11n
30
(K11n
30
)
1
Arc Sequences
4 1 7 2 8 10 4 1 11 6 9
Solving Sequence
1,4
2
5,8
9 7 3 11 10 6
c
1
c
4
c
8
c
7
c
3
c
11
c
9
c
6
c
2
, c
5
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
4
+ b
3
+ 3b
2
+ 2b + 1, a, u 1i
I
u
2
= hu
20
+ 5u
19
+ ··· 6u 1, u
19
+ 3u
18
+ ··· + 4b 8, 3u
19
+ 18u
18
+ ··· + 4a 9i
There are 2 irreducible components with 24 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
= hb
4
+ b
3
+ 3b
2
+ 2b + 1, a, u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
1
0
a
8
=
0
b
a
9
=
b
b
a
7
=
0
b
a
3
=
0
1
a
11
=
b
2
+ 1
b
2
a
10
=
b
3
2b
b
3
+ b
a
6
=
1
b
2
a
6
=
1
b
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 = 0
b = 0.395123 0.506844I
1.85594 1.41510I 10.51825 + 2.96122I
u = 1.00000
a = 0
b = 0.395123 + 0.506844I
1.85594 + 1.41510I 10.51825 2.96122I
u = 1.00000
a = 0
b = 0.10488 1.55249I
5.14581 3.16396I 8.98175 + 2.83489I
u = 1.00000
a = 0
b = 0.10488 + 1.55249I
5.14581 + 3.16396I 8.98175 2.83489I
3
II. I
u
2
=
hu
20
+5u
19
+· · ·6u 1, u
19
+3u
18
+· · ·+4b 8, 3u
19
+18u
18
+· · ·+4a 9i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
8
=
3
4
u
19
9
2
u
18
+ ··· +
71
4
u +
9
4
1
4
u
19
3
4
u
18
+ ··· +
25
4
u + 2
a
9
=
u
19
15
4
u
18
+ ··· +
23
2
u +
1
4
1
4
u
19
3
4
u
18
+ ··· +
25
4
u + 2
a
7
=
3
4
u
19
9
2
u
18
+ ··· +
71
4
u +
9
4
2u
19
29
4
u
18
+ ··· +
23
2
u +
11
4
a
3
=
u
2
+ 1
u
2
a
11
=
1
8
u
19
1
2
u
18
+ ···
3
8
u +
17
8
1
8
u
19
+
1
2
u
18
+ ···
13
8
u
1
8
a
10
=
1
4
u
19
+
5
4
u
18
+ ··· +
7
4
u
5
2
5
2
u
19
+ 10u
18
+ ··· 13u 3
a
6
=
1
1
8
u
19
1
2
u
18
+ ··· +
21
8
u +
1
8
a
6
=
1
1
8
u
19
1
2
u
18
+ ··· +
21
8
u +
1
8
(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 = 1.81202 0.09860I
a = 1.124742 + 0.057142I
b = 1.163488 + 0.628217I
14.0917 3.7151I 12.67457 + 3.10159I
u = 1.81202 + 0.09860I
a = 1.124742 0.057142I
b = 1.163488 0.628217I
14.0917 + 3.7151I 12.67457 3.10159I
u = 1.73062
a = 1.07368
b = 0.672482
10.3113 7.59679
u = 1.71507 0.27164I
a = 1.083576 + 0.166912I
b = 0.46653 + 1.62043I
7.04125 9.73657I 8.64627 + 5.28115I
u = 1.71507 + 0.27164I
a = 1.083576 0.166912I
b = 0.46653 1.62043I
7.04125 + 9.73657I 8.64627 5.28115I
u = 1.67897 0.22535I
a = 1.055754 0.143916I
b = 0.293817 1.327323I
6.07483 3.49044I 7.50331 + 0.69756I
u = 1.67897 + 0.22535I
a = 1.055754 + 0.143916I
b = 0.293817 + 1.327323I
6.07483 + 3.49044I 7.50331 0.69756I
u = 0.494818 0.034941I
a = 0.071730 1.233387I
b = 0.12366 1.50920I
6.00682 + 3.10793I 1.19914 2.44206I
u = 0.494818 + 0.034941I
a = 0.071730 + 1.233387I
b = 0.12366 + 1.50920I
6.00682 3.10793I 1.19914 + 2.44206I
u = 0.098700 0.173726I
a = 1.16971 1.93438I
b = 0.407505 0.376718I
0.333685 + 1.164937I 4.31355 5.64475I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.098700 + 0.173726I
a = 1.16971 + 1.93438I
b = 0.407505 + 0.376718I
0.333685 1.164937I 4.31355 + 5.64475I
u = 0.593945 0.749573I
a = 0.901534 0.870244I
b = 0.062242 + 1.190030I
1.80617 0.15475I 5.78761 0.24947I
u = 0.593945 + 0.749573I
a = 0.901534 + 0.870244I
b = 0.062242 1.190030I
1.80617 + 0.15475I 5.78761 + 0.24947I
u = 0.723331
a = 0.811887
b = 0.0840139
1.09578 8.64202
u = 0.742600 0.805837I
a = 0.823991 + 0.896190I
b = 0.361535 1.366300I
1.34574 + 5.46019I 7.11600 5.63427I
u = 0.742600 + 0.805837I
a = 0.823991 0.896190I
b = 0.361535 + 1.366300I
1.34574 5.46019I 7.11600 + 5.63427I
u = 1.049033 0.433248I
a = 0.533800 + 0.720846I
b = 0.786836 0.158628I
3.47404 + 1.25358I 14.5901 1.6218I
u = 1.049033 + 0.433248I
a = 0.533800 0.720846I
b = 0.786836 + 0.158628I
3.47404 1.25358I 14.5901 + 1.6218I
u = 1.41765 0.08558I
a = 0.075401 + 0.848821I
b = 0.300271 + 1.184317I
0.40356 2.62035I 6.94831 + 3.53102I
u = 1.41765 + 0.08558I
a = 0.075401 0.848821I
b = 0.300271 1.184317I
0.40356 + 2.62035I 6.94831 3.53102I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
20
+ 5u
19
+ ··· 6u 1)
c
2
(u + 1)
4
(u
20
+ 27u
19
+ ··· 12u + 1)
c
3
, c
7
u
4
(u
20
+ u
19
+ ··· 40u 16)
c
4
(u + 1)
4
(u
20
+ 5u
19
+ ··· 6u 1)
c
5
(u
4
u
3
+ 3u
2
2u + 1)(u
20
+ 2u
19
+ ··· 2u + 1)
c
6
(u
4
u
3
+ u
2
+ 1)(u
20
+ 2u
19
+ ··· 2u + 1)
c
8
, c
9
(u
4
u
3
+ 3u
2
2u + 1)(u
20
+ 6u
19
+ ··· 6u + 1)
c
10
(u
4
+ u
3
+ u
2
+ 1)(u
20
+ 2u
19
+ ··· 2u + 1)
c
11
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
20
+ 6u
19
+ ··· 6u + 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(y
20
27y
19
+ ··· + 12y + 1)
c
2
(y 1)
4
(y
20
63y
19
+ ··· + 564y + 1)
c
3
, c
7
y
4
(y
20
27y
19
+ ··· + 960y + 256)
c
5
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
20
42y
19
+ ··· 6y + 1)
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
20
+ 6y
19
+ ··· 6y + 1)
c
8
, c
9
, c
11
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
20
+ 18y
19
+ ··· 142y + 1)
8