11n
183
(K11n
183
)
1
Arc Sequences
5 7 1 9 3 9 3 11 5 7 6
Solving Sequence
6,9 3,7
8 2 5 10 1 4 11
c
6
c
7
c
2
c
5
c
9
c
1
c
4
c
11
c
3
, c
8
, c
10
Representation Ideals
I =
10
\
i=1
I
u
i
I
u
1
= hu 1, a + 1, b + 1i
I
u
2
= hu
2
2u 1, a + 1, b u + 4i
I
u
3
= hu
2
u + 1, a + u, b + 1i
I
u
4
= hu
2
u + 3, b + 2, 3a + u + 2i
I
u
5
= hu
4
+ 4u
3
+ 9u
2
+ 10u + 13, u
3
3u
2
+ 3b 3u + 2, 7u
3
15u
2
+ 39a 24u + 8i
I
u
6
= hu
6
u
5
2u
4
2u
3
2u
2
4u 1, a 1, 18u
5
+ 24u
4
+ 28u
3
+ 23u
2
+ 11b + 32u + 54i
I
u
7
= hu
7
+ u
6
+ 2u
5
3u
4
+ 5u
3
3u
2
+ 4u 1, a 1, u
6
+ 3u
5
+ 8u
4
+ 4u
3
+ 4u
2
+ 9b 4u + 5i
I
u
8
= hu
8
2u
7
+ 2u
6
2u
5
+ 3u
4
6u
3
+ 8u
2
4u + 1,
5u
7
+ 20u
6
21u
5
+ 23u
4
32u
3
+ 36u
2
+ 29b 83u + 41,
54u
7
71u
6
+ 47u
5
57u
4
+ 102u
3
238u
2
+ 29a + 241u 2i
I
u
9
= hu
2
+ u + 1, b, a ui
I
u
10
= hu
2
+ u + 1, a + u, b + 1i
There are 10 irreducible components with 36 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 1, a + 1, b + 1i
(i) Arc colorings
a
6
=
0
1
a
9
=
1
1
a
3
=
1
0
a
7
=
1
0
a
8
=
1
0
a
2
=
1
0
a
5
=
1
1
a
10
=
1
1
a
1
=
0
1
a
4
=
1
1
a
11
=
0
1
a
11
=
0
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 = 1.00000
a = 1.00000
b = 1.00000
3.28987 12.0000
3
II. I
u
2
= hu
2
2u 1, a + 1, b u + 4i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
u 4
a
3
=
1
0
a
7
=
u
u + 1
a
8
=
1
u + 1
a
2
=
u
2
a
5
=
u
u
a
10
=
u
2u 3
a
1
=
0
u 2
a
4
=
1
2u + 5
a
11
=
0
u 2
a
11
=
0
u 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 = 0.414214
a = 1.00000
b = 4.41421
8.22467 52.0000
u = 2.41421
a = 1.00000
b = 1.58579
8.22467 52.0000
5
III. I
u
3
= hu
2
u + 1, a + u, b + 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
u
1
a
3
=
1
0
a
7
=
1
1
a
8
=
0
1
a
2
=
0
1
a
5
=
u
u
a
10
=
0
u 1
a
1
=
u + 1
u
a
4
=
0
u 1
a
11
=
u + 1
0
a
11
=
u + 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 = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 1.00000
0 9.00000
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 1.00000
0 9.00000
7
IV. I
u
4
= hu
2
u + 3, b + 2, 3a + u + 2i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
3
u
2
3
2
a
3
=
1
0
a
7
=
2
3
u +
5
3
u + 2
a
8
=
1
3
u
1
3
u + 2
a
2
=
7
3
u
1
3
3u 1
a
5
=
u
u
a
10
=
5
3
u
11
3
2u 5
a
1
=
1
3
u
1
3
u 1
a
4
=
1
3
u +
1
3
u 2
a
11
=
1
3
u
1
3
1
a
11
=
1
3
u
1
3
1
(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 = 0.50000 1.65831I
a = 0.833333 + 0.552771I
b = 2.00000
3.28987 6.00000
u = 0.50000 + 1.65831I
a = 0.833333 0.552771I
b = 2.00000
3.28987 6.00000
9
V. I
u
5
=
hu
4
+4u
3
+9u
2
+10u+13, u
3
3u
2
+3b3u+2, 7u
3
15u
2
+39a24u+8i
(i) Arc colorings
a
6
=
0
u
a
9
=
0.179487u
3
+ 0.384615u
2
+ 0.615385u 0.205128
1
3
u
3
+ u
2
+ u
2
3
a
3
=
1
0
a
7
=
0.0256410u
3
0.769231u
2
1.23077u 2.25641
u
2
2u 3
a
8
=
0.0256410u
3
+ 0.230769u
2
+ 0.769231u + 0.743590
u
2
2u 3
a
2
=
0.256410u
3
0.307692u
2
0.692308u + 3.56410
u
2
2u + 4
a
5
=
u
u
a
10
=
1.17949u
3
+ 2.38462u
2
+ 2.61538u 0.205128
4
3
u
3
+ 3u
2
+ 3u
2
3
a
1
=
0.0769231u
3
0.307692u
2
0.692308u 0.769231
1
3
u
3
u
2
2u
1
3
a
4
=
0.0256410u
3
0.230769u
2
0.769231u 0.743590
2
3
u
3
3u
2
6u
17
3
a
11
=
0.0769231u
3
0.307692u
2
0.692308u 0.769231
1
3
u
3
u
2
u
1
3
a
11
=
0.0769231u
3
0.307692u
2
0.692308u 0.769231
1
3
u
3
u
2
u
1
3
(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 = 1.86603 1.50000I
a = 0.214941 0.976627I
b = 0.732051
13.1595 6.00000
u = 1.86603 + 1.50000I
a = 0.214941 + 0.976627I
b = 0.732051
13.1595 6.00000
u = 0.13397 1.50000I
a = 0.984171 0.177219I
b = 2.73205
13.1595 6.00000
u = 0.13397 + 1.50000I
a = 0.984171 + 0.177219I
b = 2.73205
13.1595 6.00000
11
VI.
I
u
6
= hu
6
u
5
2u
4
2u
3
2u
2
4u 1, a 1, 18u
5
+ 24u
4
+ · · · + 11b + 54i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
1.63636u
5
2.18182u
4
+ ··· 2.90909u 4.90909
a
3
=
1
0
a
7
=
u
0.545455u
5
0.727273u
4
+ ··· 0.636364u 1.63636
a
8
=
0.545455u
5
+ 0.727273u
4
+ ··· 0.363636u + 1.63636
0.545455u
5
0.727273u
4
+ ··· 0.636364u 1.63636
a
2
=
0.181818u
5
0.0909091u
4
+ ··· + 0.545455u + 1.54545
0.818182u
5
1.09091u
4
+ ··· 1.45455u 2.45455
a
5
=
u
u
a
10
=
0.181818u
5
0.0909091u
4
+ ··· + 0.545455u + 1.54545
1.45455u
5
2.27273u
4
+ ··· 2.36364u 4.36364
a
1
=
0.181818u
5
0.0909091u
4
+ ··· + 1.54545u + 1.54545
0.818182u
5
1.09091u
4
+ ··· 0.454545u 2.45455
a
4
=
0.545455u
5
0.727273u
4
+ ··· + 0.363636u 1.63636
1.90909u
5
+ 2.54545u
4
+ ··· + 3.72727u + 6.72727
a
11
=
0.181818u
5
0.0909091u
4
+ ··· + 1.54545u + 1.54545
0.909091u
5
1.54545u
4
+ ··· 1.72727u 2.72727
a
11
=
0.181818u
5
0.0909091u
4
+ ··· + 1.54545u + 1.54545
0.909091u
5
1.54545u
4
+ ··· 1.72727u 2.72727
(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 = 0.920485 0.648681I
a = 1.00000
b = 1.57440 + 0.45622I
1.59057 4.74950I 3.95625 + 7.59808I
u = 0.920485 + 0.648681I
a = 1.00000
b = 1.57440 0.45622I
1.59057 + 4.74950I 3.95625 7.59808I
u = 0.280929
a = 1.00000
b = 4.21688
8.11594 21.9125
u = 0.346535 1.017671I
a = 1.00000
b = 0.232332 + 0.032661I
1.59057 + 4.74950I 3.95625 7.59808I
u = 0.346535 + 1.017671I
a = 1.00000
b = 0.232332 0.032661I
1.59057 4.74950I 3.95625 + 7.59808I
u = 2.42883
a = 1.00000
b = 1.60341
8.11594 21.9125
13
VII. I
u
7
= hu
7
+ u
6
+ 2u
5
3u
4
+ 5u
3
3u
2
+ 4u 1, a 1, u
6
+ 3u
5
+
8u
4
+ 4u
3
+ 4u
2
+ 9b 4u + 5i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
1
9
u
6
1
3
u
5
+ ··· +
4
9
u
5
9
a
3
=
1
0
a
7
=
u
2
9
u
6
+
2
3
u
5
+ ··· +
10
9
u +
1
9
a
8
=
2
9
u
6
2
3
u
5
+ ···
19
9
u
1
9
2
9
u
6
+
2
3
u
5
+ ··· +
10
9
u +
1
9
a
2
=
4
9
u
6
+
1
3
u
5
+ ···
7
9
u +
11
9
4
9
u
6
1
3
u
5
+ ··· +
7
9
u
2
9
a
5
=
u
u
a
10
=
4
9
u
6
+
1
3
u
5
+ ···
7
9
u +
11
9
1
3
u
6
1
3
u
4
+ ···
1
3
u
1
3
a
1
=
8
9
u
6
+
2
3
u
5
+ ···
23
9
u +
13
9
u
a
4
=
2
9
u
6
+
2
3
u
5
+ ··· +
19
9
u +
1
9
u
2
a
11
=
8
9
u
6
+
2
3
u
5
+ ···
23
9
u +
13
9
4
9
u
6
1
3
u
5
+ ··· +
7
9
u
2
9
a
11
=
8
9
u
6
+
2
3
u
5
+ ···
23
9
u +
13
9
4
9
u
6
1
3
u
5
+ ··· +
7
9
u
2
9
(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 = 1.26560 1.56709I
a = 1.00000
b = 2.15520 + 0.47338I
14.4836 11.4109I 7.74903 + 4.57488I
u = 1.26560 + 1.56709I
a = 1.00000
b = 2.15520 0.47338I
14.4836 + 11.4109I 7.74903 4.57488I
u = 0.134406 0.899226I
a = 1.00000
b = 0.731913 0.364971I
10.56253 1.19923I 2.68829 + 5.87566I
u = 0.134406 + 0.899226I
a = 1.00000
b = 0.731913 + 0.364971I
10.56253 + 1.19923I 2.68829 5.87566I
u = 0.285988
a = 1.00000
b = 0.481841
0.666622 14.5175
u = 0.757011 0.685123I
a = 1.00000
b = 1.317633 + 0.418968I
1.68375 + 3.49152I 15.3039 5.7802I
u = 0.757011 + 0.685123I
a = 1.00000
b = 1.317633 0.418968I
1.68375 3.49152I 15.3039 + 5.7802I
15
VIII. I
u
8
=
hu
8
2u
7
+· · ·4u+1, 5u
7
+20u
6
+· · ·+29b+41, 54u
7
71u
6
+· · ·+29a2i
(i) Arc colorings
a
6
=
0
u
a
9
=
1.86207u
7
+ 2.44828u
6
+ ··· 8.31034u + 0.0689655
0.172414u
7
0.689655u
6
+ ··· + 2.86207u 1.41379
a
3
=
1
0
a
7
=
2.72414u
7
+ 4.89655u
6
+ ··· 18.6207u + 6.13793
0.310345u
7
+ 0.241379u
6
+ ··· 0.551724u 0.655172
a
8
=
2.41379u
7
+ 4.65517u
6
+ ··· 18.0690u + 6.79310
0.310345u
7
+ 0.241379u
6
+ ··· 0.551724u 0.655172
a
2
=
2.58621u
7
+ 4.34483u
6
+ ··· 15.9310u + 6.20690
0.620690u
7
+ 0.482759u
6
+ ··· 1.10345u 0.310345
a
5
=
u
u
a
10
=
2u
7
+ 3u
6
2u
5
+ 2u
4
4u
3
+ 9u
2
10u + 1
0.0344828u
7
0.137931u
6
+ ··· + 1.17241u 0.482759
a
1
=
2.72414u
7
+ 4.89655u
6
+ ··· 17.6207u + 6.13793
0.758621u
7
+ 1.03448u
6
+ ··· 2.79310u 0.379310
a
4
=
2.41379u
7
4.65517u
6
+ ··· + 18.0690u 6.79310
1.10345u
7
1.41379u
6
+ ··· + 5.51724u 0.448276
a
11
=
2.72414u
7
+ 4.89655u
6
+ ··· 17.6207u + 6.13793
0.655172u
7
+ 0.620690u
6
+ ··· 2.27586u + 0.172414
a
11
=
2.72414u
7
+ 4.89655u
6
+ ··· 17.6207u + 6.13793
0.655172u
7
+ 0.620690u
6
+ ··· 2.27586u + 0.172414
(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 = 0.945027 0.945027I
a = 0.856078 0.415941I
b = 1.52909 + 0.25707I
0.82247 3.66386I 8.00000 + 2.00000I
u = 0.945027 + 0.945027I
a = 0.856078 + 0.415941I
b = 1.52909 0.25707I
0.82247 + 3.66386I 8.00000 2.00000I
u = 0.326993 0.326993I
a = 2.47411 + 1.20209I
b = 0.529086 0.742934I
0.82247 + 3.66386I 8.00000 2.00000I
u = 0.326993 + 0.326993I
a = 2.47411 1.20209I
b = 0.529086 + 0.742934I
0.82247 3.66386I 8.00000 + 2.00000I
u = 0.415941 1.202093I
a = 0.326993 0.158875I
b = 0.529086 0.742934I
0.82247 + 3.66386I 8.00000 2.00000I
u = 0.415941 + 1.202093I
a = 0.326993 + 0.158875I
b = 0.529086 + 0.742934I
0.82247 3.66386I 8.00000 + 2.00000I
u = 1.202093 0.415941I
a = 0.945027 0.459159I
b = 1.52909 0.25707I
0.82247 + 3.66386I 8.00000 2.00000I
u = 1.202093 + 0.415941I
a = 0.945027 + 0.459159I
b = 1.52909 + 0.25707I
0.82247 3.66386I 8.00000 + 2.00000I
17
IX. I
u
9
= hu
2
+ u + 1, b, a ui
(i) Arc colorings
a
6
=
0
u
a
9
=
u
0
a
3
=
1
0
a
7
=
1
u
a
8
=
u 1
u
a
2
=
u + 1
u + 1
a
5
=
u
u
a
10
=
u + 1
1
a
1
=
u + 1
u + 1
a
4
=
u + 1
u
a
11
=
u + 1
1
a
11
=
u + 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
18
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
9
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0
3.28987 6.00000
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0
3.28987 6.00000
19
X. I
u
10
= hu
2
+ u + 1, a + u, b + 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
u
1
a
3
=
1
0
a
7
=
1
2u + 1
a
8
=
2u 2
2u + 1
a
2
=
2u + 2
3
a
5
=
u
u
a
10
=
2u 2
u 3
a
1
=
u 1
u
a
4
=
2u + 2
u 1
a
11
=
u 1
2
a
11
=
u 1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
20
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
10
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 1.00000
9.86960 9.00000
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 1.00000
9.86960 9.00000
21
XI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
u
2
(2 + u)
2
(u 1)(1 + u)
8
(u
2
u + 1)(u
2
+ u + 7)(10 + 2u + u
2
)
2
(u
4
2u
3
+ 2)
2
(u
7
4u
6
+ ··· + 120u 24)
c
2
, c
4
, c
7
c
9
u(u + 1)
2
(u
2
2)(u
2
+ 3)(u
2
u + 1)(u
2
u + 7)
2
(u
2
+ 3u + 5)
(u
3
u
2
1)
2
(u
4
+ u
2
1)
2
(u
7
+ u
6
+ ··· 2u + 2)
c
3
, c
5
(u + 1)(u
2
u + 1)
2
(u
2
u + 3)(u
2
+ u + 1)(u
2
+ 2u 1)
(u
4
+ 4u
3
+ 9u
2
+ 10u + 13)(u
6
u
5
2u
4
2u
3
2u
2
4u 1)
(u
7
+ u
6
+ 2u
5
3u
4
+ 5u
3
3u
2
+ 4u 1)
(u
8
+ 2u
7
+ 2u
6
+ 2u
5
+ 3u
4
+ 6u
3
+ 8u
2
+ 4u + 1)
c
6
, c
8
(u 1)(u
2
2u 1)(u
2
u + 1)(u
2
u + 3)(u
2
+ u + 1)
2
(u
4
+ 4u
3
+ 9u
2
+ 10u + 13)(u
6
u
5
2u
4
2u
3
2u
2
4u 1)
(u
7
+ u
6
+ 2u
5
3u
4
+ 5u
3
3u
2
+ 4u 1)
(u
8
2u
7
+ 2u
6
2u
5
+ 3u
4
6u
3
+ 8u
2
4u + 1)
c
10
u
2
(2 + u)
2
(1 + u)
2
(1 + u)
7
(u
2
u + 1)(u
2
u + 7)(10 + 2u + u
2
)
2
(u
4
+ 2u
3
+ 2)
2
(u
7
4u
6
+ ··· + 120u 24)
c
11
u
3
(u 1)
4
(u + 1)
6
(u
2
+ 3)(u
3
3u
2
+ 4u 1)
2
(u
4
2u
2
+ 5)
2
(u
7
+ 7u
6
+ 28u
5
+ 69u
4
+ 106u
3
+ 96u
2
+ 48u + 8)
22
XII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
10
y
2
(y 4)
2
(y 1)
9
(y
2
+ y + 1)(y
2
+ 13y + 49)(y
2
+ 16y + 100)
2
(y
4
4y
3
+ 4y
2
+ 4)
2
(y
7
+ 32y
6
+ 390y
5
+ 1996y
4
+ 2385y
3
7060y
2
+ 5088y 576)
c
2
, c
4
, c
7
c
9
y(y 2)
2
(y 1)
2
(y + 3)
2
(y
2
+ y 1)
4
(y
2
+ y + 1)(y
2
+ y + 25)
(y
2
+ 13y + 49)
2
(y
3
y
2
2y 1)
2
(y
7
+ 15y
6
+ 90y
5
+ 207y
4
+ 88y
3
144y
2
+ 44y 4)
c
3
, c
5
, c
6
c
8
(y 1)(y
2
6y + 1)(y
2
+ y + 1)
3
(y
2
+ 5y + 9)
(y
4
+ 2y
3
+ ··· + 134y + 169)(y
6
5y
5
+ ··· 12y + 1)
(y
7
+ 3y
6
+ 20y
5
+ 25y
4
+ 25y
3
+ 25y
2
+ 10y 1)
(y
8
+ 2y
6
+ 3y
4
+ 22y
2
+ 1)
c
11
y
3
(y 1)
10
(y + 3)
2
(y
2
2y + 5)
4
(y
3
y
2
+ 10y 1)
2
(y
7
+ 7y
6
+ 30y
5
73y
4
+ 564y
3
144y
2
+ 768y 64)
23