11n
78
(K11n
78
)
1
Arc Sequences
4 1 8 2 11 9 10 4 7 1 6
Solving Sequence
1,4 2,6,9
7 8 3 11 5 10
c
1
c
6
c
8
c
3
c
11
c
5
c
10
c
2
, c
4
, c
7
, c
9
Representation Ideals
I =
8
\
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
= ha, d, c 1, b u, u
2
+ u 1i
I
u
4
= ha, b u, d + 2u 1, u
3
+ c 1, u
4
u
3
+ 2u 1i
I
u
5
= hb, c, u 1, a + 1, d + 1i
I
u
6
= hd, c 1, b
4
b
3
+ 2b 1, b
3
2b
2
+ b + a, b
3
+ b + u 1i
I
u
7
= hu
8
+ u
7
3u
6
u
5
+ 3u
4
4u
3
3u
2
+ 4u + 4, 3u
7
+ 7u
6
u
5
3u
4
+ 5u
3
8u
2
+ 4a 13u 8,
u
7
+ 3u
6
u
5
3u
4
+ 5u
3
2u
2
+ 4b 7u 2, u
7
7u
6
7u
5
+ 7u
4
u
3
2u
2
+ 8c + 23u + 14,
u
7
3u
6
3u
5
+ 3u
4
+ 3u
3
6u
2
+ 4d + 7u + 14i
I
u
8
= h−u
7
+ 4u
5
2u
4
4u
3
+ 7u
2
+ 2d u 1, u
8
u
7
+ 6u
6
+ 4u
5
12u
4
u
3
+ 8u
2
+ 2c 8u + 1,
u
10
2u
9
+ 6u
8
+ 12u
7
14u
6
21u
5
+ 21u
4
+ 5u
3
22u
2
+ 2a + 10u,
u
11
+ 2u
10
6u
9
12u
8
+ 13u
7
+ 21u
6
17u
5
7u
4
+ 18u
3
3u
2
u 1,
u
10
2u
9
+ 5u
8
+ 12u
7
8u
6
23u
5
+ 9u
4
+ 16u
3
15u
2
+ 4b 3u + 2i
I
v
1
= ha, d, c 1, v 1, b + 1i
There are 9 irreducible components with 32 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
There are 1 irreducible components of dim
C
= 1 for 11n
78
2
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
9
=
0
d
a
7
=
a
d + 1
a
8
=
0
d
a
3
=
0
1
a
11
=
a + 1
1
a
5
=
1
0
a
10
=
a
1
a
10
=
a
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
3
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = ···
a = ···
b = ···
c = ···
d = ···
1.64493 8.27908 + 0.88933I
4
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
9
=
0
1
a
7
=
0
1
a
8
=
0
1
a
3
=
0
1
a
11
=
1
1
a
5
=
1
0
a
10
=
0
1
a
10
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(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
6
III. I
u
3
= ha, d, c 1, b u, u
2
+ u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u + 1
a
6
=
0
u
a
9
=
1
0
a
7
=
u
u
a
8
=
1
u 1
a
3
=
u
u + 1
a
11
=
1
u + 1
a
5
=
u
u + 1
a
10
=
u
u + 1
a
10
=
u
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.61803
a = 0
b = 1.61803
c = 1.00000
d = 0
8.88264 10.0000
u = 0.618034
a = 0
b = 0.618034
c = 1.00000
d = 0
0.986960 10.0000
8
IV. I
u
4
= ha, b u, d + 2u 1, u
3
+ c 1, u
4
u
3
+ 2u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
0
u
a
9
=
u
3
+ 1
2u + 1
a
7
=
u
2
1
u
3
2u
2
+ 1
a
8
=
u
3
+ 1
u
3
+ u
2
u
a
3
=
u
2
+ 1
u
2
a
11
=
1
u
2
a
5
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
2
a
10
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.15372
a = 0
b = 1.15372
c = 0.535687
d = 3.30744
0.986960 10.0000
u = 0.535687
a = 0
b = 0.535687
c = 1.15372
d = 0.0713748
0.986960 10.0000
u = 0.809017 0.981593I
a = 0
b = 0.809017 0.981593I
c = 0.809017 0.981593I
d = 0.61803 + 1.96319I
8.88264 10.0000
u = 0.809017 + 0.981593I
a = 0
b = 0.809017 + 0.981593I
c = 0.809017 + 0.981593I
d = 0.61803 1.96319I
8.88264 10.0000
10
V. I
u
5
= 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
9
=
0
1
a
7
=
1
1
a
8
=
0
1
a
3
=
0
1
a
11
=
1
0
a
5
=
1
0
a
10
=
1
0
a
10
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
c = 0
d = 1.00000
3.28987 12.0000
12
VI. I
u
6
= hd, c 1, b
4
b
3
+ 2b 1, b
3
2b
2
+ b + a, b
3
+ b + u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
b
3
b + 1
a
2
=
1
b
3
+ b
a
6
=
b
3
+ 2b
2
b
b
a
9
=
1
0
a
7
=
b
3
+ 2b
2
2b
b
a
8
=
1
b
3
b
a
3
=
b
3
b + 1
b
3
+ b
a
11
=
b
3
b
2
+ 2b
b
2
a
5
=
b
3
+ b 1
b
3
+ b
a
10
=
b
3
2b
2
+ 2b
b
2
a
10
=
b
3
2b
2
+ 2b
b
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
13
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 0.618034
a = 5.35155
b = 1.15372
c = 1.00000
d = 0
0.986960 10.0000
u = 0.618034
a = 0.115487
b = 0.535687
c = 1.00000
d = 0
0.986960 10.0000
u = 1.61803
a = 0.381966 1.213316I
b = 0.809017 0.981593I
c = 1.00000
d = 0
8.88264 10.0000
u = 1.61803
a = 0.381966 + 1.213316I
b = 0.809017 + 0.981593I
c = 1.00000
d = 0
8.88264 10.0000
14
VII. I
u
7
= hu
8
+ u
7
+ · · · + 4u + 4, 3u
7
+ 7u
6
+ · · · + 4a 8, u
7
+ 3u
6
+ · · · +
4b 2, u
7
7u
6
+ · · · + 8c + 14, u
7
3u
6
+ · · · + 4d + 14i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
3
4
u
7
7
4
u
6
+ ··· +
13
4
u + 2
1
4
u
7
3
4
u
6
+ ··· +
7
4
u +
1
2
a
9
=
1
8
u
7
+
7
8
u
6
+ ···
23
8
u
7
4
1
4
u
7
+
3
4
u
6
+ ···
7
4
u
7
2
a
7
=
3
8
u
7
9
8
u
6
+ ··· +
21
8
u +
7
4
1
2
u
7
+
1
2
u
6
+ ···
1
2
u + 2
a
8
=
1
8
u
7
+
7
8
u
6
+ ···
23
8
u
7
4
1
4
u
7
3
4
u
6
+ ··· +
7
4
u
1
2
a
3
=
u
2
+ 1
u
2
a
11
=
7
8
u
7
+
13
8
u
6
+ ···
33
8
u
7
4
1
2
u
7
+
1
2
u
6
+ ··· 2u
2
3
2
u
a
5
=
u
u
3
+ u
a
10
=
3
8
u
7
+
9
8
u
6
+ ···
21
8
u
7
4
1
2
u
7
+
1
2
u
6
+ ··· 2u
2
3
2
u
a
10
=
3
8
u
7
+
9
8
u
6
+ ···
21
8
u
7
4
1
2
u
7
+
1
2
u
6
+ ··· 2u
2
3
2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
15
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
1(vol +
1CS) Cusp shape
u = 1.59604 0.21793I
a = 0.115083 1.406861I
b = 1.066121 0.864054I
c = 1.001829 0.150682I
d = 0.166370 1.242008I
8.06290 6.78371I 8.56010 + 4.72374I
u = 1.59604 + 0.21793I
a = 0.115083 + 1.406861I
b = 1.066121 + 0.864054I
c = 1.001829 + 0.150682I
d = 0.166370 + 1.242008I
8.06290 + 6.78371I 8.56010 4.72374I
u = 0.695289 0.428533I
a = 0.542307 + 0.680462I
b = 0.566121 + 0.458821I
c = 0.623998 + 0.858133I
d = 1.59064 + 1.68383I
2.62917 1.45022I 7.43990 + 4.72374I
u = 0.695289 + 0.428533I
a = 0.542307 0.680462I
b = 0.566121 0.458821I
c = 0.623998 0.858133I
d = 1.59064 1.68383I
2.62917 + 1.45022I 7.43990 4.72374I
u = 0.529919 1.081983I
a = 0.865083 + 0.577452I
b = 1.066121 + 0.864054I
c = 0.913781 + 0.999915I
d = 0.49027 2.22033I
8.06290 + 6.78371I 8.56010 4.72374I
u = 0.529919 + 1.081983I
a = 0.865083 0.577452I
b = 1.066121 0.864054I
c = 0.913781 0.999915I
d = 0.49027 + 2.22033I
8.06290 6.78371I 8.56010 + 4.72374I
16
Solution to I
u
7
1(vol +
1CS) Cusp shape
u = 1.261410 0.030288I
a = 1.29231 + 1.30385I
b = 0.566121 + 0.458821I
c = 0.035950 0.685854I
d = 0.085459 0.705514I
2.62917 1.45022I 7.43990 + 4.72374I
u = 1.261410 + 0.030288I
a = 1.29231 1.30385I
b = 0.566121 0.458821I
c = 0.035950 + 0.685854I
d = 0.085459 + 0.705514I
2.62917 + 1.45022I 7.43990 4.72374I
17
VIII. I
u
8
= h−u
7
+ 4u
5
+ · · · + 2d 1, u
8
u
7
+ · · · + 2c + 1, u
10
2u
9
+ · · · + 2a + 10u, u
11
+ 2u
10
+ · · · u 1, u
10
2u
9
+ · · · + 4b + 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
1
2
u
10
+ u
9
+ ··· + 11u
2
5u
1
4
u
10
+
1
2
u
9
+ ··· +
3
4
u
1
2
a
9
=
1
2
u
8
+
1
2
u
7
+ ··· + 4u
1
2
1
2
u
7
2u
5
+ ··· +
1
2
u +
1
2
a
7
=
1
2
u
10
+
1
2
u
9
+ ··· + 7u
2
9
2
u
1
4
u
10
+
1
2
u
9
+ ··· +
1
4
u
1
2
a
8
=
1
2
u
8
+
1
2
u
7
+ ··· + 4u
1
2
1
2
u
10
1
2
u
9
+ ··· +
1
2
u +
1
2
a
3
=
u
2
+ 1
u
2
a
11
=
1
2
u
10
3
4
u
9
+ ··· +
19
4
u +
3
4
1
4
u
9
+
5
4
u
7
+ ··· +
1
4
u +
3
4
a
5
=
u
u
3
+ u
a
10
=
1
2
u
10
1
2
u
9
+ ··· 7u
2
+
9
2
u
1
4
u
9
+
5
4
u
7
+ ··· +
1
4
u +
3
4
a
10
=
1
2
u
10
1
2
u
9
+ ··· 7u
2
+
9
2
u
1
4
u
9
+
5
4
u
7
+ ··· +
1
4
u +
3
4
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
18
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
1(vol +
1CS) Cusp shape
u = 1.72035 0.28600I
a = 0.434360 + 0.920646I
b = 0.532201 + 1.268138I
c = 1.088658 0.174544I
d = 0.48395 1.70806I
17.3830 4.9116I 11.49209 + 1.65700I
u = 1.72035 + 0.28600I
a = 0.434360 0.920646I
b = 0.532201 1.268138I
c = 1.088658 + 0.174544I
d = 0.48395 + 1.70806I
17.3830 + 4.9116I 11.49209 1.65700I
u = 1.60901 0.41639I
a = 0.194428 1.371426I
b = 1.29448 0.81734I
c = 1.048638 + 0.270416I
d = 0.25822 + 2.28759I
14.9243 12.3125I 9.62929 + 5.75829I
u = 1.60901 + 0.41639I
a = 0.194428 + 1.371426I
b = 1.29448 + 0.81734I
c = 1.048638 0.270416I
d = 0.25822 2.28759I
14.9243 + 12.3125I 9.62929 5.75829I
u = 0.162723 0.277330I
a = 0.22673 + 2.50982I
b = 0.937916 + 0.171871I
c = 0.96637 1.53134I
d = 0.659155 0.471284I
1.66390 0.61823I 3.63835 + 1.22407I
u = 0.162723 + 0.277330I
a = 0.22673 2.50982I
b = 0.937916 0.171871I
c = 0.96637 + 1.53134I
d = 0.659155 + 0.471284I
1.66390 + 0.61823I 3.63835 1.22407I
19
Solution to I
u
8
1(vol +
1CS) Cusp shape
u = 0.552760 0.641799I
a = 0.712390 + 0.815288I
b = 0.792159 + 0.569904I
c = 0.940583 0.816704I
d = 0.129470 + 0.907870I
0.79689 + 3.53286I 6.46290 7.08687I
u = 0.552760 + 0.641799I
a = 0.712390 0.815288I
b = 0.792159 0.569904I
c = 0.940583 + 0.816704I
d = 0.129470 0.907870I
0.79689 3.53286I 6.46290 + 7.08687I
u = 0.590824
a = 0.0396568
b = 0.563771
c = 1.04963
d = 0.0234302
0.987118 9.97442
u = 1.64391 0.11631I
a = 0.234439 1.284061I
b = 0.962808 0.959946I
c = 0.077846 1.022104I
d = 0.53475 2.08362I
10.83445 + 3.51232I 10.06687 2.29315I
u = 1.64391 + 0.11631I
a = 0.234439 + 1.284061I
b = 0.962808 + 0.959946I
c = 0.077846 + 1.022104I
d = 0.53475 + 2.08362I
10.83445 3.51232I 10.06687 + 2.29315I
20
IX. 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
9
=
1
0
a
7
=
1
1
a
8
=
1
0
a
3
=
1
0
a
11
=
1
1
a
5
=
1
0
a
10
=
0
1
a
10
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
21
(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
22
X. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
u(u 1)
2
(u
2
+ u 1)
3
(u
4
u
3
+ 2u 1)
(u
8
+ u
7
+ ··· + 4u + 4)( u
11
+ 2u
10
+ ··· u 1)
c
2
u(u + 1)
2
(u
2
+ 3u + 1)
3
(u
4
+ u
3
+ 2u
2
+ 4u + 1)
(u
8
+ 7u
7
+ 17u
6
+ 17u
5
+ 19u
4
+ 50u
3
+ 65u
2
+ 40u + 16)
(u
11
+ 16u
10
+ ··· 5u + 1)
c
3
, c
8
u
3
(u
2
+ u 1)
5
(u
4
3u
3
+ 3u
2
2u + 2)
2
(u
11
+ 2u
10
u
9
8u
8
11u
7
+ 46u
5
+ 76u
4
+ 32u
3
12u
2
16u 8)
c
5
u(u 1)(u + 1)(u
2
u 1)(u
4
+ u
3
2u 1)
2
(u
4
+ u
3
u + 1)
2
(u
11
+ 2u
10
+ u
9
2u
8
+ 5u
6
+ 7u
5
6u
4
13u
3
3u
2
+ 8u + 4)
c
6
, c
7
u(u 1)(u + 1)(u
2
+ u 1)
3
(u
4
u
3
+ 2u 1)
(u
8
+ u
7
+ ··· + 4u + 4)( u
11
+ 2u
10
+ ··· u 1)
c
9
u(u + 1)
2
(u
2
+ u 1)
3
(u
4
u
3
+ 2u 1)
(u
8
+ u
7
+ ··· + 4u + 4)( u
11
+ 2u
10
+ ··· u 1)
c
10
u(u + 1)
2
(u
2
+ 3u + 1)(u
4
+ u
3
+ 2u
2
+ 4u + 1)
2
(1 + u + 4u
2
+ u
3
+ u
4
)
2
(u
11
+ 2u
10
+ ··· + 88u + 16)
c
11
u(u + 1)
2
(u
2
u 1)(u
4
+ u
3
2u 1)
2
(u
4
+ u
3
u + 1)
2
(u
11
+ 2u
10
+ u
9
2u
8
+ 5u
6
+ 7u
5
6u
4
13u
3
3u
2
+ 8u + 4)
23
XI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
6
c
7
, c
9
y(y 1)
2
(y
2
3y + 1)
3
(y
4
y
3
+ 2y
2
4y + 1)
(y
8
7y
7
+ 17y
6
17y
5
+ 19y
4
50y
3
+ 65y
2
40y + 16)
(y
11
16y
10
+ ··· 5y 1)
c
2
y(y 1)
2
(y
2
7y + 1)
3
(y
4
+ 3y
3
2y
2
12y + 1)
(y
8
15y
7
+ 89y
6
213y
5
+ 343y
4
846y
3
+ 833y
2
+ 480y + 256)
(y
11
36y
10
+ ··· 93y 1)
c
3
, c
8
y
3
(y
2
3y + 1)
5
(y
4
3y
3
+ y
2
+ 8y + 4)
2
(y
11
6y
10
+ ··· + 64y 64)
c
5
, c
11
y(y 1)
2
(y
2
3y + 1)(y
4
y
3
+ 2y
2
4y + 1)
2
(1 y + 4y
2
y
3
+ y
4
)
2
(y
11
2y
10
+ ··· + 88y 16)
c
10
y(y 1)
2
(y
2
7y + 1)(y
4
+ 3y
3
2y
2
12y + 1)
2
(1 + 7y + 16y
2
+ 7y
3
+ y
4
)
2
(y
11
+ 14y
10
+ ··· + 2336y 256)
24