11n
76
(K11n
76
)
1
Arc Sequences
4 1 8 2 11 9 3 6 7 1 6
Solving Sequence
1,4 2,6,8
9 3 7 10 11 5
c
1
c
8
c
3
c
7
c
9
c
11
c
5
c
2
, c
4
, c
6
, c
10
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, c + 1, b u, d u + 1, u
2
u 1i
I
u
4
= ha, b u, u
3
+ c + 1, u
4
+ u
3
2u 1, u
3
u
2
+ d + u + 2i
I
u
5
= hb, c, d 1, u 1, a + 1i
I
u
6
= hu
3
+ u
2
+ a, u
3
+ c + 1, u
4
+ u
3
2u 1, u
3
+ b + u + 1, u
3
u
2
+ d + u + 2i
I
u
7
= h−b
7
+ b
6
+ b
5
+ b
4
b
3
4b
2
+ 2d + 3b, b
8
b
7
3b
6
+ b
5
+ 3b
4
+ 4b
3
3b
2
4b + 4,
b
7
b
6
+ 5b
5
+ b
4
b
3
6b
2
b + 4a + 6, 3b
7
5b
6
3b
5
+ b
4
+ 7b
3
+ 10b
2
+ 4c 13b + 2,
b
7
3b
6
b
5
+ 3b
4
+ 5b
3
+ 2b
2
7b + 4u + 2i
I
u
8
= hu
11
+ 2u
10
+ u
9
2u
8
+ 5u
6
+ 7u
5
6u
4
13u
3
3u
2
+ 8u + 4,
91u
10
+ 128u
9
57u
8
158u
7
+ 192u
6
+ 531u
5
+ 335u
4
640u
3
741u
2
+ 596a + 635u + 492,
9u
10
+ 124u
9
+ 145u
8
32u
7
112u
6
+ 249u
5
+ 413u
4
24u
3
797u
2
+ 596b 209u + 458,
363u
10
+ 432u
9
+ ··· + 1192c + 1586, 281u
10
+ 428u
9
+ ··· + 596d + 1850i
I
v
1
= ha, d, v 1, c + 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
76
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
8
=
0
d
a
9
=
a
d 1
a
3
=
0
1
a
7
=
0
d
a
10
=
a
1
a
11
=
a + 1
1
a
5
=
1
0
a
5
=
1
0
(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 10.39250 4.42322I
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
8
=
0
1
a
9
=
0
1
a
3
=
0
1
a
7
=
0
1
a
10
=
0
1
a
11
=
1
1
a
5
=
1
0
a
5
=
1
0
(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, c + 1, b u, d u + 1, 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
8
=
1
u 1
a
9
=
1
2u
a
3
=
u
u + 1
a
7
=
u
u 2
a
10
=
u
u 1
a
11
=
1
u + 1
a
5
=
u
u 1
a
5
=
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 = 0.618034
a = 0
b = 0.618034
c = 1.00000
d = 1.61803
0.986960 10.0000
u = 1.61803
a = 0
b = 1.61803
c = 1.00000
d = 0.618034
8.88264 10.0000
8
IV. I
u
4
= ha, b u, u
3
+ c + 1, u
4
+ u
3
2u 1, u
3
u
2
+ d + u + 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
0
u
a
8
=
u
3
1
u
3
+ u
2
u 2
a
9
=
u
3
1
1
a
3
=
u
2
+ 1
u
2
a
7
=
u
2
1
u
3
1
a
10
=
u
2
1
u
2
a
11
=
1
u
2
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(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 = 0.809017 0.981593I
a = 0
b = 0.809017 0.981593I
c = 0.809017 0.981593I
d = 0.30902 + 1.58825I
8.88264 10.0000
u = 0.809017 + 0.981593I
a = 0
b = 0.809017 + 0.981593I
c = 0.809017 + 0.981593I
d = 0.30902 1.58825I
8.88264 10.0000
u = 0.535687
a = 0
b = 0.535687
c = 1.15372
d = 1.33107
0.986960 10.0000
u = 1.15372
a = 0
b = 1.15372
c = 0.535687
d = 0.286961
0.986960 10.0000
10
V. I
u
5
= hb, c, d 1, u 1, a + 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
3
=
0
1
a
7
=
0
1
a
10
=
1
0
a
11
=
1
0
a
5
=
1
0
a
5
=
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
0 0
12
VI. I
u
6
=
hu
3
+u
2
+a, u
3
+c+1, u
4
+u
3
2u1, u
3
+b+u+1, u
3
u
2
+d+u+2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
u
3
u
2
u
3
u 1
a
8
=
u
3
1
u
3
+ u
2
u 2
a
9
=
u
3
2u 2
1
a
3
=
u
2
+ 1
u
2
a
7
=
u
2
1
u
3
1
a
10
=
2u
3
3u 2
u
3
+ u
a
11
=
u
3
+ 2u + 2
u
3
u
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(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 = 0.809017 0.981593I
a = 1.50000 0.60666I
b = 1.61803
c = 0.809017 0.981593I
d = 0.30902 + 1.58825I
8.88264 10.0000
u = 0.809017 + 0.981593I
a = 1.50000 + 0.60666I
b = 1.61803
c = 0.809017 + 0.981593I
d = 0.30902 1.58825I
8.88264 10.0000
u = 0.535687
a = 0.133240
b = 0.618034
c = 1.15372
d = 1.33107
0.986960 10.0000
u = 1.15372
a = 2.86676
b = 0.618034
c = 0.535687
d = 0.286961
0.986960 10.0000
14
VII. I
u
7
= h−b
7
+ 2d + · · · 4b
2
+ 3b, b
8
b
7
+ · · · 4b + 4, b
7
b
6
+
· · · + 4a + 6, 3b
7
+ 4c + · · · 13b + 2, b
7
+ 4u + · · · 7b + 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
4
b
7
+
3
4
b
6
+ ··· +
7
4
b
1
2
a
2
=
1
1
2
b
7
+
1
2
b
6
+ ··· 2b
2
+
3
2
b
a
6
=
1
4
b
7
+
1
4
b
6
+ ··· +
1
4
b
3
2
b
a
8
=
3
4
b
7
+
5
4
b
6
+ ··· +
13
4
b
1
2
1
2
b
7
1
2
b
6
+ ··· + 2b
2
3
2
b
a
9
=
1
4
b
7
1
4
b
6
+ ···
5
4
b +
7
2
1
a
3
=
1
2
b
7
1
2
b
6
+ ···
3
2
b + 1
1
2
b
7
+
1
2
b
6
+ ··· 2b
2
+
3
2
b
a
7
=
5
4
b
7
+
11
4
b
6
+ ··· +
23
4
b
3
2
1
4
b
7
3
4
b
6
+ ···
7
4
b +
1
2
a
10
=
1
2
b
7
+
1
2
b
6
+ ···
1
2
b
3
+
1
2
b
b
2
a
11
=
1
2
b
7
1
2
b
6
+ ··· + b
2
1
2
b
b
2
a
5
=
1
4
b
7
3
4
b
6
+ ···
7
4
b +
1
2
b
3
+ b
a
5
=
1
4
b
7
3
4
b
6
+ ···
7
4
b +
1
2
b
3
+ b
(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 = 0.566121 + 0.458821I
a = 3.15073 + 0.42077I
b = 1.261410 0.030288I
c = 0.676097 + 0.978318I
d = 0.109976 0.519497I
2.62917 1.45022I 7.43990 + 4.72374I
u = 0.566121 0.458821I
a = 3.15073 0.42077I
b = 1.261410 + 0.030288I
c = 0.676097 0.978318I
d = 0.109976 + 0.519497I
2.62917 + 1.45022I 7.43990 4.72374I
u = 1.066121 + 0.864054I
a = 0.664045 0.626793I
b = 0.529919 1.081983I
c = 0.676097 0.978318I
d = 0.39002 + 1.84237I
8.06290 + 6.78371I 8.56010 4.72374I
u = 1.066121 0.864054I
a = 0.664045 + 0.626793I
b = 0.529919 + 1.081983I
c = 0.676097 + 0.978318I
d = 0.39002 1.84237I
8.06290 6.78371I 8.56010 + 4.72374I
u = 0.566121 + 0.458821I
a = 0.518493 0.826005I
b = 0.695289 0.428533I
c = 0.676097 + 0.978318I
d = 0.109976 0.519497I
2.62917 1.45022I 7.43990 + 4.72374I
u = 0.566121 0.458821I
a = 0.518493 + 0.826005I
b = 0.695289 + 0.428533I
c = 0.676097 0.978318I
d = 0.109976 + 0.519497I
2.62917 + 1.45022I 7.43990 4.72374I
16
Solution to I
u
7
1(vol +
1CS) Cusp shape
u = 1.066121 0.864054I
a = 1.29629 1.03203I
b = 1.59604 0.21793I
c = 0.676097 + 0.978318I
d = 0.39002 1.84237I
8.06290 6.78371I 8.56010 + 4.72374I
u = 1.066121 + 0.864054I
a = 1.29629 + 1.03203I
b = 1.59604 + 0.21793I
c = 0.676097 0.978318I
d = 0.39002 + 1.84237I
8.06290 + 6.78371I 8.56010 4.72374I
17
VIII. I
u
8
= hu
11
+ 2u
10
+ · · · + 8u + 4, 91u
10
+ 128u
9
+ · · · + 596a +
492, 9u
10
+ 124u
9
+ · · · + 596b + 458, 363u
10
+ 432u
9
+ · · · + 1192c +
1586, 281u
10
+ 428u
9
+ · · · + 596d + 1850i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
0.152685u
10
0.214765u
9
+ ··· 1.06544u 0.825503
0.0151007u
10
0.208054u
9
+ ··· + 0.350671u 0.768456
a
8
=
0.304530u
10
0.362416u
9
+ ··· 0.594799u 1.33054
0.471477u
10
0.718121u
9
+ ··· 1.77349u 3.10403
a
9
=
0.203020u
10
0.241611u
9
+ ··· 0.229866u 1.05369
0.293624u
10
0.489933u
9
+ ··· 1.12584u 2.16443
a
3
=
u
2
+ 1
u
2
a
7
=
0.141779u
10
0.342282u
9
+ ··· 0.596477u 1.15940
0.387584u
10
0.506711u
9
+ ··· 0.666107u 2.05705
a
10
=
0.0763423u
10
+ 0.107383u
9
+ ··· + 0.282718u 0.337248
0.177852u
10
0.228188u
9
+ ··· 0.647651u 0.939597
a
11
=
0.101510u
10
+ 0.120805u
9
+ ··· + 0.364933u + 1.27685
0.177852u
10
+ 0.228188u
9
+ ··· + 0.647651u + 0.939597
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(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 = 1.29448 0.81734I
a = 0.94038 1.17343I
b = 1.60901 0.41639I
c = 0.565710 1.030625I
d = 0.32965 + 2.04536I
14.9243 12.3125I 9.62929 + 5.75829I
u = 1.29448 + 0.81734I
a = 0.94038 + 1.17343I
b = 1.60901 + 0.41639I
c = 0.565710 + 1.030625I
d = 0.32965 2.04536I
14.9243 + 12.3125I 9.62929 5.75829I
u = 0.792159 0.569904I
a = 0.651009 + 0.677715I
b = 0.552760 + 0.641799I
c = 0.777399 0.751448I
d = 0.82545 + 1.53098I
0.79689 3.53286I 6.46290 + 7.08687I
u = 0.792159 + 0.569904I
a = 0.651009 0.677715I
b = 0.552760 0.641799I
c = 0.777399 + 0.751448I
d = 0.82545 1.53098I
0.79689 + 3.53286I 6.46290 7.08687I
u = 0.563771
a = 0.0415597
b = 0.590824
c = 1.10000
d = 1.42161
0.987118 9.97442
u = 0.532201 1.268138I
a = 1.281381 0.156138I
b = 1.72035 + 0.28600I
c = 0.923024 + 1.050141I
d = 0.04222 1.42193I
17.3830 + 4.9116I 11.49209 1.65700I
19
Solution to I
u
8
1(vol +
1CS) Cusp shape
u = 0.532201 + 1.268138I
a = 1.281381 + 0.156138I
b = 1.72035 0.28600I
c = 0.923024 1.050141I
d = 0.04222 + 1.42193I
17.3830 4.9116I 11.49209 + 1.65700I
u = 0.937916 0.171871I
a = 0.769041 + 0.361555I
b = 0.162723 + 0.277330I
c = 0.373639 + 0.482953I
d = 0.160409 0.252652I
1.66390 + 0.61823I 3.63835 1.22407I
u = 0.937916 + 0.171871I
a = 0.769041 0.361555I
b = 0.162723 0.277330I
c = 0.373639 0.482953I
d = 0.160409 + 0.252652I
1.66390 0.61823I 3.63835 + 1.22407I
u = 0.962808 0.959946I
a = 1.36057 0.80757I
b = 1.64391 0.11631I
c = 0.996447 + 0.742266I
d = 0.437522 0.637453I
10.83445 + 3.51232I 10.06687 2.29315I
u = 0.962808 + 0.959946I
a = 1.36057 + 0.80757I
b = 1.64391 + 0.11631I
c = 0.996447 0.742266I
d = 0.437522 + 0.637453I
10.83445 3.51232I 10.06687 + 2.29315I
20
IX. I
v
1
= ha, d, v 1, c + 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
3
=
1
0
a
7
=
1
0
a
10
=
0
1
a
11
=
1
1
a
5
=
1
0
a
5
=
1
0
(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
3.28987 12.0000
22
X. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
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)
c
2
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
3
, c
7
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
4
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)
c
5
, c
11
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
6
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
8
, 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)
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)
23
XI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
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
2
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)
c
3
, c
7
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
6
, c
8
c
9
, c
11
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
10
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)
24