10
16
(K10a
115
)
1
Arc Sequences
6 8 9 10 2 1 5 4 3 7
Solving Sequence
3,10
9 4 5 8 2 6 1 7
c
9
c
3
c
4
c
8
c
2
c
5
c
1
c
7
c
6
, c
10
Representation Ideals
I = I
u
1
I
u
1
= hu
23
+ u
22
+ ··· + 2u + 1i
There are 1 irreducible components with 23 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
23
+ u
22
+ · · · + 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
9
=
u
u
a
4
=
u
2
+ 1
u
2
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
u
3
+ 2u
u
3
+ u
a
2
=
u
6
3u
4
2u
2
+ 1
u
6
2u
4
u
2
a
6
=
u
16
+ 7u
14
+ 19u
12
+ 22u
10
+ 3u
8
14u
6
6u
4
+ 4u
2
+ 1
u
16
+ 6u
14
+ 14u
12
+ 14u
10
+ 2u
8
6u
6
2u
4
+ 2u
2
a
1
=
u
19
+ 8u
17
+ 26u
15
+ 40u
13
+ 19u
11
24u
9
30u
7
+ 9u
3
u
21
+ 9u
19
+ ··· + 3u
3
+ u
a
7
=
u
9
4u
7
5u
5
+ 3u
u
11
5u
9
8u
7
3u
5
+ 3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
22
4u
21
36u
20
32u
19
132u
18
104u
17
236u
16
164u
15
160u
14
100u
13
+
108u
12
+ 44u
11
+ 224u
10
+ 84u
9
+ 48u
8
+ 28u
7
80u
6
+ 8u
5
24u
4
+ 8u
2
16u 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.833290 0.100941I
12.38016 5.22748I 9.66631 + 3.33432I
u = 0.833290 + 0.100941I
12.38016 + 5.22748I 9.66631 3.33432I
u = 0.717171
2.00773 4.01170
u = 0.385250 1.158797I
9.14246 + 0.83337I 6.62647 + 0.43888I
u = 0.385250 + 1.158797I
9.14246 0.83337I 6.62647 0.43888I
u = 0.367126 1.335770I
7.87123 9.54664I 5.28748 + 5.57899I
u = 0.367126 + 1.335770I
7.87123 + 9.54664I 5.28748 5.57899I
u = 0.298297 1.284851I
2.00599 3.66457I 0.82434 + 2.67133I
u = 0.298297 + 1.284851I
2.00599 + 3.66457I 0.82434 2.67133I
u = 0.223948 0.344528I
0.140168 0.925919I 2.94249 + 7.44214I
u = 0.223948 + 0.344528I
0.140168 + 0.925919I 2.94249 7.44214I
u = 0.039073 1.333054I
4.96840 1.68040I 2.82272 + 4.29991I
u = 0.039073 + 1.333054I
4.96840 + 1.68040I 2.82272 4.29991I
u = 0.116711 1.367359I
1.46467 + 3.53591I 1.36507 3.24061I
u = 0.116711 + 1.367359I
1.46467 3.53591I 1.36507 + 3.24061I
u = 0.313551 1.193895I
0.817157 + 0.745308I 5.08009 + 0.73522I
u = 0.313551 + 1.193895I
0.817157 0.745308I 5.08009 0.73522I
u = 0.337825 1.317982I
0.16340 + 7.25342I 3.09734 7.25802I
u = 0.337825 + 1.317982I
0.16340 7.25342I 3.09734 + 7.25802I
u = 0.458650 0.443050I
7.11725 + 1.68405I 6.35516 3.83025I
u = 0.458650 + 0.443050I
7.11725 1.68405I 6.35516 + 3.83025I
u = 0.778833 0.078232I
4.21185 + 3.22031I 8.22079 4.90443I
u = 0.778833 + 0.078232I
4.21185 3.22031I 8.22079 + 4.90443I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
c
10
(u
23
+ u
22
+ ··· 2u 1)
c
2
, c
4
(u
23
+ u
22
+ ··· + 4u 5)
c
3
, c
8
, c
9
(u
23
+ u
22
+ ··· + 2u + 1)
c
7
(u
23
+ 7u
22
+ ··· + 40u + 17)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
c
10
(y
23
+ 27y
22
+ ··· 4y 1)
c
2
, c
4
(y
23
17y
22
+ ··· 144y 25)
c
3
, c
8
, c
9
(y
23
+ 19y
22
+ ··· 4y 1)
c
7
(y
23
9y
22
+ ··· + 1260y 289)
5