11n
65
(K11n
65
)
1
Arc Sequences
4 1 8 2 9 11 3 6 5 1 6
Solving Sequence
5,9
6
2,10
4 1 8 3 11 7
c
5
c
9
c
4
c
1
c
8
c
3
c
11
c
6
c
2
, c
7
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 1, a 1, 2b 1i
I
u
2
= ha
6
+ 2a
4
+ a
2
+ 1, a
2
+ u + 1, a
4
a
3
+ a
2
+ b a + 1i
I
u
3
= ha
10
+ a
8
4a
7
+ 3a
6
+ 9a
5
+ 35a
4
+ 46a
3
+ 54a
2
+ 31a + 17,
2904a
9
+ 26497b + ··· 126739a 123626, 7289a
9
+ 26497u + ··· + 122144a + 3595i
I
u
4
= hu
13
+ 2u
12
2u
11
7u
10
u
9
+ 9u
8
+ 16u
7
+ 18u
6
2u
5
33u
4
24u
3
+ 10u
2
+ 17u + 4,
73u
12
+ 2060u
11
+ ··· + 9382b + 3126, 1771u
12
+ 2038u
11
+ ··· + 18764a 26529i
There are 4 irreducible components with 30 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, a 1, 2b 1i
(i) Arc colorings
a
5
=
0
1
a
9
=
1
1
2
a
6
=
1
3
2
a
2
=
1
0
a
10
=
1
1
2
a
4
=
1
1
a
1
=
0
1
a
8
=
2
2
a
3
=
1
1
a
11
=
1
1
2
a
7
=
2
2
a
7
=
2
2
(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 = 1.00000
a = 1.00000
b = 0.500000
0 2.25000
3
II. I
u
2
= ha
6
+ 2a
4
+ a
2
+ 1, a
2
+ u + 1, a
4
a
3
+ a
2
+ b a + 1i
(i) Arc colorings
a
5
=
0
a
2
1
a
9
=
a
a
4
+ a
3
a
2
+ a 1
a
6
=
a
4
a
2
a
3
a
2
a
2
=
1
0
a
10
=
a
a
5
a
4
a
3
a
2
1
a
4
=
a
2
1
a
2
1
a
1
=
a
4
2a
2
a
4
2a
2
1
a
8
=
0
a
5
+ 2a
3
+ a
a
3
=
a
2
1
a
4
+ a
2
1
a
11
=
a
4
2a
2
+ a
a
5
2a
4
a
3
3a
2
2
a
7
=
a
5
a
3
+ a
a
5
2a
3
a
7
=
a
5
a
3
+ a
a
5
2a
3
(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 = 0.877439 0.744862I
a = 0.562280 0.662359I
b = 0.33764 1.56228I
0.26574 2.82812I 0.49024 + 2.97945I
u = 0.877439 + 0.744862I
a = 0.562280 + 0.662359I
b = 0.33764 + 1.56228I
0.26574 + 2.82812I 0.49024 2.97945I
u = 0.754878
a = 1.32472I
b = 2.32472 + 1.00000I
4.40332 7.01951
u = 0.754878
a = 1.32472I
b = 2.32472 1.00000I
4.40332 7.01951
u = 0.877439 + 0.744862I
a = 0.562280 0.662359I
b = 0.337641 0.437720I
0.26574 + 2.82812I 0.49024 2.97945I
u = 0.877439 0.744862I
a = 0.562280 + 0.662359I
b = 0.337641 + 0.437720I
0.26574 2.82812I 0.49024 + 2.97945I
5
III. I
u
3
= ha
10
+ a
8
+ · · · + 31a + 17, 2904a
9
+ 26497b + · · · 126739a
123626, 7289a
9
+ 26497u + · · · + 122144a + 3595i
(i) Arc colorings
a
5
=
0
0.275088a
9
+ 0.169944a
8
+ ··· 4.60973a 0.135676
a
9
=
a
0.109597a
9
+ 0.297505a
8
+ ··· + 4.78315a + 4.66566
a
6
=
0.0831037a
9
0.0499679a
8
+ ··· 0.591765a 2.88904
0.395894a
9
+ 0.0816696a
8
+ ··· + 9.55331a + 0.833755
a
2
=
1
0
a
10
=
a
0.191116a
9
+ 0.213005a
8
+ ··· + 1.67970a + 3.74756
a
4
=
0.275088a
9
+ 0.169944a
8
+ ··· 4.60973a 0.135676
0.275088a
9
+ 0.169944a
8
+ ··· 4.60973a 0.135676
a
1
=
0.0540061a
9
0.0815187a
8
+ ··· 0.0439672a 0.429256
0.0540061a
9
0.0815187a
8
+ ··· 0.0439672a 1.42926
a
8
=
0.335095a
9
+ 0.260520a
8
+ ··· 3.00532a 1.43650
0.434879a
9
+ 0.392271a
8
+ ··· 3.47824a 0.910329
a
3
=
0.0300034a
9
+ 0.0452881a
8
+ ··· + 0.802204a + 1.34959
0.0840095a
9
+ 0.126807a
8
+ ··· + 0.846171a + 1.77884
a
11
=
0.231045a
9
+ 0.0663471a
8
+ ··· + 4.85523a + 1.77714
0.0852172a
9
+ 0.437408a
8
+ ··· + 6.81462a + 5.03476
a
7
=
0.562177a
9
0.358003a
8
+ ··· + 6.09858a + 4.73152
0.761747a
9
0.621504a
8
+ ··· + 7.04442a + 4.67917
a
7
=
0.562177a
9
0.358003a
8
+ ··· + 6.09858a + 4.73152
0.761747a
9
0.621504a
8
+ ··· + 7.04442a + 4.67917
(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 = 1.41878 + 0.21917I
a = 0.857206 0.694663I
b = 0.424829 1.135914I
9.16243 + 4.40083I 2.74431 3.49859I
u = 1.41878 0.21917I
a = 0.857206 + 0.694663I
b = 0.424829 + 1.135914I
9.16243 4.40083I 2.74431 + 3.49859I
u = 0.309916 + 0.549911I
a = 0.63676 1.28488I
b = 1.194296 0.377316I
3.61897 1.53058I 1.48489 + 4.43065I
u = 0.309916 0.549911I
a = 0.63676 + 1.28488I
b = 1.194296 + 0.377316I
3.61897 + 1.53058I 1.48489 4.43065I
u = 1.21774
a = 0.241441 0.921544I
b = 1.32946 2.82997I
5.69095 0.518857
u = 1.21774
a = 0.241441 + 0.921544I
b = 1.32946 + 2.82997I
5.69095 0.518857
u = 1.41878 0.21917I
a = 0.107707 1.316559I
b = 0.07567 2.70164I
9.16243 4.40083I 2.74431 + 3.49859I
u = 1.41878 + 0.21917I
a = 0.107707 + 1.316559I
b = 0.07567 + 2.70164I
9.16243 + 4.40083I 2.74431 3.49859I
u = 0.309916 0.549911I
a = 1.84312 0.94403I
b = 0.134669 + 0.397335I
3.61897 + 1.53058I 1.48489 4.43065I
u = 0.309916 + 0.549911I
a = 1.84312 + 0.94403I
b = 0.134669 0.397335I
3.61897 1.53058I 1.48489 + 4.43065I
7
IV. I
u
4
= hu
13
+ 2u
12
+ · · · + 17u + 4, 73u
12
+ 2060u
11
+ · · · + 9382b +
3126, 1771u
12
+ 2038u
11
+ · · · + 18764a 26529i
(i) Arc colorings
a
5
=
0
u
a
9
=
0.0943829u
12
0.108612u
11
+ ··· 1.63462u + 1.41382
0.00778086u
12
0.219569u
11
+ ··· 1.81827u 0.333191
a
6
=
0.0286186u
12
+ 0.0747175u
11
+ ··· + 1.33500u 0.647783
0.0254743u
12
+ 0.129823u
11
+ ··· + 1.98721u + 0.131955
a
2
=
1
0
a
10
=
0.0943829u
12
0.108612u
11
+ ··· 1.63462u + 1.41382
0.0613942u
12
0.0681091u
11
+ ··· 0.833191u 0.0125773
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
8
=
0.165530u
12
+ 0.0820721u
11
+ ··· + 0.716052u + 2.62940
0.302601u
12
0.125240u
11
+ ··· + 0.199531u + 0.341505
a
3
=
u
4
u
2
+ 1
u
4
a
11
=
0.259913u
12
+ 0.190684u
11
+ ··· + 2.35067u + 2.21557
0.310382u
12
+ 0.0943296u
11
+ ··· + 2.01780u + 0.674696
a
7
=
0.0407163u
12
+ 0.00170539u
11
+ ··· + 0.0811128u 1.92848
0.204327u
12
0.156363u
11
+ ··· 2.40578u 1.33873
a
7
=
0.0407163u
12
+ 0.00170539u
11
+ ··· + 0.0811128u 1.92848
0.204327u
12
0.156363u
11
+ ··· 2.40578u 1.33873
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.46956 0.59251I
a = 0.127735 + 1.153868I
b = 0.38673 + 2.67516I
17.2166 11.4167I 1.78764 + 5.02800I
u = 1.46956 + 0.59251I
a = 0.127735 1.153868I
b = 0.38673 2.67516I
17.2166 + 11.4167I 1.78764 5.02800I
u = 0.955186 0.433947I
a = 0.903676 + 0.296124I
b = 0.260785 + 0.621147I
0.903643 0.585016I 4.40140 + 1.35233I
u = 0.955186 + 0.433947I
a = 0.903676 0.296124I
b = 0.260785 0.621147I
0.903643 + 0.585016I 4.40140 1.35233I
u = 0.869334 0.624757I
a = 0.151013 0.644177I
b = 0.213331 1.067375I
1.00399 3.84064I 5.54977 + 8.01840I
u = 0.869334 + 0.624757I
a = 0.151013 + 0.644177I
b = 0.213331 + 1.067375I
1.00399 + 3.84064I 5.54977 8.01840I
u = 0.326480
a = 1.72892
b = 0.334640
0.885241 11.4836
u = 0.028967 1.273931I
a = 1.368728 + 0.204713I
b = 0.246365 + 0.208175I
12.48262 + 4.81706I 0.19074 2.27482I
u = 0.028967 + 1.273931I
a = 1.368728 0.204713I
b = 0.246365 0.208175I
12.48262 4.81706I 0.19074 + 2.27482I
u = 0.933504 0.177892I
a = 0.093973 + 0.315785I
b = 0.475505 + 0.639800I
1.65953 + 0.62739I 3.40176 1.52650I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.933504 + 0.177892I
a = 0.093973 0.315785I
b = 0.475505 0.639800I
1.65953 0.62739I 3.40176 + 1.52650I
u = 1.49484 0.63529I
a = 0.404896 1.067512I
b = 0.45097 2.35583I
17.0497 + 2.0233I 2.18781 0.87077I
u = 1.49484 + 0.63529I
a = 0.404896 + 1.067512I
b = 0.45097 + 2.35583I
17.0497 2.0233I 2.18781 + 0.87077I
10
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)(u
3
+ u
2
1)
2
(u
5
+ u
4
2u
3
u
2
+ u 1)
2
(u
13
+ 2u
12
+ ··· + 17u + 4)
c
2
(u + 1)(u
3
+ u
2
+ 2u + 1)
2
(u
5
+ 5u
4
+ 8u
3
+ 3u
2
u + 1)
2
(u
13
+ 8u
12
+ ··· + 209u + 16)
c
3
, c
7
u(u
5
u
4
+ 2u
3
u
2
+ u 1)
2
(u
6
3u
4
+ 2u
2
+ 1)
(u
13
+ 3u
12
+ ··· 2u 8)
c
4
(u + 1)(u
3
+ u
2
1)
2
(u
5
+ u
4
2u
3
u
2
+ u 1)
2
(u
13
+ 2u
12
+ ··· + 17u + 4)
c
5
, c
6
(u + 1)(u
2
+ 1)
3
(u
10
+ 3u
9
+ ··· + 32u + 17)(u
13
u
12
+ ··· + u 1)
c
8
, c
9
, c
11
(u 1)(u
2
+ 1)
3
(u
10
+ 3u
9
+ ··· + 32u + 17)(u
13
u
12
+ ··· + u 1)
c
10
(u + 1)
7
(u
10
+ 11u
9
+ ··· + 1016u + 289)
(u
13
+ 19u
12
+ ··· 13u 1)
11
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)(y
3
y
2
+ 2y 1)
2
(y
5
5y
4
+ 8y
3
3y
2
y 1)
2
(y
13
8y
12
+ ··· + 209y 16)
c
2
(y 1)(y
3
+ 3y
2
+ 2y 1)
2
(y
5
9y
4
+ 32y
3
35y
2
5y 1)
2
(y
13
4y
12
+ ··· + 22817y 256)
c
3
, c
7
y(y
3
3y
2
+ 2y + 1)
2
(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
2
(y
13
3y
12
+ ··· + 180y 64)
c
5
, c
6
, c
8
c
9
, c
11
(y 1)(y + 1)
6
(y
10
+ 11y
9
+ ··· + 1016y + 289)
(y
13
+ 19y
12
+ ··· 13y 1)
c
10
(y 1)
7
(y
10
25y
9
+ ··· + 78660y + 83521)
(y
13
49y
12
+ ··· 61y 1)
12