11n
63
(K11n
63
)
1
Arc Sequences
4 1 7 2 11 10 4 5 7 6 9
Solving Sequence
5,11 2,6
4 1 3 10 7 9 8
c
5
c
4
c
1
c
2
c
10
c
6
c
9
c
8
c
3
, c
7
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
4
+ a
3
+ 3a
2
+ 2a + 1, b + 1, u 1i
I
u
2
= hu
23
5u
22
+ ··· 4u + 1, u
5
u
3
+ 2u
2
+ b + u, u
22
4u
21
+ ··· + 8a 1i
There are 2 irreducible components with 27 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
= ha
4
+ a
3
+ 3a
2
+ 2a + 1, b + 1, u 1i
(i) Arc colorings
a
5
=
0
1
a
11
=
a
1
a
2
=
1
0
a
6
=
a
2
a + 1
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
10
=
a
3
+ a
a
2
a 1
a
7
=
a
3
+ a
2
+ 2a + 1
a
3
+ a
2
+ 2a + 1
a
9
=
a
3
+ a
2
+ 2a + 1
0
a
8
=
a
3
+ a
2
+ 2a + 1
a
3
+ a
2
+ 2a + 1
a
8
=
a
3
+ a
2
+ 2a + 1
a
3
+ a
2
+ 2a + 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 = 1.00000
a = 0.395123 0.506844I
b = 1.00000
1.85594 1.41510I 11.17855 + 5.62908I
u = 1.00000
a = 0.395123 + 0.506844I
b = 1.00000
1.85594 + 1.41510I 11.17855 5.62908I
u = 1.00000
a = 0.10488 1.55249I
b = 1.00000
5.14581 3.16396I 6.32145 + 1.65351I
u = 1.00000
a = 0.10488 + 1.55249I
b = 1.00000
5.14581 + 3.16396I 6.32145 1.65351I
3
II.
I
u
2
= hu
23
5u
22
+ · · · 4u + 1, u
5
u
3
+ 2u
2
+ b + u, u
22
4u
21
+ · · · + 8a 1i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
8
u
22
+
1
2
u
21
+ ···
19
8
u +
1
8
u
5
+ u
3
2u
2
u
a
2
=
1
0
a
6
=
1.87500u
22
+ 7.75000u
21
+ ··· 6.12500u + 2.12500
3
8
u
22
+
3
2
u
21
+ ···
1
8
u +
3
8
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
10
=
13
2
u
22
+
57
2
u
21
+ ··· 27u +
19
2
3u
22
+
51
4
u
21
+ ···
23
2
u +
15
4
a
7
=
23
4
u
22
+
107
4
u
21
+ ···
109
4
u + 13
3
2
u
22
+
33
4
u
21
+ ···
19
2
u +
21
4
a
9
=
8u
22
+
145
4
u
21
+ ··· 34u +
63
4
2.75000u
22
+ 11.5000u
21
+ ··· 9.25000u + 4.25000
a
8
=
8u
22
+
145
4
u
21
+ ··· 34u +
63
4
15
4
u
22
+
71
4
u
21
+ ···
65
4
u + 8
a
8
=
8u
22
+
145
4
u
21
+ ··· 34u +
63
4
15
4
u
22
+
71
4
u
21
+ ···
65
4
u + 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 = 1.215142 0.309680I
a = 0.330209 1.218290I
b = 2.00634 + 0.40260I
5.91283 1.29853I 3.45106 + 0.05233I
u = 1.215142 + 0.309680I
a = 0.330209 + 1.218290I
b = 2.00634 0.40260I
5.91283 + 1.29853I 3.45106 0.05233I
u = 1.095323 0.135004I
a = 0.204764 0.422200I
b = 1.183021 + 0.002185I
1.35992 0.76790I 3.34761 1.39618I
u = 1.095323 + 0.135004I
a = 0.204764 + 0.422200I
b = 1.183021 0.002185I
1.35992 + 0.76790I 3.34761 + 1.39618I
u = 0.566410
a = 0.541479
b = 0.198647
0.900453 11.8238
u = 0.417445 0.531363I
a = 1.043713 + 0.336011I
b = 0.888143 0.622056I
0.92312 1.99790I 2.34638 + 5.92992I
u = 0.417445 + 0.531363I
a = 1.043713 0.336011I
b = 0.888143 + 0.622056I
0.92312 + 1.99790I 2.34638 5.92992I
u = 0.402570 0.803293I
a = 1.64032 + 0.74276I
b = 2.51077 0.76252I
8.92549 3.15334I 1.05029 + 3.26062I
u = 0.402570 + 0.803293I
a = 1.64032 0.74276I
b = 2.51077 + 0.76252I
8.92549 + 3.15334I 1.05029 3.26062I
u = 0.437297 0.133257I
a = 1.30520 + 0.72293I
b = 0.725742 + 0.312150I
0.68293 + 1.66090I 3.45266 4.83485I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.437297 + 0.133257I
a = 1.30520 0.72293I
b = 0.725742 0.312150I
0.68293 1.66090I 3.45266 + 4.83485I
u = 0.588229 0.289273I
a = 1.19623 + 2.00856I
b = 0.977734 + 0.785272I
6.06133 + 3.74831I 0.03467 4.58469I
u = 0.588229 + 0.289273I
a = 1.19623 2.00856I
b = 0.977734 0.785272I
6.06133 3.74831I 0.03467 + 4.58469I
u = 0.843663 1.134513I
a = 0.895592 + 0.767748I
b = 2.03691 1.63946I
16.6150 2.3845I 0.450202 + 0.532296I
u = 0.843663 + 1.134513I
a = 0.895592 0.767748I
b = 2.03691 + 1.63946I
16.6150 + 2.3845I 0.450202 0.532296I
u = 0.871131 1.017531I
a = 0.289757 + 0.671513I
b = 0.689207 0.674028I
7.83203 0.28979I 2.09012 + 1.34957I
u = 0.871131 + 1.017531I
a = 0.289757 0.671513I
b = 0.689207 + 0.674028I
7.83203 + 0.28979I 2.09012 1.34957I
u = 0.962074 0.924080I
a = 0.337189 + 0.383823I
b = 0.592636 + 0.031782I
4.95941 + 3.41645I 6.52166 2.22573I
u = 0.962074 + 0.924080I
a = 0.337189 0.383823I
b = 0.592636 0.031782I
4.95941 3.41645I 6.52166 + 2.22573I
u = 1.065840 0.921777I
a = 0.648152 0.199540I
b = 1.92048 + 0.20940I
7.20945 + 7.39071I 3.32519 6.20381I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.065840 + 0.921777I
a = 0.648152 + 0.199540I
b = 1.92048 0.20940I
7.20945 7.39071I 3.32519 + 6.20381I
u = 1.14545 0.94090I
a = 0.781632 0.794577I
b = 3.30624 + 0.28844I
15.6090 + 9.9000I 1.58759 4.90312I
u = 1.14545 + 0.94090I
a = 0.781632 + 0.794577I
b = 3.30624 0.28844I
15.6090 9.9000I 1.58759 + 4.90312I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
23
+ 5u
22
+ ··· 4u 1)
c
2
(u + 1)
4
(u
23
+ 5u
22
+ ··· + 14u + 1)
c
3
, c
7
u
4
(u
23
+ u
22
+ ··· + 24u + 16)
c
4
(u + 1)
4
(u
23
+ 5u
22
+ ··· 4u 1)
c
5
, c
6
(u
4
u
3
+ 3u
2
2u + 1)(u
23
+ 2u
22
+ ··· + 12u
3
1)
c
8
(u
4
+ u
3
+ u
2
+ 1)(u
23
+ 2u
22
+ ··· + 2u 1)
c
9
, c
10
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
23
+ 2u
22
+ ··· + 12u
3
1)
c
11
(u
4
+ u
3
+ u
2
+ 1)(u
23
+ 8u
22
+ ··· + 168u + 49)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(y
23
5y
22
+ ··· + 14y 1)
c
2
(y 1)
4
(y
23
+ 31y
22
+ ··· + 14y 1)
c
3
, c
7
y
4
(y
23
+ 27y
22
+ ··· 2240y 256)
c
5
, c
6
, c
9
c
10
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
23
+ 28y
22
+ ··· + 60y
2
1)
c
8
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
23
28y
22
+ ··· 40y
2
1)
c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
23
16y
22
+ ··· + 77224y 2401)
9