11n
77
(K11n
77
)
1
Arc Sequences
4 1 8 2 11 9 10 4 7 5 10
Solving Sequence
2,4
5
1,8 3,10
7 9 11 6
c
4
c
1
c
3
c
7
c
9
c
11
c
5
c
2
, c
6
, c
8
, c
10
Representation Ideals
I =
7
\
i=1
I
u
i
\
I
v
1
I
u
1
= ha, u 1, d + c + 1, cb + 1i
I
u
2
= ha, c, u 1, d + 1, b + 1i
I
u
3
= ha, c 1, u 1, d + 1, b + 1i
I
u
4
= hd + 1, u
2
+ b, u
4
u
3
+ a + 2u, u
5
+ 2u
4
+ 2u
3
u
2
u 1, u
4
2u
3
u
2
+ c + 2u + 1i
I
u
5
= hd + 1, u
5
u
3
3u
2
+ 4, 3u
4
+ 2u
3
+ u
2
+ 8a 3u 10, u
4
6u
3
+ u
2
+ 4b + u + 14,
u
4
+ 2u
3
+ 3u
2
+ 8c 5u 10i
I
u
6
= h−u
2
+ b, u
4
u
3
+ a + 2u, u
4
+ 2u
3
+ 3u
2
+ d + u, u
5
+ 2u
4
+ 2u
3
u
2
u 1,
u
4
2u
3
3u
2
+ c u 1i
I
u
7
= hu
5
+ u
4
+ 3u 1, u
4
2u
3
2u
2
+ 2a 3, u
4
+ u
3
+ u
2
+ 2d + u, u
4
+ u
3
+ u
2
+ 2b 3u + 2,
u
4
u
3
u
2
+ 2c u 2i
I
v
1
= hb, c, v 1, a + 1, d + 1i
There are 8 irreducible components with 23 representations.
There are 1 irreducible components of dim
C
= 1 for 11n
77
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, u 1, d + c + 1, cb + 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
1
a
5
=
1
1
a
1
=
1
1
a
8
=
0
b
a
3
=
0
1
a
10
=
c
c 1
a
7
=
c
c + b + 1
a
9
=
0
b
a
11
=
c + 1
c 2
a
6
=
c
c 1
a
6
=
c
c 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 = ···
a = ···
b = ···
c = ···
d = ···
4.93480 17.8631 + 0.3965I
3
II. I
u
2
= ha, c, u 1, d + 1, b + 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
1
a
5
=
1
1
a
1
=
1
1
a
8
=
0
1
a
3
=
0
1
a
10
=
0
1
a
7
=
0
1
a
9
=
0
1
a
11
=
1
2
a
6
=
0
1
a
6
=
0
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 = 1.00000
a = 0
b = 1.00000
c = 0
d = 1.00000
3.28987 12.0000
5
III. I
u
3
= ha, c 1, u 1, d + 1, b + 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
1
a
5
=
1
1
a
1
=
1
1
a
8
=
0
1
a
3
=
0
1
a
10
=
1
1
a
7
=
1
0
a
9
=
0
1
a
11
=
1
1
a
6
=
1
1
a
6
=
1
1
(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 = 1.00000
a = 0
b = 1.00000
c = 1.00000
d = 1.00000
3.28987 12.0000
7
IV. I
u
4
= hd + 1, u
2
+ b, u
4
u
3
+ a + 2u, u
5
+ 2u
4
+ 2u
3
u
2
u
1, u
4
2u
3
+ · · · + c + 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
1
=
1
u
2
a
8
=
u
4
+ u
3
2u
u
2
a
3
=
u
2
+ 1
u
4
a
10
=
u
4
+ 2u
3
+ u
2
2u 1
1
a
7
=
u
4
+ 2u
3
+ u
2
2u 1
u
2
+ u
a
9
=
u
4
+ u
3
2u
u
3
+ u
2
1
a
11
=
2u
4
+ 3u
3
+ 2u
2
3u 1
u
4
u
3
u
2
+ u 1
a
6
=
2u
4
4u
3
2u
2
+ 3u + 2
u
4
+ u
3
u
2
2u
a
6
=
2u
4
4u
3
2u
2
+ 3u + 2
u
4
+ u
3
u
2
2u
(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 = 1.09977 1.12945I
a = 1.089060 0.727199I
b = 0.06615 + 2.48427I
c = 0.723387 0.900340I
d = 1.00000
8.62005 4.12490I 10.89396 + 2.15443I
u = 1.09977 + 1.12945I
a = 1.089060 + 0.727199I
b = 0.06615 2.48427I
c = 0.723387 + 0.900340I
d = 1.00000
8.62005 + 4.12490I 10.89396 2.15443I
u = 0.317129 0.618084I
a = 0.891355 + 1.065135I
b = 0.281458 + 0.392024I
c = 0.05854 + 1.50680I
d = 1.00000
1.43849 1.10891I 9.63452 + 2.04112I
u = 0.317129 + 0.618084I
a = 0.891355 1.065135I
b = 0.281458 0.392024I
c = 0.05854 1.50680I
d = 1.00000
1.43849 + 1.10891I 9.63452 2.04112I
u = 0.833800
a = 0.604590
b = 0.695222
c = 0.329693
d = 1.00000
4.49352 20.9430
9
V. I
u
5
= hd + 1, u
5
u
3
3u
2
+ 4, 3u
4
+ 2u
3
+ · · · + 8a 10, u
4
6u
3
+
· · · + 4b + 14, u
4
+ 2u
3
+ · · · + 8c 10i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
1
=
1
u
2
a
8
=
3
8
u
4
1
4
u
3
+ ··· +
3
8
u +
5
4
1
4
u
4
+
3
2
u
3
+ ···
1
4
u
7
2
a
3
=
u
2
+ 1
u
4
a
10
=
1
8
u
4
1
4
u
3
+ ··· +
5
8
u +
5
4
1
a
7
=
1
8
u
4
1
4
u
3
+ ··· +
5
8
u +
5
4
u
3
3
a
9
=
3
8
u
4
1
4
u
3
+ ··· +
3
8
u +
5
4
1
4
u
4
+
1
2
u
3
+ ··· +
5
4
u
5
2
a
11
=
3
8
u
4
1
4
u
3
+ ··· +
1
8
u +
1
4
1
2
u
4
1
2
u
2
+
1
2
u
a
6
=
1
4
u
4
+
3
4
u
2
3
4
u 2
1
4
u
4
1
2
u
3
+ ··· +
5
4
u +
5
2
a
6
=
1
4
u
4
+
3
4
u
2
3
4
u 2
1
4
u
4
1
2
u
3
+ ··· +
5
4
u +
5
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.10870
a = 0.454684
b = 5.19662
c = 0.247946
d = 1.00000
4.49352 20.9430
u = 0.71268 1.30259I
a = 1.077607 + 0.878534I
b = 1.36547 0.90407I
c = 0.688192 1.014894I
d = 1.00000
8.62005 + 4.12490I 10.89396 2.15443I
u = 0.71268 + 1.30259I
a = 1.077607 0.878534I
b = 1.36547 + 0.90407I
c = 0.688192 + 1.014894I
d = 1.00000
8.62005 4.12490I 10.89396 + 2.15443I
u = 1.267024 0.176417I
a = 0.195051 + 0.728580I
b = 0.76716 1.46223I
c = 0.687834 + 0.444359I
d = 1.00000
1.43849 + 1.10891I 9.63452 2.04112I
u = 1.267024 + 0.176417I
a = 0.195051 0.728580I
b = 0.76716 + 1.46223I
c = 0.687834 0.444359I
d = 1.00000
1.43849 1.10891I 9.63452 + 2.04112I
11
VI. I
u
6
= h−u
2
+ b, u
4
u
3
+ a + 2u, u
4
+ 2u
3
+ 3u
2
+ d + u, u
5
+ 2u
4
+
2u
3
u
2
u 1, u
4
2u
3
+ · · · + c 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
1
=
1
u
2
a
8
=
u
4
+ u
3
2u
u
2
a
3
=
u
2
+ 1
u
4
a
10
=
u
4
+ 2u
3
+ 3u
2
+ u + 1
u
4
2u
3
3u
2
u
a
7
=
u
4
+ 2u
3
+ 3u
2
+ u + 1
u
2
2u 2
a
9
=
u
4
+ u
3
2u
u
3
+ u
2
1
a
11
=
u
4
+ 2u
3
+ 2u
2
+ u + 1
u
4
2u
3
2u
2
u
a
6
=
2u
2
3u 1
2u
2
+ 2u + 1
a
6
=
2u
2
3u 1
2u
2
+ 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 1.09977 1.12945I
a = 1.089060 0.727199I
b = 0.06615 + 2.48427I
c = 0.708233 + 0.679852I
d = 1.70823 0.67985I
8.62005 4.12490I 10.89396 + 2.15443I
u = 1.09977 + 1.12945I
a = 1.089060 + 0.727199I
b = 0.06615 2.48427I
c = 0.708233 0.679852I
d = 1.70823 + 0.67985I
8.62005 + 4.12490I 10.89396 2.15443I
u = 0.317129 0.618084I
a = 0.891355 + 1.065135I
b = 0.281458 + 0.392024I
c = 0.427158 + 0.436597I
d = 0.572842 0.436597I
1.43849 1.10891I 9.63452 + 2.04112I
u = 0.317129 + 0.618084I
a = 0.891355 1.065135I
b = 0.281458 0.392024I
c = 0.427158 0.436597I
d = 0.572842 + 0.436597I
1.43849 + 1.10891I 9.63452 2.04112I
u = 0.833800
a = 0.604590
b = 0.695222
c = 5.56215
d = 4.56215
4.49352 20.9430
13
VII. I
u
7
= hu
5
+ u
4
+ 3u 1, u
4
2u
3
2u
2
+ 2a 3, u
4
+ u
3
+ u
2
+ 2d +
u, u
4
+ u
3
+ u
2
+ 2b 3u + 2, u
4
u
3
+ · · · + 2c 2i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
1
=
1
u
2
a
8
=
1
2
u
4
+ u
3
+ u
2
+
3
2
1
2
u
4
1
2
u
3
+ ··· +
3
2
u 1
a
3
=
u
2
+ 1
u
4
a
10
=
1
2
u
4
+
1
2
u
3
+
1
2
u
2
+
1
2
u + 1
1
2
u
4
1
2
u
3
1
2
u
2
1
2
u
a
7
=
1
2
u
4
+
1
2
u
3
+
1
2
u
2
+
1
2
u + 1
1
2
u
4
1
2
a
9
=
1
2
u
4
+ u
3
+ u
2
+
3
2
1
2
u
3
1
2
u
2
+
1
2
u
1
2
a
11
=
1
2
u
4
+
1
2
u
3
1
2
u
2
+
1
2
u + 1
1
2
u
4
1
2
u
3
+
1
2
u
2
1
2
u
a
6
=
1
2
u
3
1
2
u
2
1
2
u
1
2
1
2
u
3
+
1
2
u
2
1
2
u +
1
2
a
6
=
1
2
u
3
1
2
u
2
1
2
u
1
2
1
2
u
3
+
1
2
u
2
1
2
u +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
14
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
1(vol +
1CS) Cusp shape
u = 1.29342 0.87939I
a = 1.053698 + 0.587871I
b = 1.62557 2.63626I
c = 0.961274 + 0.877470I
d = 1.96127 0.87747I
6.61272 11.96038I 13.0958 + 6.1649I
u = 1.29342 + 0.87939I
a = 1.053698 0.587871I
b = 1.62557 + 2.63626I
c = 0.961274 0.877470I
d = 1.96127 + 0.87747I
6.61272 + 11.96038I 13.0958 6.1649I
u = 0.328197
a = 1.64887
b = 0.585038
c = 1.24143
d = 0.241431
0.709220 14.2105
u = 0.629322 0.921686I
a = 0.878131 0.946140I
b = 1.41808 1.17248I
c = 0.159441 0.670889I
d = 1.159441 + 0.670889I
1.14410 + 3.50618I 10.79893 4.59139I
u = 0.629322 + 0.921686I
a = 0.878131 + 0.946140I
b = 1.41808 + 1.17248I
c = 0.159441 + 0.670889I
d = 1.159441 0.670889I
1.14410 3.50618I 10.79893 + 4.59139I
15
VIII. I
v
1
= hb, c, v 1, a + 1, d + 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
1
0
a
5
=
1
0
a
1
=
1
0
a
8
=
1
0
a
3
=
1
0
a
10
=
0
1
a
7
=
1
1
a
9
=
1
0
a
11
=
1
1
a
6
=
0
1
a
6
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
16
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 1.00000
b = 0
c = 0
d = 1.00000
3.28987 12.0000
17
IX. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
, c
7
u(u 1)
2
(u
5
u
3
3u
2
+ 4)(u
5
+ u
4
+ 3u 1)
(u
5
+ 2u
4
+ 2u
3
u
2
u 1)
2
c
2
, c
11
u(u + 1)
2
(u
5
+ 6u
3
+ u
2
u + 1)
2
(u
5
+ u
4
+ 6u
3
2u
2
+ 9u + 1)
(u
5
+ 2u
4
+ u
3
+ 9u
2
+ 24u + 16)
c
3
, c
8
u
3
(u
5
u
4
+ 5u
3
u
2
+ 2u + 2)
3
(u
5
+ 4u
4
+ 8u
3
+ 8u
2
4)
c
4
, c
10
u(u 1)(u + 1)(u
5
u
3
3u
2
+ 4)(u
5
+ u
4
+ 3u 1)
(u
5
+ 2u
4
+ 2u
3
u
2
u 1)
2
c
5
, c
9
u(u + 1)
2
(u
5
u
3
3u
2
+ 4)(u
5
+ u
4
+ 3u 1)
(u
5
+ 2u
4
+ 2u
3
u
2
u 1)
2
18
X. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
5
c
6
, c
7
, c
9
c
10
y(y 1)
2
(y
5
+ 6y
3
y
2
y 1)
2
(y
5
2y
4
+ y
3
9y
2
+ 24y 16)
(y
5
y
4
+ 6y
3
+ 2y
2
+ 9y 1)
c
2
, c
11
y(y 1)
2
(y
5
2y
4
+ 13y
3
97y
2
+ 288y 256)
(y
5
+ 11y
4
+ 58y
3
+ 102y
2
+ 85y 1)
(y
5
+ 12y
4
+ 34y
3
13y
2
y 1)
2
c
3
, c
8
y
3
(y
5
32y
2
+ 64y 16)(y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y 4)
3
19