11n
133
(K11n
133
)
1
Arc Sequences
5 1 10 11 2 9 4 5 1 8 7
Solving Sequence
1,5
2 3
6,9
7 10 8 11 4
c
1
c
2
c
5
c
6
c
9
c
8
c
11
c
4
c
3
, c
7
, c
10
Representation Ideals
I =
8
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b + u + 1, a u 1i
I
u
2
= hu
5
5u
4
+ 7u
3
2u
2
+ u 1, a 1, u
2
+ b + ui
I
u
3
= hu
5
+ 3u
4
+ u
3
2u
2
u + 1, a + 1, u
2
+ b + ui
I
u
4
= hu
6
4u
5
u
4
+ 18u
3
9u
2
20u + 16, u
5
+ 4u
4
u
3
8u
2
+ 4b + 7u,
u
5
+ 3u
4
+ 3u
3
11u
2
+ 4a 3u + 9i
I
u
5
= hb
2
+ b + 1, a + 1, u 1i
I
u
6
= hb
2
+ b + 1, u 1, b + a 1i
I
u
7
= hb
6
4b
5
b
4
+ 18b
3
9b
2
20b + 16, a 1, b
5
+ 2b
4
+ 7b
3
12b
2
11b + 4u + 16i
I
u
8
= hb
6
+ 7b
5
+ 18b
4
+ 27b
3
+ 38b
2
+ 11b + 1, 3b
5
21b
4
53b
3
75b
2
105b + 8u 17,
7b
5
+ 47b
4
+ 113b
3
+ 161b
2
+ 233b + 8a + 31i
There are 8 irreducible components with 34 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 + u + 1, a u 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u 1
a
3
=
u + 2
u 1
a
6
=
u
u 1
a
9
=
u + 1
u 1
a
7
=
0
1
a
10
=
2u + 2
u 1
a
8
=
u + 1
1
a
11
=
1
1
a
4
=
u
0
a
4
=
u
0
(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 = 0.500000 + 0.866025I
1.64493 + 2.02988I 9.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.500000 0.866025I
1.64493 2.02988I 9.00000 + 3.46410I
3
II. I
u
2
= hu
5
5u
4
+ 7u
3
2u
2
+ u 1, a 1, u
2
+ b + ui
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
9
=
1
u
2
u
a
7
=
u
2
2u
u
4
3u
3
+ u
2
+ u
a
10
=
u
2
+ u + 1
u
2
u
a
8
=
1
u
a
11
=
u
3
3u
2
+ u + 1
u
4
+ 2u
3
+ u
2
u
a
4
=
u
3
2u
2
u + 1
u
3
+ 2u
2
a
4
=
u
3
2u
2
u + 1
u
3
+ 2u
2
(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 = 0.181543 0.487016I
a = 1.00000
b = 0.022683 + 0.663845I
1.11858 + 1.59084I 0.80256 2.24828I
u = 0.181543 + 0.487016I
a = 1.00000
b = 0.022683 0.663845I
1.11858 1.59084I 0.80256 + 2.24828I
u = 0.668174
a = 1.00000
b = 0.221718
1.20060 7.90698
u = 2.34746 0.17191I
a = 1.00000
b = 3.13354 0.63521I
18.6126 + 10.9920I 8.84907 4.91483I
u = 2.34746 + 0.17191I
a = 1.00000
b = 3.13354 + 0.63521I
18.6126 10.9920I 8.84907 + 4.91483I
5
III. I
u
3
= hu
5
+ 3u
4
+ u
3
2u
2
u + 1, a + 1, u
2
+ b + ui
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
9
=
1
u
2
u
a
7
=
u
2
2u
u
4
3u
3
u
2
+ u
a
10
=
u
2
+ u 1
u
2
u
a
8
=
1
u
a
11
=
u
3
+ 3u
2
+ u 1
u
4
+ 2u
3
u
2
u
a
4
=
u
3
2u
2
+ u + 1
u
3
+ 2u
2
a
4
=
u
3
2u
2
+ u + 1
u
3
+ 2u
2
(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 = 2.30777
a = 1.00000
b = 3.01803
16.3055 8.85728
u = 0.921567 0.544227I
a = 1.00000
b = 0.368464 0.458856I
2.41702 7.42796I 6.48635 + 7.69371I
u = 0.921567 + 0.544227I
a = 1.00000
b = 0.368464 + 0.458856I
2.41702 + 7.42796I 6.48635 7.69371I
u = 0.575451 0.217130I
a = 1.00000
b = 0.859450 + 0.467025I
2.58971 + 1.95896I 10.08501 4.98123I
u = 0.575451 + 0.217130I
a = 1.00000
b = 0.859450 0.467025I
2.58971 1.95896I 10.08501 + 4.98123I
7
IV. I
u
4
= hu
6
4u
5
u
4
+ 18u
3
9u
2
20u + 16, u
5
+ 4u
4
u
3
8u
2
+
4b + 7u, u
5
+ 3u
4
+ 3u
3
11u
2
+ 4a 3u + 9i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
9
=
1
4
u
5
3
4
u
4
+ ··· +
3
4
u
9
4
1
4
u
5
u
4
+
1
4
u
3
+ 2u
2
7
4
u
a
7
=
3
16
u
5
+
1
2
u
4
+ ···
21
16
u + 1
1
4
u
5
+ u
4
1
4
u
3
3u
2
+
7
4
u + 1
a
10
=
1
4
u
4
u
3
+
3
4
u
2
+
5
2
u
9
4
1
4
u
5
u
4
+
1
4
u
3
+ 2u
2
7
4
u
a
8
=
1
4
u
5
3
4
u
4
+ ··· +
3
4
u
9
4
1
4
u
5
+
1
2
u
4
+ ···
11
4
u + 4
a
11
=
1
8
u
5
1
4
u
4
+ ··· +
19
8
u
11
4
1
4
u
5
u
4
+
5
4
u
3
+ u
2
15
4
u + 2
a
4
=
3
16
u
5
+
1
2
u
4
+ ···
29
16
u + 4
1
4
u
5
1
2
u
4
+ ··· +
11
4
u 3
a
4
=
3
16
u
5
+
1
2
u
4
+ ···
29
16
u + 4
1
4
u
5
1
2
u
4
+ ··· +
11
4
u 3
(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 = 1.48721 0.12793I
a = 0.106812 0.438266I
b = 0.17052 1.66828I
2.25297 4.59885I 9.12391 + 5.59727I
u = 1.48721 + 0.12793I
a = 0.106812 + 0.438266I
b = 0.17052 + 1.66828I
2.25297 + 4.59885I 9.12391 5.59727I
u = 1.054941 0.264726I
a = 0.240575 0.570430I
b = 0.170516 0.063771I
2.25297 + 0.53909I 9.12391 + 1.33093I
u = 1.054941 + 0.264726I
a = 0.240575 + 0.570430I
b = 0.170516 + 0.063771I
2.25297 0.53909I 9.12391 1.33093I
u = 2.43227 0.39265I
a = 0.883763 0.142671I
b = 3.15897 + 0.86603I
16.8782 + 2.0299I 10.75217 3.46410I
u = 2.43227 + 0.39265I
a = 0.883763 + 0.142671I
b = 3.15897 0.86603I
16.8782 2.0299I 10.75217 + 3.46410I
9
V. I
u
5
= hb
2
+ b + 1, a + 1, u 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
1
0
a
9
=
1
b
a
7
=
b 1
b + 1
a
10
=
b 1
b
a
8
=
1
b + 1
a
11
=
b 1
b
a
4
=
b
0
a
4
=
b
0
(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 = 1.00000
a = 1.00000
b = 0.500000 0.866025I
1.64493 2.02988I 9.00000 + 3.46410I
u = 1.00000
a = 1.00000
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 9.00000 3.46410I
11
VI. I
u
6
= hb
2
+ b + 1, u 1, b + a 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
1
0
a
9
=
b + 1
b
a
7
=
0
b + 1
a
10
=
1
b
a
8
=
b + 1
1
a
11
=
1
b
a
4
=
1
b + 1
a
4
=
1
b + 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 = 1.00000
a = 0.500000 0.866025I
b = 0.500000 0.866025I
1.64493 2.02988I 9.00000 + 3.46410I
u = 1.00000
a = 0.500000 + 0.866025I
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 9.00000 3.46410I
13
VII. I
u
7
=
hb
6
4b
5
b
4
+18b
3
9b
2
20b+16, a1, b
5
+2b
4
+7b
3
12b
2
11b+4u+16i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
4
b
5
1
2
b
4
+ ··· +
11
4
b 4
a
2
=
1
1
4
b
5
+
1
2
b
4
+ ···
5
4
b + 1
a
3
=
1
4
b
5
1
2
b
4
+ ··· +
3
2
b
2
+
5
4
b
1
4
b
5
+
1
2
b
4
+ ···
5
4
b + 1
a
6
=
1
4
b
5
+
1
2
b
4
+ ···
11
4
b + 4
1
4
b
5
+
1
2
b
4
+ ··· +
1
4
b 1
a
9
=
1
b
a
7
=
1
2
b
5
+
3
2
b
4
+
5
2
b
3
8b
2
4b + 9
1
4
b
5
+ b
4
+ ··· +
1
4
b + 3
a
10
=
b + 1
b
a
8
=
1
1
4
b
5
1
2
b
4
+ ··· +
9
4
b 1
a
11
=
1
4
b
5
+
1
2
b
4
+ ···
15
4
b + 4
1
4
b
5
+ b
4
+ ··· +
1
4
b + 2
a
4
=
1
2
b
3
3
2
b
2
1
2
b + 3
1
4
b
5
1
2
b
4
+ ··· +
11
4
b 3
a
4
=
1
2
b
3
3
2
b
2
1
2
b + 3
1
4
b
5
1
2
b
4
+ ··· +
11
4
b 3
(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 = 0.102785 + 0.665457I
a = 1.00000
b = 1.48721 0.12793I
2.25297 0.53909I 9.12391 1.33093I
u = 0.102785 0.665457I
a = 1.00000
b = 1.48721 + 0.12793I
2.25297 + 0.53909I 9.12391 + 1.33093I
u = 0.102785 0.665457I
a = 1.00000
b = 1.054941 0.264726I
2.25297 + 4.59885I 9.12391 5.59727I
u = 0.102785 + 0.665457I
a = 1.00000
b = 1.054941 + 0.264726I
2.25297 4.59885I 9.12391 + 5.59727I
u = 2.20557
a = 1.00000
b = 2.43227 0.39265I
16.8782 2.0299I 10.75217 + 3.46410I
u = 2.20557
a = 1.00000
b = 2.43227 + 0.39265I
16.8782 + 2.0299I 10.75217 3.46410I
15
VIII. I
u
8
= hb
6
+ 7b
5
+ 18b
4
+ 27b
3
+ 38b
2
+ 11b + 1, 3b
5
+ 8u + · · ·
105b 17, 7b
5
+ 47b
4
+ · · · + 8a + 31i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
3
8
b
5
+
21
8
b
4
+ ··· +
105
8
b +
17
8
a
2
=
1
1
8
b
5
+
3
4
b
4
+ ··· +
5
2
b
2
+
23
8
b
a
3
=
1
8
b
5
3
4
b
4
+ ···
23
8
b + 1
1
8
b
5
+
3
4
b
4
+ ··· +
5
2
b
2
+
23
8
b
a
6
=
3
8
b
5
21
8
b
4
+ ···
105
8
b
17
8
5
8
b
5
+
33
8
b
4
+ ··· +
151
8
b +
25
8
a
9
=
7
8
b
5
47
8
b
4
+ ···
233
8
b
31
8
b
a
7
=
7
8
b
5
+ 6b
4
+ ··· +
233
8
b +
13
4
1
2
b
5
+
27
8
b
4
+ ··· + 17b +
25
8
a
10
=
7
8
b
5
47
8
b
4
+ ···
241
8
b
31
8
b
a
8
=
7
8
b
5
47
8
b
4
+ ···
233
8
b
31
8
3
8
b
5
21
8
b
4
+ ···
105
8
b
17
8
a
11
=
1
8
b
5
7
8
b
4
+ ···
51
8
b
15
8
1
2
b
5
+
27
8
b
4
+ ··· + 17b +
17
8
a
4
=
1
4
b
5
7
4
b
4
+ ···
41
4
b
5
4
1
8
b
5
+
7
8
b
4
+ ··· +
39
8
b +
3
8
a
4
=
1
4
b
5
7
4
b
4
+ ···
41
4
b
5
4
1
8
b
5
+
7
8
b
4
+ ··· +
39
8
b +
3
8
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
16
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
1(vol +
1CS) Cusp shape
u = 2.20557
a = 1.102785 0.178028I
b = 3.15897 0.86603I
16.8782 2.0299I 10.75217 + 3.46410I
u = 2.20557
a = 1.102785 + 0.178028I
b = 3.15897 + 0.86603I
16.8782 + 2.0299I 10.75217 3.46410I
u = 0.102785 + 0.665457I
a = 0.52491 + 2.15379I
b = 0.17052 1.66828I
2.25297 4.59885I 9.12391 + 5.59727I
u = 0.102785 0.665457I
a = 0.52491 2.15379I
b = 0.17052 + 1.66828I
2.25297 + 4.59885I 9.12391 5.59727I
u = 0.102785 0.665457I
a = 0.62769 + 1.48834I
b = 0.170516 0.063771I
2.25297 + 0.53909I 9.12391 + 1.33093I
u = 0.102785 + 0.665457I
a = 0.62769 1.48834I
b = 0.170516 + 0.063771I
2.25297 0.53909I 9.12391 1.33093I
17
IX. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
2
+ u + 1)(u
3
2u
2
1)
4
(u
5
+ 3u
4
+ u
3
2u
2
u + 1)
(u
5
+ 5u
4
+ ··· + u + 1)(u
6
+ 4u
5
+ ··· + 20u + 16)
c
2
(u + 1)
4
(u
2
u + 1)(u
3
+ 4u
2
4u + 1)
4
(u
5
+ 7u
4
+ 11u
3
+ 12u
2
+ 5u + 1)(u
5
+ 11u
4
+ 31u
3
3u + 1)
(u
6
+ 18u
5
+ 127u
4
+ 434u
3
+ 769u
2
+ 688u + 256)
c
3
, c
5
, c
8
(u + 1)
4
(u
2
u + 1)(u
3
2u
2
1)
4
(u
5
3u
4
+ u
3
+ 2u
2
u 1)
(u
5
+ 5u
4
+ ··· + u + 1)(u
6
+ 4u
5
+ ··· + 20u + 16)
c
4
, c
7
, c
11
(u + 1)
2
(u
2
u + 1)
8
(u
5
+ u
4
+ ··· u
2
1)(u
5
+ 3u
4
+ ··· + 3u
2
1)
(u
6
+ 5u
5
+ 12u
4
+ 15u
3
+ 11u
2
+ 4u + 4)
c
6
, c
9
(u
2
u + 1)
3
(u
5
6u
4
+ 9u
3
+ 4u + 1)(u
5
4u
4
+ 3u
3
1)
(u
6
4u
5
u
4
+ 18u
3
9u
2
20u + 16)
(u
6
+ 7u
5
+ 18u
4
+ 27u
3
+ 38u
2
+ 11u + 1)
2
c
10
u
4
(u
2
+ 3u + 3)(u
3
u
2
u + 2)
4
(u
5
+ 2u
4
5u
2
6u 3)
(u
5
+ 4u
4
+ ··· 2u 1)(u
6
+ 8u
5
+ ··· + 22u + 4)
18
X. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
5
c
8
(y 1)
4
(y
2
+ y + 1)(1 4y 4y
2
+ y
3
)
4
(y
5
11y
4
+ ··· 3y 1)
(y
5
7y
4
+ 11y
3
12y
2
+ 5y 1)
(y
6
18y
5
+ 127y
4
434y
3
+ 769y
2
688y + 256)
c
2
(y 1)
4
(y
2
+ y + 1)(y
3
24y
2
+ 8y 1)
4
(y
5
59y
4
+ ··· + 9y 1)(y
5
27y
4
+ ··· + y 1)
(y
6
70y
5
+ 2043y
4
17286y
3
+ 59201y
2
79616y + 65536)
c
4
, c
7
, c
11
(y 1)
2
(y
2
+ y + 1)
8
(y
5
+ y
4
+ 3y
3
+ y
2
2y 1)
(y
5
+ y
4
+ ··· + 6y 1)(y
6
y
5
+ ··· + 72y + 16)
c
6
, c
9
(y
2
+ y + 1)
3
(y
5
18y
4
+ 89y
3
+ 84y
2
+ 16y 1)
(y
5
10y
4
+ 9y
3
8y
2
1)
(y
6
18y
5
+ 127y
4
434y
3
+ 769y
2
688y + 256)
(y
6
13y
5
+ 22y
4
+ 487y
3
+ 886y
2
45y + 1)
2
c
10
y
4
(y
2
3y + 9)(4 + 5y 3y
2
+ y
3
)
4
(y
5
4y
4
+ ··· + 6y 9)
(y
5
4y
4
+ 24y
3
17y
2
+ 6y 1)
(y
6
6y
5
+ 79y
4
302y
3
+ 457y
2
76y + 16)
19