10
34
(K10a
19
)
1
Arc Sequences
4 8 5 2 1 10 9 3 7 6
Solving Sequence
2,8
3 9 7 10 6 1 5 4
c
2
c
8
c
7
c
9
c
6
c
10
c
5
c
3
c
1
, c
4
Representation Ideals
I = I
u
1
I
u
1
= hu
18
u
17
+ ··· u + 1i
There are 1 irreducible components with 18 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
18
u
17
u
16
+ 2u
15
+ 6u
14
7u
13
5u
12
+ 10u
11
+ 11u
10
15u
9
7u
8
+ 14u
7
+ 6u
6
10u
5
2u
4
+ 6u
3
u
2
u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u
2
a
9
=
u
u
3
+ u
a
7
=
u
3
u
5
u
3
+ u
a
10
=
u
5
+ u
u
7
+ u
5
2u
3
+ u
a
6
=
u
7
2u
3
u
9
u
7
+ 3u
5
2u
3
+ u
a
1
=
u
9
+ 3u
5
+ u
u
11
+ u
9
4u
7
+ 3u
5
3u
3
+ u
a
5
=
u
11
4u
7
3u
3
u
13
u
11
+ 5u
9
4u
7
+ 6u
5
3u
3
+ u
a
4
=
u
13
2u
11
+ 5u
9
8u
7
+ 6u
5
6u
3
+ u
u
13
u
11
+ 5u
9
4u
7
+ 6u
5
3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
16
+ 4u
15
+ 4u
14
8u
13
24u
12
+ 24u
11
+ 20u
10
36u
9
44u
8
+ 40u
7
+ 28u
6
40u
5
24u
4
+ 16u
3
+ 8u
2
12u + 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.944193 0.919683I
12.31671 + 3.38380I 0.20360 2.27447I
u = 0.944193 + 0.919683I
12.31671 3.38380I 0.20360 + 2.27447I
u = 0.885264 0.680601I
5.44176 + 6.61296I 1.60438 7.00860I
u = 0.885264 + 0.680601I
5.44176 6.61296I 1.60438 + 7.00860I
u = 0.727312 0.096422I
1.138658 + 0.137643I 9.21435 0.51404I
u = 0.727312 + 0.096422I
1.138658 0.137643I 9.21435 + 0.51404I
u = 0.718345 0.757260I
5.99819 1.29789I 3.32252 + 0.68135I
u = 0.718345 + 0.757260I
5.99819 + 1.29789I 3.32252 0.68135I
u = 0.275451 0.493368I
1.67574 + 0.60080I 4.05524 0.52802I
u = 0.275451 + 0.493368I
1.67574 0.60080I 4.05524 + 0.52802I
u = 0.784251 0.644550I
2.41237 2.42038I 1.45127 + 3.59982I
u = 0.784251 + 0.644550I
2.41237 + 2.42038I 1.45127 3.59982I
u = 0.816176 0.315615I
0.00395 3.50386I 4.01768 + 8.20647I
u = 0.816176 + 0.315615I
0.00395 + 3.50386I 4.01768 8.20647I
u = 0.932919 0.939980I
16.3133 + 1.5857I 3.06627 0.65832I
u = 0.932919 + 0.939980I
16.3133 1.5857I 3.06627 + 0.65832I
u = 0.966316 0.920631I
16.2022 8.4223I 2.83851 + 5.16445I
u = 0.966316 + 0.920631I
16.2022 + 8.4223I 2.83851 5.16445I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u
18
+ u
17
+ ··· + 3u + 1)
c
2
, c
8
(u
18
+ u
17
+ ··· + u + 1)
c
3
(u
18
+ 11u
17
+ ··· + 3u + 1)
c
5
, c
6
, c
7
c
9
, c
10
(u
18
+ 3u
17
+ ··· + 3u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
18
11y
17
+ ··· 3y + 1)
c
2
, c
8
(y
18
3y
17
+ ··· 3y + 1)
c
3
(y
18
7y
17
+ ··· + y + 1)
c
5
, c
6
, c
7
c
9
, c
10
(y
18
+ 25y
17
+ ··· + 9y + 1)
5