10
146
(K10n
23
)
1
Arc Sequences
7 5 7 9 2 4 1 6 3 2
Solving Sequence
2,7
1
4,8
3 6 5 10 9
c
1
c
7
c
3
c
6
c
5
c
10
c
9
c
2
, c
4
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
4
+ 4u
3
+ 5u
2
+ 2u + 1, u
2
+ b + 2u, u
3
3u
2
+ a 2u + 1i
I
u
2
= hu
10
9u
9
+ 37u
8
89u
7
+ 137u
6
143u
5
+ 122u
4
124u
3
+ 136u
2
96u + 32,
2u
8
11u
7
+ 27u
6
35u
5
+ 27u
4
19u
3
+ 31u
2
+ 4b 30u + 12,
5u
9
+ 37u
8
121u
7
+ 229u
6
277u
5
+ 243u
4
218u
3
+ 252u
2
+ 32a 208u + 112i
I
u
3
= ha
10
+ a
9
+ 2a
8
+ 2a
7
+ 4a
6
+ 4a
5
+ 9a
4
+ 7a
3
+ 8a
2
+ 4a + 1, u + 1,
35a
9
+ 26a
8
+ 66a
7
+ 49a
6
+ 118a
5
+ 111a
4
+ 269a
3
+ 153a
2
+ 47b + 195a + 63i
There are 3 irreducible components with 24 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
4
+ 4u
3
+ 5u
2
+ 2u + 1, u
2
+ b + 2u, u
3
3u
2
+ a 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
u
3
+ 3u
2
+ 2u 1
u
2
2u
a
1
=
u
3
+ 4u
2
+ 5u + 2
u
2
2u 1
a
4
=
0
u
a
8
=
u
3
+ 3u
2
+ 3u + 1
u
2
2u 1
a
3
=
u
3
+ 4u
2
+ 5u + 2
1
a
6
=
u
3
+ 3u
2
+ 2u 1
u
3
3u
2
3u 1
a
5
=
u 2
u
3
3u
2
3u 1
a
10
=
u
3
+ 3u
2
+ 3u + 1
u
2
2u 1
a
9
=
u
2
3u 2
2u
2
3u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 8u + 8
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.86603 0.50000I
a = 0.133975 0.500000I
b = 0.500000 0.866025I
3.28987 2.02988I 6.00000 + 3.46410I
u = 1.86603 + 0.50000I
a = 0.133975 + 0.500000I
b = 0.500000 + 0.866025I
3.28987 + 2.02988I 6.00000 3.46410I
u = 0.133975 0.500000I
a = 1.86603 0.50000I
b = 0.500000 + 0.866025I
3.28987 + 2.02988I 6.00000 3.46410I
u = 0.133975 + 0.500000I
a = 1.86603 + 0.50000I
b = 0.500000 0.866025I
3.28987 2.02988I 6.00000 + 3.46410I
3
II. I
u
2
= hu
10
9u
9
+ · · · 96u + 32, 2u
8
11u
7
+ · · · + 4b + 12, 5u
9
+
37u
8
+ · · · + 32a + 112i
(i) Arc colorings
a
2
=
1
0
a
7
=
5
32
u
9
37
32
u
8
+ ··· +
13
2
u
7
2
1
2
u
8
+
11
4
u
7
+ ··· +
15
2
u 3
a
1
=
1
32
u
9
7
32
u
8
+ ···
11
8
u
2
+
3
2
u
1
16
u
9
7
16
u
8
+ ··· + 3u 1
a
4
=
0
u
a
8
=
3
8
u
9
+
41
16
u
8
+ ···
47
4
u +
7
2
11
16
u
9
69
16
u
8
+ ··· +
23
2
u 2
a
3
=
1
32
u
9
7
32
u
8
+ ···
11
8
u
2
+
3
2
u
1
16
u
9
7
16
u
8
+ ··· + 3u 1
a
6
=
5
32
u
9
37
32
u
8
+ ··· +
13
2
u
7
2
1
4
u
9
+ 2u
8
+ ···
23
2
u + 5
a
5
=
3
32
u
9
+
27
32
u
8
+ ··· 5u +
3
2
1
4
u
9
+ 2u
8
+ ···
23
2
u + 5
a
10
=
3
32
u
9
21
32
u
8
+ ··· +
9
2
u 1
1
16
u
9
7
16
u
8
+ ··· + 3u 1
a
9
=
3
16
u
9
25
16
u
8
+ ··· + 13u 5
3
8
u
8
+
17
8
u
7
+ ··· + 8u 4
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1
4
u
9
9
4
u
8
+
33
4
u
7
69
4
u
6
+
93
4
u
5
91
4
u
4
+
39
2
u
3
20u
2
+ 20u 14
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.625089 0.778917I
a = 0.337181 + 0.501974I
b = 0.741866 + 0.796341I
1.60483 1.51336I 1.256588 + 0.171947I
u = 0.625089 + 0.778917I
a = 0.337181 0.501974I
b = 0.741866 0.796341I
1.60483 + 1.51336I 1.256588 0.171947I
u = 0.722559 0.567039I
a = 0.875847 0.108998I
b = 0.429682 + 0.277960I
1.23090 + 1.07704I 4.33290 2.58024I
u = 0.722559 + 0.567039I
a = 0.875847 + 0.108998I
b = 0.429682 0.277960I
1.23090 1.07704I 4.33290 + 2.58024I
u = 0.94514 1.33248I
a = 0.599946 + 0.015806I
b = 1.077562 0.740596I
1.74604 + 4.90489I 2.53483 7.39457I
u = 0.94514 + 1.33248I
a = 0.599946 0.015806I
b = 1.077562 + 0.740596I
1.74604 4.90489I 2.53483 + 7.39457I
u = 1.66770 0.84950I
a = 0.715358 + 0.272045I
b = 0.25937 + 1.52583I
7.19127 + 3.97850I 1.38540 2.06163I
u = 1.66770 + 0.84950I
a = 0.715358 0.272045I
b = 0.25937 1.52583I
7.19127 3.97850I 1.38540 + 2.06163I
u = 1.78968 0.93001I
a = 0.654078 0.303601I
b = 0.36963 1.73551I
6.44324 + 10.56102I 0.07312 6.56398I
u = 1.78968 + 0.93001I
a = 0.654078 + 0.303601I
b = 0.36963 + 1.73551I
6.44324 10.56102I 0.07312 + 6.56398I
5
III. I
u
3
= ha
10
+ a
9
+ · · · + 4a + 1, u + 1, 35a
9
+ 47b + · · · + 195a + 63i
(i) Arc colorings
a
2
=
1
0
a
7
=
a
0.744681a
9
0.553191a
8
+ ··· 4.14894a 1.34043
a
1
=
0.191489a
9
+ 0.0851064a
8
+ ··· + 1.63830a + 1.74468
0.191489a
9
+ 0.0851064a
8
+ ··· + 1.63830a 0.255319
a
4
=
0
1
a
8
=
0.851064a
9
0.489362a
8
+ ··· 4.17021a 1.53191
a
3
a
a
3
=
a
2
0.191489a
9
0.0851064a
8
+ ··· 1.63830a 1.74468
a
6
=
a
0.744681a
9
0.553191a
8
+ ··· 5.14894a 1.34043
a
5
=
0.744681a
9
0.553191a
8
+ ··· 4.14894a 1.34043
0.744681a
9
0.553191a
8
+ ··· 5.14894a 1.34043
a
10
=
0.382979a
9
+ 0.170213a
8
+ ··· + 3.27660a + 1.48936
0.191489a
9
+ 0.0851064a
8
+ ··· + 1.63830a 0.255319
a
9
=
0.723404a
9
+ 0.765957a
8
+ ··· + 3.74468a + 1.70213
0.978723a
9
+ 0.212766a
8
+ ··· + 5.59574a + 1.36170
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
240
47
a
9
232
47
a
8
372
47
a
7
336
47
a
6
836
47
a
5
788
47
a
4
1764
47
a
3
1264
47
a
2
1176
47
a
338
47
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.100317 0.776746I
b = 0.05818 1.69128I
7.66842 + 3.33174I 1.91874 2.36228I
u = 1.00000
a = 1.100317 + 0.776746I
b = 0.05818 + 1.69128I
7.66842 3.33174I 1.91874 + 2.36228I
u = 1.00000
a = 0.336131 0.264954I
b = 0.233677 + 0.885557I
1.47006 2.21397I 0.88568 + 4.22289I
u = 1.00000
a = 0.336131 + 0.264954I
b = 0.233677 0.885557I
1.47006 + 2.21397I 0.88568 4.22289I
u = 1.00000
a = 0.208142 1.063000I
b = 0.416284
4.17205 7.60884
u = 1.00000
a = 0.208142 + 1.063000I
b = 0.416284
4.17205 7.60884
u = 1.00000
a = 0.102454 1.150511I
b = 0.233677 0.885557I
1.47006 + 2.21397I 0.88568 4.22289I
u = 1.00000
a = 0.102454 + 1.150511I
b = 0.233677 + 0.885557I
1.47006 2.21397I 0.88568 + 4.22289I
u = 1.00000
a = 1.042136 0.914533I
b = 0.05818 1.69128I
7.66842 + 3.33174I 1.91874 2.36228I
u = 1.00000
a = 1.042136 + 0.914533I
b = 0.05818 + 1.69128I
7.66842 3.33174I 1.91874 + 2.36228I
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)
2
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
2
(u
10
+ 3u
9
+ 9u
8
+ 16u
7
+ 24u
6
+ 25u
5
+ 21u
4
+ 4u
3
3u
2
3u + 4)
c
2
, c
3
, c
5
c
6
(u
2
+ 1)
2
(u
10
+ u
8
u
7
+ 5u
6
+ 2u
4
3u
3
+ 2u
2
+ 1)
(u
10
+ u
9
+ 2u
8
+ 2u
7
+ 4u
6
+ 4u
5
+ 9u
4
+ 7u
3
+ 8u
2
+ 4u + 1)
c
4
(u
4
u
2
+ 1)(u
5
u
4
+ u
2
+ u 1)
2
(u
10
+ 3u
9
+ 3u
8
2u
7
6u
6
3u
5
+ 3u
4
+ 4u
3
+ 3u
2
+ 3u + 2)
c
7
(u
2
+ u + 1)
2
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
2
(u
10
+ 3u
9
+ 9u
8
+ 16u
7
+ 24u
6
+ 25u
5
+ 21u
4
+ 4u
3
3u
2
3u + 4)
c
8
(u
4
2u
3
+ 2u
2
4u + 4)
(u
10
+ u
9
+ 2u
8
2u
7
+ 6u
6
+ 10u
5
+ 11u
4
+ 27u
3
+ 6u
2
+ 10u + 29)
(u
10
+ 2u
9
+ 9u
8
+ 7u
7
+ 30u
6
+ 6u
5
+ 41u
4
+ 22u
2
+ 4)
c
9
(u
4
+ 2u
3
+ 2u
2
+ 4u + 4)
(u
10
+ u
9
+ 2u
8
2u
7
+ 6u
6
+ 10u
5
+ 11u
4
+ 27u
3
+ 6u
2
+ 10u + 29)
(u
10
+ 2u
9
+ 9u
8
+ 7u
7
+ 30u
6
+ 6u
5
+ 41u
4
+ 22u
2
+ 4)
c
10
(u
2
u + 1)
2
(u
5
+ 7u
4
+ 16u
3
+ 13u
2
+ 3u 1)
2
(u
10
+ 9u
9
+ ··· 33u + 16)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y
2
+ y + 1)
2
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y 1)
2
(y
10
+ 9y
9
+ ··· 33y + 16)
c
2
, c
3
, c
5
c
6
(y + 1)
4
(y
10
+ 2y
9
+ 11y
8
+ 13y
7
+ 33y
6
+ 20y
5
+ 26y
4
+ 9y
3
+ 8y
2
+ 4y + 1)
(y
10
+ 3y
9
+ 8y
8
+ 22y
7
+ 38y
6
+ 54y
5
+ 77y
4
+ 71y
3
+ 26y
2
+ 1)
c
4
(y
2
y + 1)
2
(y
5
y
4
+ 4y
3
3y
2
+ 3y 1)
2
(y
10
3y
9
+ 9y
8
16y
7
+ 24y
6
25y
5
+ 21y
4
4y
3
3y
2
+ 3y + 4)
c
8
, c
9
(y
4
4y
2
+ 16)(y
10
+ 3y
9
+ ··· + 248y + 841)
(y
10
+ 14y
9
+ ··· + 176y + 16)
c
10
(y
2
+ y + 1)
2
(y
5
17y
4
+ 80y
3
59y
2
+ 35y 1)
2
(y
10
15y
9
+ ··· + 5343y + 256)
9