10
160
(K10n
33
)
1
Arc Sequences
7 9 7 2 8 1 3 5 4 2
Solving Sequence
3,9 2,5
4 10 1 8 6 7
c
2
c
4
c
9
c
10
c
8
c
5
c
7
c
1
, c
3
, c
6
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
6
3u
5
+ 6u
4
12u
3
+ 10u
2
10u + 1, u
5
+ 2u
4
3u
3
+ 9u
2
+ 5b 3u + 4,
13u
5
+ 38u
4
74u
3
+ 148u
2
+ 5a 115u + 111i
I
u
2
= hu
5
+ u
3
2u
2
1, a + 1, u
4
+ b 2ui
I
u
3
= hu
9
+ 6u
7
+ 3u
6
+ 14u
5
+ 10u
4
+ 13u
3
+ 7u
2
u 1, a + 1,
36u
8
22u
7
+ 219u
6
18u
5
+ 468u
4
+ 74u
3
+ 334u
2
+ 47b + 27u 76i
There are 3 irreducible components with 20 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
6
3u
5
+ 6u
4
12u
3
+ 10u
2
10u + 1, u
5
+ 2u
4
3u
3
+ 9u
2
+
5b 3u + 4, 13u
5
+ 38u
4
+ · · · + 5a + 111i
(i) Arc colorings
a
3
=
1
0
a
9
=
13
5
u
5
38
5
u
4
+ ··· + 23u
111
5
1
5
u
5
2
5
u
4
+ ··· +
3
5
u
4
5
a
2
=
8
5
u
5
24
5
u
4
+ ··· +
78
5
u
78
5
2
5
u
4
+ u
3
+ ··· +
11
5
u
4
5
a
5
=
0
u
a
4
=
13
5
u
5
+
38
5
u
4
+ ··· 23u +
121
5
1
a
10
=
12u
5
174
5
u
4
+ ··· +
522
5
u
523
5
4
5
u
5
12
5
u
4
+ ··· +
34
5
u
24
5
a
1
=
4u
5
57
5
u
4
+ ··· +
166
5
u
169
5
1
5
u
4
2
5
u
2
+
2
5
u
8
5
a
8
=
13
5
u
5
38
5
u
4
+ ··· + 23u
111
5
1
a
6
=
5.60000u
5
+ 15.8000u
4
+ ··· 45.6000u + 48.6000
1
5
u
5
+
4
5
u
4
+ ···
14
5
u +
13
5
a
7
=
13
5
u
5
38
5
u
4
+ ··· + 23u
116
5
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =
4
5
u
5
8
5
u
4
+
12
5
u
3
36
5
u
2
+
12
5
u
86
5
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.06205 1.68872I
a = 0.674823 0.211887I
b = 0.877439 0.744862I
3.55561 + 2.82812I 10.49024 2.97945I
u = 0.06205 + 1.68872I
a = 0.674823 + 0.211887I
b = 0.877439 + 0.744862I
3.55561 2.82812I 10.49024 + 2.97945I
u = 0.110715
a = 19.9974
b = 0.754878
7.69319 17.0195
u = 0.399691 1.126439I
a = 1.348884 0.423535I
b = 0.877439 + 0.744862I
3.55561 2.82812I 10.49024 + 2.97945I
u = 0.399691 + 1.126439I
a = 1.348884 + 0.423535I
b = 0.877439 0.744862I
3.55561 + 2.82812I 10.49024 2.97945I
u = 2.21400
a = 0.0500065
b = 0.754878
7.69319 17.0195
3
II. I
u
2
= hu
5
+ u
3
2u
2
1, a + 1, u
4
+ b 2ui
(i) Arc colorings
a
3
=
1
0
a
9
=
1
u
4
+ 2u
a
2
=
u
4
+ 2u + 1
u
4
u
3
+ u + 2
a
5
=
0
u
a
4
=
u
4
+ u
3
+ u
2
u 1
u
3
+ u 2
a
10
=
u
3
u
2
2u 1
1
a
1
=
u
3
u + 1
u
3
u + 1
a
8
=
1
u
4
u
2
+ 2u
a
6
=
u
u + 1
a
7
=
u
4
u
2
+ 2u 1
u
4
u
2
+ 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
4
+ u
3
2u 10
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.517256 1.205545I
a = 1.00000
b = 0.885210 + 0.546617I
1.44657 + 3.45949I 7.29654 5.67761I
u = 0.517256 + 1.205545I
a = 1.00000
b = 0.885210 0.546617I
1.44657 3.45949I 7.29654 + 5.67761I
u = 0.064504 0.703751I
a = 1.00000
b = 0.361950 1.318329I
1.57933 1.42206I 9.07660 + 1.47974I
u = 0.064504 + 0.703751I
a = 1.00000
b = 0.361950 + 1.318329I
1.57933 + 1.42206I 9.07660 1.47974I
u = 1.16352
a = 1.00000
b = 0.494320
6.84525 5.25373
5
III. I
u
3
= hu
9
+ 6u
7
+ · · · u 1, a + 1, 36u
8
22u
7
+ · · · + 47b 76i
(i) Arc colorings
a
3
=
1
0
a
9
=
1
0.765957u
8
+ 0.468085u
7
+ ··· 0.574468u + 1.61702
a
2
=
0.765957u
8
+ 0.468085u
7
+ ··· 0.574468u + 2.61702
0.0425532u
8
+ 0.0851064u
7
+ ··· 1.46809u + 0.0212766
a
5
=
0
u
a
4
=
0.851064u
8
0.297872u
7
+ ··· + 2.63830u 1.57447
0.0851064u
8
+ 0.170213u
7
+ ··· + 2.06383u 0.957447
a
10
=
0.106383u
8
+ 0.212766u
7
+ ··· 1.17021u 1.44681
0.127660u
8
+ 0.744681u
7
+ ··· + 0.404255u 0.0638298
a
1
=
0.808511u
8
0.382979u
7
+ ··· 0.893617u 2.59574
0.468085u
8
0.0638298u
7
+ ··· + 1.85106u 0.765957
a
8
=
1
0.765957u
8
+ 0.468085u
7
+ ··· 0.574468u + 1.61702
a
6
=
u
0.468085u
8
+ 0.0638298u
7
+ ··· + 0.148936u + 0.765957
a
7
=
0.765957u
8
+ 0.468085u
7
+ ··· 0.574468u + 0.617021
0.765957u
8
+ 0.468085u
7
+ ··· 0.574468u + 1.61702
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
213
47
u
8
+
44
47
u
7
1284
47
u
6
340
47
u
5
2910
47
u
4
1370
47
u
3
2454
47
u
2
853
47
u +
58
47
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.470631 0.112677I
a = 1.00000
b = 0.204797 1.087895I
2.64060 + 1.65275I 0.59079 4.28210I
u = 0.470631 + 0.112677I
a = 1.00000
b = 0.204797 + 1.087895I
2.64060 1.65275I 0.59079 + 4.28210I
u = 0.406410 1.188169I
a = 1.00000
b = 0.647333 + 0.135453I
0.87559 + 2.35950I 4.89060 1.18144I
u = 0.406410 + 1.188169I
a = 1.00000
b = 0.647333 0.135453I
0.87559 2.35950I 4.89060 + 1.18144I
u = 0.05862 1.56131I
a = 1.00000
b = 1.12278 + 1.21739I
13.58819 0.68871I 9.49454 + 0.10018I
u = 0.05862 + 1.56131I
a = 1.00000
b = 1.12278 1.21739I
13.58819 + 0.68871I 9.49454 0.10018I
u = 0.322071
a = 1.00000
b = 0.531326
0.846327 11.7230
u = 0.65739 1.73551I
a = 1.00000
b = 1.20035 1.05816I
14.0726 9.2039I 9.16258 + 4.28229I
u = 0.65739 + 1.73551I
a = 1.00000
b = 1.20035 + 1.05816I
14.0726 + 9.2039I 9.16258 4.28229I
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u
5
+ u
4
2u
3
u
2
+ u + 1)(u
6
+ u
5
4u
4
4u
3
+ 4u
2
+ 8u 7)
(u
9
u
8
7u
7
+ 7u
6
+ 13u
5
13u
4
+ 2u
3
+ u
2
+ 1)
c
2
(u
3
u
2
+ 1)
2
(u
5
+ 2u
4
+ 3u
3
+ 2u
2
1)
(u
9
+ 5u
8
+ 13u
7
+ 20u
6
+ 22u
5
+ 18u
4
+ 11u
3
5u 2)
c
3
(u 1)
6
(u
5
+ 2u
3
+ u
2
+ 1)
(u
9
+ 7u
8
+ 26u
7
+ 61u
6
+ 103u
5
+ 129u
4
+ 125u
3
+ 86u
2
+ 40u + 8)
c
5
(u
5
+ u
3
2u
2
1)(u
6
+ 3u
5
+ 6u
4
+ 12u
3
+ 10u
2
+ 10u + 1)
(u
9
+ 6u
7
3u
6
+ 14u
5
10u
4
+ 13u
3
7u
2
u + 1)
c
6
(u
5
u
4
2u
3
+ u
2
+ u 1)(u
6
+ u
5
4u
4
4u
3
+ 4u
2
+ 8u 7)
(u
9
u
8
7u
7
+ 7u
6
+ 13u
5
13u
4
+ 2u
3
+ u
2
+ 1)
c
7
(u 1)
6
(u
5
+ 2u
3
u
2
1)
(u
9
+ 7u
8
+ 26u
7
+ 61u
6
+ 103u
5
+ 129u
4
+ 125u
3
+ 86u
2
+ 40u + 8)
c
8
(u
5
+ u
3
+ 2u
2
+ 1)(u
6
+ 3u
5
+ 6u
4
+ 12u
3
+ 10u
2
+ 10u + 1)
(u
9
+ 6u
7
3u
6
+ 14u
5
10u
4
+ 13u
3
7u
2
u + 1)
c
9
(u
5
2u
3
+ 3u
2
2u + 1)(u
6
+ u
5
6u
4
+ 8u
3
+ 2u
2
6u 11)
(u
9
11u
7
+ 2u
6
+ 35u
5
32u
4
+ 47u
3
+ 8u
2
+ u + 13)
c
10
(u
5
5u
4
+ 8u
3
7u
2
+ 3u 1)
(u
6
+ 9u
5
+ 32u
4
+ 78u
3
+ 136u
2
+ 120u + 49)
(u
9
+ 15u
8
+ 89u
7
+ 253u
6
+ 325u
5
+ 129u
4
+ 16u
3
25u
2
2u + 1)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
6
(y
5
5y
4
+ 8y
3
7y
2
+ 3y 1)
(y
6
9y
5
+ 32y
4
78y
3
+ 136y
2
120y + 49)
(y
9
15y
8
+ 89y
7
253y
6
+ 325y
5
129y
4
+ 16y
3
+ 25y
2
2y 1)
c
2
(y
3
y
2
+ 2y 1)
2
(y
5
+ 2y
4
+ y
3
+ 4y 1)
(y
9
+ y
8
+ 13y
7
+ 14y
6
+ 40y
5
+ 50y
4
19y
3
38y
2
+ 25y 4)
c
3
, c
7
(y 1)
6
(y
5
+ 4y
4
+ ··· 2y 1)(y
9
+ 3y
8
+ ··· + 224y 64)
c
5
, c
8
(y
5
+ 2y
4
+ y
3
4y
2
4y 1)
(y
6
+ 3y
5
16y
4
82y
3
128y
2
80y + 1)
(y
9
+ 12y
8
+ 64y
7
+ 185y
6
+ 290y
5
+ 210y
4
+ 7y
3
55y
2
+ 15y 1)
c
9
(y
5
4y
4
y
2
2y 1)(y
6
13y
5
+ ··· 80y + 121)
(y
9
22y
8
+ ··· 207y 169)
c
10
(y
5
9y
4
11y
2
5y 1)
(y
6
17y
5
108y
4
+ 558y
3
+ 2912y
2
1072y + 2401)
(y
9
47y
8
+ ··· + 54y 1)
9