9
20
(K9a
19
)
1
Arc Sequences
6 9 7 2 1 8 4 3 5
Solving Sequence
3,7
4 8 9 2 5 6 1
c
3
c
7
c
8
c
2
c
4
c
6
c
1
c
5
, c
9
Representation Ideals
I = I
u
1
I
u
1
= hu
20
+ u
19
+ ··· 2u 1i
There are 1 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
20
+ u
19
5u
18
6u
17
+ 11u
16
+ 16u
15
10u
14
22u
13
2u
12
+
13u
11
+ 13u
10
+ 4u
9
9u
8
10u
7
+ 4u
5
+ 3u
4
+ u
3
u
2
2u 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
8
=
u
u
3
+ u
a
9
=
u
3
u
3
+ u
a
2
=
u
6
u
4
+ 1
u
6
2u
4
+ u
2
a
5
=
u
14
+ 3u
12
4u
10
+ u
8
+ 2u
6
2u
4
+ 1
u
14
+ 4u
12
7u
10
+ 6u
8
2u
6
+ u
2
a
6
=
u
3
u
5
u
3
+ u
a
1
=
u
14
+ 3u
12
4u
10
+ u
8
+ 2u
6
2u
4
+ 1
u
16
+ 4u
14
8u
12
+ 8u
10
4u
8
a
1
=
u
14
+ 3u
12
4u
10
+ u
8
+ 2u
6
2u
4
+ 1
u
16
+ 4u
14
8u
12
+ 8u
10
4u
8
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
19
24u
17
4u
16
+ 64u
15
+ 20u
14
84u
13
44u
12
+ 36u
11
+
44u
10
+ 44u
9
8u
8
60u
7
24u
6
+ 16u
5
+ 16u
4
+ 12u
3
8u 14
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.205802 0.505812I
10.02014 9.64430I 11.65468 + 6.20543I
u = 1.205802 + 0.505812I
10.02014 + 9.64430I 11.65468 6.20543I
u = 1.170971 0.421653I
4.43833 2.14390I 9.45592 + 0.24308I
u = 1.170971 + 0.421653I
4.43833 + 2.14390I 9.45592 0.24308I
u = 0.912041 0.514968I
2.98499 4.84109I 7.63163 + 6.37981I
u = 0.912041 + 0.514968I
2.98499 + 4.84109I 7.63163 6.37981I
u = 0.733657
0.976841 10.9388
u = 0.529602 0.535861I
1.94274 + 0.58469I 5.20205 0.00910I
u = 0.529602 + 0.535861I
1.94274 0.58469I 5.20205 + 0.00910I
u = 0.113113 0.821783I
6.78373 + 4.79919I 8.69810 3.09464I
u = 0.113113 + 0.821783I
6.78373 4.79919I 8.69810 + 3.09464I
u = 0.092790 0.716473I
0.91595 1.80448I 4.82463 + 3.70058I
u = 0.092790 + 0.716473I
0.91595 + 1.80448I 4.82463 3.70058I
u = 0.774874 0.460321I
1.25618 + 1.94645I 1.05320 4.81876I
u = 0.774874 + 0.460321I
1.25618 1.94645I 1.05320 + 4.81876I
u = 1.06181
6.53321 13.8998
u = 1.174856 0.481002I
4.01054 + 6.27316I 8.10015 6.54347I
u = 1.174856 + 0.481002I
4.01054 6.27316I 8.10015 + 6.54347I
u = 1.224932 0.393654I
10.81795 0.63661I 12.96035 0.16989I
u = 1.224932 + 0.393654I
10.81795 + 0.63661I 12.96035 + 0.16989I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
9
(u
20
+ u
19
+ ··· 2u 1)
c
2
, c
8
(u
20
+ 3u
19
+ ··· + 12u + 1)
c
3
, c
7
(u
20
+ u
19
+ ··· 2u 1)
c
4
(u
20
+ 3u
19
+ ··· 2u + 5)
c
6
(u
20
+ 11u
19
+ ··· + 2u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
9
(y
20
19y
19
+ ··· 2y + 1)
c
2
, c
8
(y
20
+ 17y
19
+ ··· 62y + 1)
c
3
, c
7
(y
20
11y
19
+ ··· 2y + 1)
c
4
(y
20
7y
19
+ ··· 274y + 25)
c
6
(y
20
3y
19
+ ··· 6y + 1)
5