10
54
(K10a
48
)
1
Arc Sequences
8 5 6 10 3 9 1 7 4 2
Solving Sequence
2,5
3
6,8
1 7 10 4 9
c
2
c
5
c
1
c
7
c
10
c
4
c
9
c
3
, c
6
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
3
b
2
+ 1, u + 1, b
2
+ ai
I
u
2
= hu
26
4u
25
+ ··· + u 1, u
25
+ u
24
+ ··· + 4a + 9, 2u
25
5u
24
+ ··· + 2b + 1i
There are 2 irreducible components with 29 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
= hb
3
b
2
+ 1, u + 1, b
2
+ ai
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
6
=
1
0
a
8
=
b
2
b
a
1
=
b
2
b
2
a
7
=
1
b
2
+ b + 1
a
10
=
0
b
2
a
4
=
0
1
a
9
=
0
b
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2b
2
+ 7b + 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.569840
b = 0.754878
0.531480 4.42382
u = 1.00000
a = 0.215080 + 1.307141I
b = 0.877439 0.744862I
4.66906 + 2.82812I 7.71191 2.59975I
u = 1.00000
a = 0.215080 1.307141I
b = 0.877439 + 0.744862I
4.66906 2.82812I 7.71191 + 2.59975I
3
II.
I
u
2
= hu
26
4u
25
+· · ·+u1, u
25
+u
24
+· · ·+4a+9, 2u
25
5u
24
+· · ·+2b+1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
6
=
u
u
3
+ u
a
8
=
1
4
u
25
1
4
u
24
+ ···
15
2
u
9
4
u
25
+
5
2
u
24
+ ··· +
3
2
u
1
2
a
1
=
3
4
u
25
+
3
4
u
24
+ ···
1
2
u +
3
4
7
4
u
25
+
19
4
u
24
+ ···
1
2
u
9
4
a
7
=
u
6
+ 3u
4
+ 2u
3
2u
2
4u 1
1
4
u
25
+
3
4
u
24
+ ··· +
11
4
u
2
1
4
a
10
=
5
2
u
25
+
11
2
u
24
+ ··· u
3
2
7
4
u
25
+
19
4
u
24
+ ···
1
2
u
9
4
a
4
=
u
2
+ 1
u
4
2u
2
a
9
=
9
2
u
25
25
2
u
24
+ ··· 3u +
7
2
21
4
u
25
+
57
4
u
24
+ ···
1
2
u
19
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 14u
25
+
73
2
u
24
+ ··· +
3
2
u
19
2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.52557 0.02102I
a = 0.86364 + 1.94054I
b = 0.920092 0.872965I
10.46155 + 3.23113I 6.21855 2.44261I
u = 1.52557 + 0.02102I
a = 0.86364 1.94054I
b = 0.920092 + 0.872965I
10.46155 3.23113I 6.21855 + 2.44261I
u = 1.256160 0.104364I
a = 0.30115 1.70965I
b = 0.714859 + 0.468666I
2.60764 + 1.82411I 3.14672 3.41167I
u = 1.256160 + 0.104364I
a = 0.30115 + 1.70965I
b = 0.714859 0.468666I
2.60764 1.82411I 3.14672 + 3.41167I
u = 0.813735
a = 0.587992
b = 0.370909
1.14285 10.2092
u = 0.646358 0.378619I
a = 0.182177 + 0.177626I
b = 0.493543 + 0.417386I
1.336672 + 0.113896I 6.51816 + 0.27618I
u = 0.646358 + 0.378619I
a = 0.182177 0.177626I
b = 0.493543 0.417386I
1.336672 0.113896I 6.51816 0.27618I
u = 0.636851 0.871335I
a = 0.299881 + 0.419254I
b = 0.890496 0.876738I
8.31406 0.26926I 5.67547 0.24692I
u = 0.636851 + 0.871335I
a = 0.299881 0.419254I
b = 0.890496 + 0.876738I
8.31406 + 0.26926I 5.67547 + 0.24692I
u = 0.583257 0.894502I
a = 0.65595 1.31235I
b = 0.942244 + 0.855193I
8.15003 + 6.14753I 5.18996 5.20017I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.583257 + 0.894502I
a = 0.65595 + 1.31235I
b = 0.942244 0.855193I
8.15003 6.14753I 5.18996 + 5.20017I
u = 0.354778 0.613616I
a = 1.50135 + 0.58086I
b = 0.846572 0.426560I
0.25133 + 3.55563I 0.67279 7.82227I
u = 0.354778 + 0.613616I
a = 1.50135 0.58086I
b = 0.846572 + 0.426560I
0.25133 3.55563I 0.67279 + 7.82227I
u = 0.035614 0.282177I
a = 1.76272 + 2.42519I
b = 0.782649 0.135062I
1.320764 0.339413I 6.54496 + 0.64162I
u = 0.035614 + 0.282177I
a = 1.76272 2.42519I
b = 0.782649 + 0.135062I
1.320764 + 0.339413I 6.54496 0.64162I
u = 0.457633 0.052615I
a = 0.90823 2.23710I
b = 0.884681 + 0.778751I
3.71424 2.93248I 1.57920 + 3.07432I
u = 0.457633 + 0.052615I
a = 0.90823 + 2.23710I
b = 0.884681 0.778751I
3.71424 + 2.93248I 1.57920 3.07432I
u = 1.42573
a = 0.690585
b = 1.05838
3.31147 2.10675
u = 1.47440 0.17499I
a = 0.446539 1.314907I
b = 1.024208 + 0.483667I
6.23030 6.31822I 4.39684 + 5.98052I
u = 1.47440 + 0.17499I
a = 0.446539 + 1.314907I
b = 1.024208 0.483667I
6.23030 + 6.31822I 4.39684 5.98052I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.53361 0.09619I
a = 0.402223 + 0.879109I
b = 0.352335 0.784080I
8.43955 1.72575I 8.31886 + 0.55186I
u = 1.53361 + 0.09619I
a = 0.402223 0.879109I
b = 0.352335 + 0.784080I
8.43955 + 1.72575I 8.31886 0.55186I
u = 1.58862 0.30654I
a = 0.05923 + 1.97842I
b = 0.996075 0.858678I
15.2731 10.5913I 6.79989 + 5.68919I
u = 1.58862 + 0.30654I
a = 0.05923 1.97842I
b = 0.996075 + 0.858678I
15.2731 + 10.5913I 6.79989 5.68919I
u = 1.60711 0.27991I
a = 0.972202 1.019591I
b = 0.851371 + 0.929645I
15.7394 4.0044I 7.52896 + 1.00327I
u = 1.60711 + 0.27991I
a = 0.972202 + 1.019591I
b = 0.851371 0.929645I
15.7394 + 4.0044I 7.52896 1.00327I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
3
u
2
+ 1)(u
26
+ 2u
25
+ ··· + 2u 1)
c
2
, c
3
(u + 1)
3
(u
26
+ 4u
25
+ ··· u 1)
c
4
, c
9
u
3
(u
26
+ u
25
+ ··· + 12u + 8)
c
5
(u 1)
3
(u
26
+ 4u
25
+ ··· u 1)
c
6
, c
10
(u
3
u
2
+ 2u 1)(u
26
+ 6u
25
+ ··· + 14u + 1)
c
7
(u
3
+ u
2
1)(u
26
+ 2u
25
+ ··· + 2u 1)
c
8
(u
3
+ u
2
+ 2u + 1)(u
26
+ 6u
25
+ ··· + 14u + 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y
3
y
2
+ 2y 1)(y
26
6y
25
+ ··· 14y + 1)
c
2
, c
3
, c
5
(y 1)
3
(y
26
28y
25
+ ··· + 9y + 1)
c
4
, c
9
y
3
(y
26
21y
25
+ ··· 272y + 64)
c
6
, c
8
, c
10
(y
3
+ 3y
2
+ 2y 1)(y
26
+ 30y
25
+ ··· 38y + 1)
9