456 0.144601I
b = 1.97688 + 0.71300I
12.58098 + 1.38651I 13.48038 0.69811I
u = 1.75327 2.19395I
a = 0.539260 + 0.494592I
b = 2.21938 0.57715I
13.7044 7.4448I 12.73851 + 2.77525I
u = 1.75327 + 2.19395I
a = 0.539260 0.494592I
b = 2.21938 + 0.57715I
13.7044 + 7.4448I 12.73851 2.77525I
u = 1.047950 0.797283I
a = 0.470008 + 0.545761I
b = 0.723376 0.590700I
3.55618 2.69456I 12.98158 + 3.53797I
u = 1.047950 + 0.797283I
a = 0.470008 0.545761I
b = 0.723376 + 0.590700I
3.55618 + 2.69456I 12.98158 3.53797I
u = 0.746948 0.623690I
a = 0.408421 + 0.283080I
b = 0.333924 0.361452I
0.58979 1.50760I 5.46704 + 3.06669I
u = 0.746948 + 0.623690I
a = 0.408421 0.283080I
b = 0.333924 + 0.361452I
0.58979 + 1.50760I 5.46704 3.06669I
u = 0.0529590
a = 25.8220
b = 1.10452
6.50002 13.6591
u = 0.116044
a = 4.00162
b = 0.568678
0.828081 11.8313
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.95280
a = 1.05363
b = 2.27195
18.3080 11.1746
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)
2
(u
2
+ u + 1)
(u
11
+ 2u
10
+ 7u
9
+ 10u
8
+ 17u
7
+ 19u
6
+ 14u
5
+ 18u
4
+ 6u
2
6u + 1)
c
2
(u
2
+ u + 1)
3
(u
11
+ 10u
10
+ ··· + 24u 1)
c
3
, c
4
, c
8
c
9
u
2
(u
2
2)
2
(u
11
+ u
10
10u
9
9u
8
+ 37u
7
+ 31u
6
54u
5
48u
4
+ 8u
3
4u + 4)
c
5
(u
2
u + 1)(u
2
+ u + 1)
2
(u
11
+ 2u
10
+ 7u
9
+ 10u
8
+ 17u
7
+ 19u
6
+ 14u
5
+ 18u
4
+ 6u
2
6u + 1)
c
6
(u
2
u + 1)
2
(u
2
+ u + 1)(u
11
+ 2u
10
+ ··· 90u 13)
c
7
(u 1)
2
(u + 1)
4
(u
11
+ 3u
10
+ ··· u 7)
c
10
(u 1)
6
(u
11
+ 23u
10
+ ··· + 225u + 49)
c
11
(u 1)
4
(u + 1)
2
(u
11
+ 3u
10
+ ··· u 7)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
11
+ 10y
10
+ ··· + 24y 1)
c
2
(y
2
+ y + 1)
3
(y
11
14y
10
+ ··· + 432y 1)
c
3
, c
4
, c
8
c
9
y
2
(y 2)
4
(y
11
21y
10
+ ··· + 16y 16)
c
6
(y
2
+ y + 1)
3
(y
11
38y
10
+ ··· + 4460y 169)
c
7
, c
11
(y 1)
6
(y
11
23y
10
+ ··· + 225y 49)
c
10
(y 1)
6
(y
11
63y
10
+ ··· 61879y 2401)
9