10
45
(K10a
25
)
1
Arc Sequences
6 1 10 9 2 3 4 5 8 7
Solving Sequence
4,9
5 8 10 3 7 1 2 6
c
4
c
8
c
9
c
3
c
7
c
10
c
2
c
6
c
1
, c
5
Representation Ideals
I = I
u
1
I
u
1
= hu
44
+ u
43
+ ··· + 2u + 1i
There are 1 irreducible components with 44 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
44
+ u
43
+ · · · + 2u + 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
5
=
1
u
2
a
8
=
u
u
3
+ u
a
10
=
u
3
u
5
+ u
3
+ u
a
3
=
u
8
+ u
6
+ u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
7
=
u
3
u
3
+ u
a
1
=
u
11
2u
9
2u
7
+ u
3
u
11
+ 3u
9
+ 4u
7
+ 3u
5
+ u
3
+ u
a
2
=
u
32
+ 7u
30
+ ··· + 2u
12
+ 1
u
32
8u
30
+ ··· 12u
8
4u
6
a
6
=
u
21
+ 4u
19
+ 9u
17
+ 12u
15
+ 12u
13
+ 10u
11
+ 9u
9
+ 6u
7
+ 3u
5
+ u
u
23
+ 5u
21
+ ··· + 2u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
43
4u
42
+ ··· 12u 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.764138 0.339961I
2.30812 + 8.68200I 3.24304 6.31705I
u = 0.764138 + 0.339961I
2.30812 8.68200I 3.24304 + 6.31705I
u = 0.721497 0.387567I
4.35508 + 1.04298I 6.65567 0.28795I
u = 0.721497 + 0.387567I
4.35508 1.04298I 6.65567 + 0.28795I
u = 0.633616 0.150714I
0.14884 2.53826I 0.24501 + 3.05915I
u = 0.633616 + 0.150714I
0.14884 + 2.53826I 0.24501 3.05915I
u = 0.603028 0.598100I
1.27162 1.94114I 1.79245 + 3.31415I
u = 0.603028 + 0.598100I
1.27162 + 1.94114I 1.79245 3.31415I
u = 0.570711 1.124896I
13.7161I 10.0128I
u = 0.570711 + 1.124896I
13.7161I 10.0128I
u = 0.568139 1.097595I
2.27286 5.97235I 3.25101 + 4.61402I
u = 0.568139 + 1.097595I
2.27286 + 5.97235I 3.25101 4.61402I
u = 0.525013 0.980467I
0.14884 2.53826I 0.24501 + 3.05915I
u = 0.525013 + 0.980467I
0.14884 + 2.53826I 0.24501 3.05915I
u = 0.494931 1.113463I
2.75664 1.75570I 3.52773 + 0.85914I
u = 0.494931 + 1.113463I
2.75664 + 1.75570I 3.52773 0.85914I
u = 0.358424 1.122989I
3.67003 5.88530I 4.74516 + 6.36553I
u = 0.358424 + 1.122989I
3.67003 + 5.88530I 4.74516 6.36553I
u = 0.326591 0.684448I
1.50871I 4.89247I
u = 0.326591 + 0.684448I
1.50871I 4.89247I
u = 0.228645 1.128367I
2.27286 + 5.97235I 3.25101 4.61402I
u = 0.228645 + 1.128367I
2.27286 5.97235I 3.25101 + 4.61402I
u = 0.219635 1.024162I
1.08070I 1.29853I
u = 0.219635 + 1.024162I
1.08070I 1.29853I
u = 0.254705 1.115021I
4.35508 1.04298I 6.65567 + 0.28795I
u = 0.254705 + 1.115021I
4.35508 + 1.04298I 6.65567 0.28795I
u = 0.323792 1.114602I
5.06451 + 1.08737I 7.53766 0.51091I
u = 0.323792 + 1.114602I
5.06451 1.08737I 7.53766 + 0.51091I
u = 0.523186 1.116827I
3.71248 + 6.51845I 4.95829 6.88419I
u = 0.523186 + 1.116827I
3.71248 6.51845I 4.95829 + 6.88419I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.560153 1.120389I
2.30812 + 8.68200I 3.24304 6.31705I
u = 0.560153 + 1.120389I
2.30812 8.68200I 3.24304 + 6.31705I
u = 0.567171 0.946930I
2.75664 1.75570I 3.52773 + 0.85914I
u = 0.567171 + 0.946930I
2.75664 + 1.75570I 3.52773 0.85914I
u = 0.570170 1.011785I
3.67003 + 5.88530I 4.74516 6.36553I
u = 0.570170 + 1.011785I
3.67003 5.88530I 4.74516 + 6.36553I
u = 0.651337 0.622116I
3.71248 + 6.51845I 4.95829 6.88419I
u = 0.651337 + 0.622116I
3.71248 6.51845I 4.95829 + 6.88419I
u = 0.662694 0.538070I
5.06451 1.08737I 7.53766 + 0.51091I
u = 0.662694 + 0.538070I
5.06451 + 1.08737I 7.53766 0.51091I
u = 0.663842 0.251119I
1.27162 1.94114I 1.79245 + 3.31415I
u = 0.663842 + 0.251119I
1.27162 + 1.94114I 1.79245 3.31415I
u = 0.737315 0.329710I
3.75579I 2.66459I
u = 0.737315 + 0.329710I
3.75579I 2.66459I
4
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
, c
5
c
8
(u
44
+ u
43
+ ··· + 2u + 1)
c
2
, c
9
(u
44
+ 21u
43
+ ··· + 2u + 1)
c
3
, c
10
(u
44
+ 5u
43
+ ··· + 82u + 13)
c
6
, c
7
(u
44
+ u
43
+ ··· + 68u + 17)
5
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
5
c
8
(y
44
+ 21y
43
+ ··· + 2y + 1)
c
2
, c
9
(y
44
+ 5y
43
+ ··· + 6y + 1)
c
3
, c
10
(y
44
+ 9y
43
+ ··· + 5314y + 169)
c
6
, c
7
(y
44
11y
43
+ ··· 4794y + 289)
6