10
40
(K10a
30
)
1
Arc Sequences
6 9 10 7 1 5 3 2 8 4
Solving Sequence
1,5
6 2 7 4 10 3 8 9
c
5
c
1
c
6
c
4
c
10
c
3
c
7
c
9
c
2
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu 1i
I
u
2
= hu
4
+ u
3
+ 1i
I
u
3
= hu
32
+ u
31
+ ··· + 2u + 1i
There are 3 irreducible components with 37 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 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
6
=
1
1
a
2
=
0
1
a
7
=
1
0
a
4
=
1
1
a
10
=
0
1
a
3
=
1
0
a
8
=
1
0
a
9
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
1.64493 6.00000
3
II. I
u
2
= hu
4
+ u
3
+ 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
6
=
u
u
a
2
=
u
2
+ 1
u
2
a
7
=
u
u
3
+ u
a
4
=
u
3
1
a
10
=
u
3
+ 1
1
a
3
=
1
0
a
8
=
u
3
+ 2u
u
3
+ u
a
9
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.018913 0.602565I
1.64493 6.00000
u = 1.018913 + 0.602565I
1.64493 6.00000
u = 0.518913 0.666610I
1.64493 6.00000
u = 0.518913 + 0.666610I
1.64493 6.00000
5
III. I
u
3
= hu
32
+ u
31
+ · · · + 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
6
=
u
u
a
2
=
u
2
+ 1
u
2
a
7
=
u
u
3
+ u
a
4
=
u
3
u
5
u
3
+ u
a
10
=
u
8
+ u
6
u
4
+ 1
u
10
2u
8
+ 3u
6
2u
4
+ u
2
a
3
=
u
13
2u
11
+ 3u
9
2u
7
u
u
15
+ 3u
13
6u
11
+ 7u
9
6u
7
+ 4u
5
2u
3
+ u
a
8
=
u
25
4u
23
+ ··· + 2u
3
+ u
u
27
+ 5u
25
+ ··· 3u
3
+ u
a
9
=
u
31
6u
29
+ ··· 18u
5
+ 6u
3
u
31
5u
29
+ ··· 2u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
30
+ 20u
28
+ 4u
27
68u
26
20u
25
+ 156u
24
+ 68u
23
276u
22
160u
21
+ 380u
20
+
292u
19
404u
18
428u
17
+ 328u
16
+ 504u
15
160u
14
496u
13
8u
12
+ 392u
11
+
124u
10
252u
9
156u
8
+ 120u
7
+ 116u
6
28u
5
64u
4
4u
3
+ 16u
2
+ 12u + 2
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.093529 0.032199I
6.73005 + 1.36697I 11.90065 0.55023I
u = 1.093529 + 0.032199I
6.73005 1.36697I 11.90065 + 0.55023I
u = 1.036494 0.686644I
0.19628 + 12.88874I 3.87677 9.41526I
u = 1.036494 + 0.686644I
0.19628 12.88874I 3.87677 + 9.41526I
u = 0.997643 0.681461I
2.66422 + 5.49753I 0.37719 4.60034I
u = 0.997643 + 0.681461I
2.66422 5.49753I 0.37719 + 4.60034I
u = 0.858258 0.694285I
2.60826 + 2.66625I 2.22295 3.31297I
u = 0.858258 + 0.694285I
2.60826 2.66625I 2.22295 + 3.31297I
u = 0.849583 0.407230I
0.10900 + 4.15286I 6.01286 7.18864I
u = 0.849583 + 0.407230I
0.10900 4.15286I 6.01286 + 7.18864I
u = 0.674958 0.742403I
3.63561 0.05779I 1.67435 0.61686I
u = 0.674958 + 0.742403I
3.63561 + 0.05779I 1.67435 + 0.61686I
u = 0.613006 0.792175I
1.06972 7.30693I 1.82356 + 4.86883I
u = 0.613006 + 0.792175I
1.06972 + 7.30693I 1.82356 4.86883I
u = 0.416995 0.648442I
0.08923 + 4.79464I 2.70911 5.61871I
u = 0.416995 + 0.648442I
0.08923 4.79464I 2.70911 + 5.61871I
u = 0.164238 0.469611I
1.64326 1.19641I 1.57525 + 0.85209I
u = 0.164238 + 0.469611I
1.64326 + 1.19641I 1.57525 0.85209I
u = 0.600521 0.762759I
0.98960 + 2.26361I 5.01894 0.67006I
u = 0.600521 + 0.762759I
0.98960 2.26361I 5.01894 + 0.67006I
u = 0.730192 0.168194I
1.169215 0.193186I 9.20830 + 0.78328I
u = 0.730192 + 0.168194I
1.169215 + 0.193186I 9.20830 0.78328I
u = 0.828553 0.741140I
5.70053 + 0.95663I 2.35494 0.97622I
u = 0.828553 + 0.741140I
5.70053 0.95663I 2.35494 + 0.97622I
u = 0.891994 0.729689I
5.50827 6.53878I 1.61404 + 6.99151I
u = 0.891994 + 0.729689I
5.50827 + 6.53878I 1.61404 6.99151I
u = 1.022973 0.630121I
3.03384 5.05352I 8.11469 + 5.31459I
u = 1.022973 + 0.630121I
3.03384 + 5.05352I 8.11469 5.31459I
u = 1.031611 0.673233I
2.26376 7.72193I 6.98438 + 5.32873I
u = 1.031611 + 0.673233I
2.26376 + 7.72193I 6.98438 5.32873I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.098860 0.059621I
4.95901 6.50568I 8.96918 + 5.51070I
u = 1.098860 + 0.059621I
4.95901 + 6.50568I 8.96918 5.51070I
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u 1)(u
4
+ u
3
+ 1)(u
32
+ u
31
+ ··· + 2u + 1)
c
2
, c
8
(u 1)(u
4
+ u
3
+ 1)(u
32
+ u
31
+ ··· u
2
+ 1)
c
3
, c
10
(u + 1)
5
(u
32
4u
31
+ ··· 28u + 4)
c
4
, c
6
(u + 1)(u
4
+ u
3
+ 2u
2
+ 1)(u
32
+ 11u
31
+ ··· + 2u + 1)
c
7
(u)(u
4
u
2
2u + 3)(u
32
+ 3u
31
+ ··· + 2u + 3)
c
9
(u + 1)(u
4
+ u
3
+ 2u
2
+ 1)(u
32
+ 15u
31
+ ··· + 2u + 1)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)(y
4
y
3
+ 2y
2
+ 1)(y
32
11y
31
+ ··· 2y + 1)
c
2
, c
8
(y 1)(y
4
y
3
+ 2y
2
+ 1)(y
32
15y
31
+ ··· 2y + 1)
c
3
, c
10
(y 1)
5
(y
32
20y
31
+ ··· 184y + 16)
c
4
, c
6
(y 1)(y
4
+ 3y
3
+ ··· + 4y + 1)(y
32
+ 21y
31
+ ··· + 2y + 1)
c
7
(y)(y
4
2y
3
+ ··· 10y + 9)(y
32
+ 5y
31
+ ··· + 164y + 9)
c
9
(y 1)(y
4
+ 3y
3
+ ··· + 4y + 1)(y
32
+ 5y
31
+ ··· + 2y + 1)
9