10
30
(K10a
34
)
1
Arc Sequences
8 9 6 10 1 3 2 7 5 4
Solving Sequence
1,8
2 7 9 3 6 4 5 10
c
1
c
7
c
8
c
2
c
6
c
3
c
5
c
10
c
4
, c
9
Representation Ideals
I = I
u
1
I
u
1
= hu
33
u
32
+ ··· u + 1i
There are 1 irreducible components with 33 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
33
u
32
+ · · · u + 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
9
=
u
3
u
5
+ u
3
+ u
a
3
=
u
6
u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
6
=
u
11
2u
9
2u
7
+ u
3
u
13
+ 3u
11
+ 5u
9
+ 4u
7
+ 2u
5
+ u
3
+ u
a
4
=
u
16
3u
14
5u
12
4u
10
u
8
+ 1
u
18
+ 4u
16
+ 9u
14
+ 12u
12
+ 11u
10
+ 8u
8
+ 6u
6
+ 4u
4
+ u
2
a
5
=
u
13
+ 2u
11
+ 3u
9
+ 2u
7
+ 2u
5
+ 2u
3
+ u
u
13
+ 3u
11
+ 5u
9
+ 4u
7
+ 2u
5
+ u
3
+ u
a
10
=
u
31
6u
29
+ ··· 6u
5
2u
3
u
31
7u
29
+ ··· 4u
5
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
32
+ 4u
31
28u
30
+ 24u
29
104u
28
+ 84u
27
256u
26
+ 200u
25
460u
24
+ 364u
23
644u
22
+532u
21
744u
20
+652u
19
756u
18
+688u
17
696u
16
+632u
15
572u
14
+512u
13
404u
12
+ 364u
11
248u
10
+ 228u
9
140u
8
+ 120u
7
68u
6
+ 52u
5
24u
4
+ 20u
3
4u
2
2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.756564 0.531512I
9.92249 + 3.59396I 1.77642 3.03909I
u = 0.756564 + 0.531512I
9.92249 3.59396I 1.77642 + 3.03909I
u = 0.750033 0.440215I
3.38108 3.30675I 2.44424 + 3.71770I
u = 0.750033 + 0.440215I
3.38108 + 3.30675I 2.44424 3.71770I
u = 0.622996 1.042518I
8.40124 + 1.63491I 0.45903 2.05852I
u = 0.622996 + 1.042518I
8.40124 1.63491I 0.45903 + 2.05852I
u = 0.595580 1.086540I
1.46905 + 8.41845I 5.65597 8.08731I
u = 0.595580 + 1.086540I
1.46905 8.41845I 5.65597 + 8.08731I
u = 0.539171 0.794585I
4.95997 + 2.19825I 0.55384 3.61625I
u = 0.539171 + 0.794585I
4.95997 2.19825I 0.55384 + 3.61625I
u = 0.471931
1.00604 9.72742
u = 0.429591 1.067312I
3.60742 + 3.47782I 12.61515 4.95314I
u = 0.429591 + 1.067312I
3.60742 3.47782I 12.61515 + 4.95314I
u = 0.135018 1.027238I
1.43040 1.50384I 9.59059 + 3.60616I
u = 0.135018 + 1.027238I
1.43040 + 1.50384I 9.59059 3.60616I
u = 0.096071 1.100995I
4.19152 + 4.53523I 5.07914 3.09222I
u = 0.096071 + 1.100995I
4.19152 4.53523I 5.07914 + 3.09222I
u = 0.326389 0.884213I
0.54661 1.45331I 5.02647 + 4.36257I
u = 0.326389 + 0.884213I
0.54661 + 1.45331I 5.02647 4.36257I
u = 0.345723 1.064989I
0.390154 0.572456I 8.31906 + 0.48605I
u = 0.345723 + 1.064989I
0.390154 + 0.572456I 8.31906 0.48605I
u = 0.481406 1.089807I
0.50606 6.56196I 6.35976 + 7.19745I
u = 0.481406 + 1.089807I
0.50606 + 6.56196I 6.35976 7.19745I
u = 0.593279 1.056962I
1.99857 4.30723I 4.15179 + 2.03529I
u = 0.593279 + 1.056962I
1.99857 + 4.30723I 4.15179 2.03529I
u = 0.601136 0.188022I
2.96939 + 2.39560I 2.36922 3.31266I
u = 0.601136 + 0.188022I
2.96939 2.39560I 2.36922 + 3.31266I
u = 0.609745 1.098974I
7.42465 11.82876I 1.93037 + 7.75337I
u = 0.609745 + 1.098974I
7.42465 + 11.82876I 1.93037 7.75337I
u = 0.722135 0.492648I
3.67259 0.72831I 1.49015 + 3.12560I
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.722135 + 0.492648I
3.67259 + 0.72831I 1.49015 3.12560I
u = 0.789032 0.436494I
9.39642 + 6.56751I 1.02440 3.41838I
u = 0.789032 + 0.436494I
9.39642 6.56751I 1.02440 + 3.41838I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u
33
+ u
32
+ ··· u 1)
c
2
(u
33
+ u
32
+ ··· + u + 1)
c
3
, c
6
(u
33
+ 5u
32
+ ··· 31u 3)
c
4
, c
9
, c
10
(u
33
+ u
32
+ ··· + 3u + 1)
c
5
(u
33
+ u
32
+ ··· + 61u 17)
c
8
(u
33
+ 15u
32
+ ··· + u 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y
33
+ 15y
32
+ ··· + y 1)
c
2
(y
33
y
32
+ ··· + 33y 1)
c
3
, c
6
(y
33
+ 27y
32
+ ··· + y 9)
c
4
, c
9
, c
10
(y
33
+ 31y
32
+ ··· + y 1)
c
5
(y
33
+ 11y
32
+ ··· 3011y 289)
c
8
(y
33
+ 7y
32
+ ··· + 17y 1)
5