11n
48
(K11n
48
)
1
Arc Sequences
6 1 7 9 2 10 11 1 6 4 8
Solving Sequence
1,6
2
3,9
10 5 8 11 7 4
c
1
c
2
c
9
c
5
c
8
c
11
c
7
c
3
c
4
, c
6
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b, a u 1i
I
u
2
= hu
2
+ u + 1, a + u + 1, b
2
2i
I
u
3
= hu
20
+ 4u
19
+ ··· + 2u + 1, 1.14153 × 10
15
u
19
4.26909 × 10
15
u
18
+ ··· + 8.56391 × 10
15
b + 1.10053 × 10
16
,
6.37825 × 10
15
u
19
+ 2.69128 × 10
16
u
18
+ ··· + 8.56391 × 10
15
a + 3.19918 × 10
16
i
There are 3 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
+ u + 1, b, a u 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
9
=
u + 1
0
a
10
=
u + 1
u
a
5
=
u
u + 1
a
8
=
u + 1
0
a
11
=
1
0
a
7
=
u + 1
0
a
4
=
u 1
u + 1
a
4
=
u 1
u + 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 = 0.500000 0.866025I
b = 0
1.64493 + 2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0
1.64493 2.02988I 3.46410I
3
II. I
u
2
= hu
2
+ u + 1, a + u + 1, b
2
2i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
9
=
u 1
b
a
10
=
u 1
b + u
a
5
=
u
u + 1
a
8
=
b u 1
b
a
11
=
bu b 1
2
a
7
=
u + 1
b
a
4
=
b u 1
bu + 3u + 1
a
4
=
b u 1
bu + 3u + 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 = 0.500000 + 0.866025I
b = 1.41421
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 1.41421
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 1.41421
3.28987 2.02988I 2.00000 + 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 1.41421
3.28987 2.02988I 2.00000 + 3.46410I
5
III. I
u
3
=
hu
20
+4u
19
+· · ·+2u+1, 1.14×10
15
u
19
4.27×10
15
u
18
+· · ·+8.56×10
15
b+
1.10 × 10
16
, 6.38 × 10
15
u
19
+ 2.69 × 10
16
u
18
+ · · · + 8.56 × 10
15
a + 3.20 × 10
16
i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
9
=
0.744782u
19
3.14258u
18
+ ··· 36.0811u 3.73565
0.133295u
19
+ 0.498498u
18
+ ··· + 1.29523u 1.28508
a
10
=
0.744782u
19
3.14258u
18
+ ··· 36.0811u 3.73565
0.169440u
19
+ 0.644676u
18
+ ··· + 2.36692u 1.12163
a
5
=
u
u
3
+ u
a
8
=
0.878077u
19
3.64108u
18
+ ··· 37.3764u 2.45056
0.133295u
19
+ 0.498498u
18
+ ··· + 1.29523u 1.28508
a
11
=
1.12163u
19
+ 4.65596u
18
+ ··· + 50.9286u + 4.61018
0.278931u
19
1.06850u
18
+ ··· 2.73563u + 1.50344
a
7
=
0.408694u
19
+ 1.81167u
18
+ ··· + 31.0989u + 4.13465
0.233980u
19
0.931452u
18
+ ··· 3.19070u + 0.665806
a
4
=
0.174714u
19
0.880217u
18
+ ··· 27.9082u 4.80045
0.258642u
19
+ 1.02897u
18
+ ··· + 6.32631u 0.537037
a
4
=
0.174714u
19
0.880217u
18
+ ··· 27.9082u 4.80045
0.258642u
19
+ 1.02897u
18
+ ··· + 6.32631u 0.537037
(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 = 1.10251 0.94498I
a = 0.757275 0.607433I
b = 0.965691 0.331710I
0.59440 + 3.60439I 2.12405 4.48047I
u = 1.10251 + 0.94498I
a = 0.757275 + 0.607433I
b = 0.965691 + 0.331710I
0.59440 3.60439I 2.12405 + 4.48047I
u = 0.729618 0.601963I
a = 0.002660 + 0.502436I
b = 1.343653 + 0.072587I
5.16928 + 2.57908I 8.86572 4.96809I
u = 0.729618 + 0.601963I
a = 0.002660 0.502436I
b = 1.343653 0.072587I
5.16928 2.57908I 8.86572 + 4.96809I
u = 0.36649 1.92873I
a = 0.003517 0.916631I
b = 1.46300 0.57660I
10.3499 + 9.9960I 0.04826 5.02986I
u = 0.36649 + 1.92873I
a = 0.003517 + 0.916631I
b = 1.46300 + 0.57660I
10.3499 9.9960I 0.04826 + 5.02986I
u = 0.342865 1.301009I
a = 0.567886 + 0.523820I
b = 0.778134 + 0.346045I
1.166155 + 0.686038I 1.05440 + 1.40687I
u = 0.342865 + 1.301009I
a = 0.567886 0.523820I
b = 0.778134 0.346045I
1.166155 0.686038I 1.05440 1.40687I
u = 0.195248 0.372946I
a = 0.774007 0.603593I
b = 0.277579 0.390975I
0.131319 + 1.058255I 2.00315 6.24655I
u = 0.195248 + 0.372946I
a = 0.774007 + 0.603593I
b = 0.277579 + 0.390975I
0.131319 1.058255I 2.00315 + 6.24655I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.007874 1.394548I
a = 0.585444 + 0.602506I
b = 1.027385 + 0.480106I
1.53618 + 5.13397I 0.62229 5.85487I
u = 0.007874 + 1.394548I
a = 0.585444 0.602506I
b = 1.027385 0.480106I
1.53618 5.13397I 0.62229 + 5.85487I
u = 0.01089 1.99331I
a = 0.051284 + 0.957095I
b = 0.076693 + 1.208715I
15.1661 + 3.6593I 2.66886 2.23636I
u = 0.01089 + 1.99331I
a = 0.051284 0.957095I
b = 0.076693 1.208715I
15.1661 3.6593I 2.66886 + 2.23636I
u = 0.030822 0.170366I
a = 0.29189 + 5.45429I
b = 1.370381 0.067940I
3.14870 + 0.11294I 1.81708 + 1.16761I
u = 0.030822 + 0.170366I
a = 0.29189 5.45429I
b = 1.370381 + 0.067940I
3.14870 0.11294I 1.81708 1.16761I
u = 0.337333 0.681420I
a = 1.15996 1.26131I
b = 0.515876 0.585061I
3.04083 + 0.89466I 2.94359 + 0.44261I
u = 0.337333 + 0.681420I
a = 1.15996 + 1.26131I
b = 0.515876 + 0.585061I
3.04083 0.89466I 2.94359 0.44261I
u = 0.36556 1.75156I
a = 0.101116 1.030466I
b = 1.39118 0.66555I
11.13935 2.93869I 0.922503 + 0.819894I
u = 0.36556 + 1.75156I
a = 0.101116 + 1.030466I
b = 1.39118 + 0.66555I
11.13935 + 2.93869I 0.922503 0.819894I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
20
+ 4u
19
+ ··· + 2u + 1)
c
2
(u
2
+ u + 1)
3
(u
20
+ 28u
19
+ ··· + 74u + 1)
c
3
(u
2
u + 1)(u
4
2u
3
+ ··· + 2u + 1)(u
20
+ 8u
18
+ ··· 16u + 41)
c
4
(u
2
u + 1)(u
4
+ 2u
3
+ ··· 2u + 1)(u
20
+ 2u
19
+ ··· 2204u + 839)
c
5
(u
2
u + 1)
3
(u
20
+ 4u
19
+ ··· + 2u + 1)
c
6
(u 1)
2
(u + 1)
4
(u
20
+ 3u
19
+ ··· 19u + 7)
c
7
, c
8
, c
11
u
2
(u
2
2)
2
(u
20
+ u
19
+ ··· + 12u + 4)
c
9
(u 1)
4
(u + 1)
2
(u
20
+ 3u
19
+ ··· 19u + 7)
c
10
(u
2
u + 1)(u
2
+ u + 1)
2
(u
20
+ 2u
19
+ ··· 2u + 1)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
20
+ 28y
19
+ ··· + 74y + 1)
c
2
(y
2
+ y + 1)
3
(y
20
68y
19
+ ··· 1654y + 1)
c
3
(y
2
+ y + 1)(y
4
+ 6y
3
+ 35y
2
+ 6y + 1)
(y
20
+ 16y
19
+ ··· + 10814y + 1681)
c
4
(y
2
+ y + 1)(y
4
+ 6y
3
+ 35y
2
+ 6y + 1)
(y
20
52y
19
+ ··· + 5336234y + 703921)
c
6
, c
9
(y 1)
6
(y
20
29y
19
+ ··· + 297y + 49)
c
7
, c
8
, c
11
y
2
(y 2)
4
(y
20
15y
19
+ ··· 48y + 16)
c
10
(y
2
+ y + 1)
3
(y
20
+ 4y
19
+ ··· + 2y + 1)
10