11n
74
(K11n
74
)
1
Arc Sequences
4 1 8 2 11 9 4 6 8 1 6
Solving Sequence
1,4 2,6,8
9 10 3 7 11 5
c
1
c
8
c
9
c
3
c
7
c
11
c
5
c
2
, c
4
, c
6
, c
10
Representation Ideals
I =
5
\
i=1
I
u
i
\
I
v
1
I
u
1
= hc, b 1, u 1, da 1i
I
u
2
= ha, c, b 1, d 1, u 1i
I
u
3
= hb, c, u 1, a + 1, d + 1i
I
u
4
= hu
3
+ 2u
2
+ 4a + 3u, u
3
+ 4b + u + 2, u
3
+ 4d + 3u 6, u
4
+ 2u
3
+ 3u
2
4u 4, u
3
+ 4u
2
+ 8c + 3u 6i
I
u
5
= h−u
3
u
2
+ 2d u + 1, u
6
+ 3u
5
+ 5u
4
+ 3u
3
+ 2a 6u + 4, u
7
+ 3u
6
+ 5u
5
+ 3u
4
u
3
7u
2
+ 3u + 1,
u
6
+ 3u
5
+ 6u
4
+ 4u
3
u
2
+ 4b 7u + 2, u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ u
2
+ 2c 5u + 3i
I
v
1
= ha, d, c 1, v 1, b + 1i
There are 6 irreducible components with 14 representations.
There are 1 irreducible components of dim
C
= 1 for 11n
74
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
= hc, b 1, u 1, da 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
6
=
a
1
a
8
=
0
d
a
9
=
a
d 1
a
10
=
a
1
a
3
=
0
1
a
7
=
0
d
a
11
=
a + 1
1
a
5
=
1
0
a
5
=
1
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 = ···
a = ···
b = ···
c = ···
d = ···
1.64493 7.01802 4.61409I
3
II. I
u
2
= ha, c, b 1, d 1, u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
6
=
0
1
a
8
=
0
1
a
9
=
0
1
a
10
=
0
1
a
3
=
0
1
a
7
=
0
1
a
11
=
1
1
a
5
=
1
0
a
5
=
1
0
(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
0 0
5
III. I
u
3
= hb, c, u 1, a + 1, d + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
6
=
1
0
a
8
=
0
1
a
9
=
1
1
a
10
=
1
0
a
3
=
0
1
a
7
=
0
1
a
11
=
1
0
a
5
=
1
0
a
5
=
1
0
(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 = 1.00000
b = 0
c = 0
d = 1.00000
3.28987 12.0000
7
IV. I
u
4
= hu
3
+ 2u
2
+ 4a + 3u, u
3
+ 4b + u + 2, u
3
+ 4d + 3u 6, u
4
+
2u
3
+ 3u
2
4u 4, u
3
+ 4u
2
+ 8c + 3u 6i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
1
4
u
3
1
2
u
2
3
4
u
1
4
u
3
1
4
u
1
2
a
8
=
1
8
u
3
1
2
u
2
3
8
u +
3
4
1
4
u
3
3
4
u +
3
2
a
9
=
1
4
u
3
1
2
u
2
3
4
u
1
4
u
3
u
2
3
4
u +
5
2
a
10
=
5
8
u
3
9
8
u
5
4
1
2
u
3
+
1
2
u
a
3
=
u
2
+ 1
u
2
a
7
=
1
8
u
3
1
2
u
2
3
8
u +
3
4
1
4
u
3
+ u
2
9
4
u +
1
2
a
11
=
1
8
u
3
+
5
8
u +
5
4
1
2
u
3
1
2
u
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(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 = 1.20711 1.83612I
a = 0.750000 0.380272I
b = 2.41421
c = 0.85355 1.29833I
d = 0.20711 + 1.83612I
17.2718 0
u = 1.20711 + 1.83612I
a = 0.750000 + 0.380272I
b = 2.41421
c = 0.85355 + 1.29833I
d = 0.20711 1.83612I
17.2718 0
u = 0.726339
a = 0.376768
b = 0.414214
c = 0.806492
d = 2.14055
2.46740 0
u = 1.14055
a = 1.87677
b = 0.414214
c = 0.513599
d = 0.273661
2.46740 0
9
V. I
u
5
= h−u
3
u
2
+ 2d u + 1, u
6
+ 3u
5
+ · · · + 2a + 4, u
7
+ 3u
6
+ · · · +
3u + 1, u
6
+ 3u
5
+ · · · + 4b + 2, u
6
+ 3u
5
+ · · · + 2c + 3i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
1
2
u
6
3
2
u
5
+ ··· + 3u 2
1
4
u
6
3
4
u
5
+ ··· +
7
4
u
1
2
a
8
=
1
2
u
6
3
2
u
5
+ ··· +
5
2
u
3
2
1
2
u
3
+
1
2
u
2
+
1
2
u
1
2
a
9
=
1
2
u
6
3
2
u
5
+ ··· + 3u 2
1
4
u
5
+
1
4
u
4
+ ··· +
1
4
u
3
4
a
10
=
1
2
u
6
u
5
+ ··· +
5
2
u 2
1
4
u
5
1
4
u
4
+ u
2
+
1
4
u
3
4
a
3
=
u
2
+ 1
u
2
a
7
=
1
2
u
6
3
2
u
5
+ ··· +
5
2
u
3
2
1
2
u
5
u
4
+ ··· + u
1
2
a
11
=
1
2
u
6
+
5
4
u
5
+ ···
11
4
u +
11
4
1
4
u
5
+
1
4
u
4
u
2
1
4
u +
3
4
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(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 = 1.56387 1.00084I
a = 0.666881 0.919602I
b = 1.84642 0.60235I
c = 0.544347 + 1.185976I
d = 0.12253 2.10558I
14.9463 10.4045I 1.17625 + 4.09895I
u = 1.56387 + 1.00084I
a = 0.666881 + 0.919602I
b = 1.84642 + 0.60235I
c = 0.544347 1.185976I
d = 0.12253 + 2.10558I
14.9463 + 10.4045I 1.17625 4.09895I
u = 0.46828 1.59550I
a = 0.812628 + 0.339128I
b = 1.45546 + 1.12941I
c = 0.97317 1.11623I
d = 0.16054 + 1.45536I
5.28066 2.46552I 0.37200 + 1.61165I
u = 0.46828 + 1.59550I
a = 0.812628 0.339128I
b = 1.45546 1.12941I
c = 0.97317 + 1.11623I
d = 0.16054 1.45536I
5.28066 + 2.46552I 0.37200 1.61165I
u = 0.222829
a = 2.65729
b = 0.869789
c = 2.06517
d = 0.592120
1.26042 8.87751
u = 0.643564 0.238013I
a = 0.317102 + 0.524945I
b = 0.543935 + 0.251709I
c = 0.396234 + 0.938255I
d = 0.079132 0.413310I
1.11796 + 1.29283I 4.63450 5.74515I
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.643564 + 0.238013I
a = 0.317102 0.524945I
b = 0.543935 0.251709I
c = 0.396234 0.938255I
d = 0.079132 + 0.413310I
1.11796 1.29283I 4.63450 + 5.74515I
11
VI. I
v
1
= ha, d, c 1, v 1, b + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
1
0
a
2
=
1
0
a
6
=
0
1
a
8
=
1
0
a
9
=
1
1
a
10
=
0
1
a
3
=
1
0
a
7
=
1
0
a
11
=
1
1
a
5
=
1
0
a
5
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
c = 1.00000
d = 0
0 0
13
VII. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
u(u 1)
2
(u
4
+ 2u
3
+ 3u
2
4u 4)
(u
7
+ 3u
6
+ 5u
5
+ 3u
4
u
3
7u
2
+ 3u + 1)
c
2
, c
9
u(u + 1)
2
(u
4
+ 2u
3
+ 17u
2
40u + 16)
(u
7
+ u
6
+ 5u
5
+ 29u
4
+ 67u
3
61u
2
+ 23u 1)
c
3
, c
7
u
3
(u
2
+ 4u + 2)
2
(u
7
6u
5
+ 4u
4
+ 32u
3
12u
2
+ 16u 8)
c
4
, c
8
u(u + 1)
2
(u
4
+ 2u
3
+ 3u
2
4u 4)
(u
7
+ 3u
6
+ 5u
5
+ 3u
4
u
3
7u
2
+ 3u + 1)
c
5
, c
11
(u)(u 1)(u + 1)(1 + 2u + u
2
)
2
(u
7
+ u
6
+ ··· 8u 4)
c
10
u(u + 1)
2
(u
2
+ 6u + 1)
2
(u
7
+ 9u
6
+ 46u
5
+ 142u
4
+ 297u
3
+ 249u
2
+ 88u + 16)
14
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
6
c
8
y(y 1)
2
(y
4
+ 2y
3
+ 17y
2
40y + 16)
(y
7
+ y
6
+ 5y
5
+ 29y
4
+ 67y
3
61y
2
+ 23y 1)
c
2
, c
9
y(y 1)
2
(y
4
+ 30y
3
+ 481y
2
1056y + 256)
(y
7
+ 9y
6
+ 101y
5
3y
4
+ 8259y
3
581y
2
+ 407y 1)
c
3
, c
7
y
3
(y
2
12y + 4)
2
(y
7
12y
6
+ 100y
5
368y
4
+ 928y
3
+ 944y
2
+ 64y 64)
c
5
, c
11
y(y 1)
2
(y
2
6y + 1)
2
(y
7
9y
6
+ 46y
5
142y
4
+ 297y
3
249y
2
+ 88y 16)
c
10
y(y 1)
2
(y
2
34y + 1)
2
(y
7
+ 11y
6
+ 154y
5
+ 2854y
4
+ 25301y
3
14273y
2
224y 256)
15