10
153
(K10n
10
)
1
Arc Sequences
9 6 10 7 3 8 5 1 3 8
Solving Sequence
4,7
5
8,10
1 3 6 9 2
c
4
c
7
c
10
c
3
c
6
c
9
c
1
c
2
, c
5
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
2
+ b 1, a + 1, u 1i
I
u
2
= hu
3
+ u
2
1, b, u
2
+ a 2u 1i
I
u
3
= hu
5
+ 5u
4
+ 7u
3
2u
2
8u + 1, u
4
2u
3
u
2
+ 2b + u + 1, u
4
+ 5u
3
+ 7u
2
+ a u 7i
There are 3 irreducible components with 10 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
2
+ b 1, a + 1, u 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
1
a
5
=
1
1
a
8
=
1
0
a
10
=
1
b
a
1
=
b 1
b
a
3
=
b + 1
b + 1
a
6
=
1
1
a
9
=
2b
b + 1
a
2
=
b + 1
b + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 9
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.61803
7.23771 9.00000
u = 1.00000
a = 1.00000
b = 0.618034
0.657974 9.00000
3
II. I
u
2
= hu
3
+ u
2
1, b, u
2
+ a 2u 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
8
=
u
u
2
+ u 1
a
10
=
u
2
+ 2u + 1
0
a
1
=
u
2
+ 3u + 1
u
2
u + 1
a
3
=
1
0
a
6
=
u
2
+ 1
u
2
a
9
=
u
2
+ 2u + 1
0
a
2
=
u
u
2
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
8u 4
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.877439 0.744862I
a = 0.539798 0.182582I
b = 0
4.66906 2.82812I 2.80443 + 4.65175I
u = 0.877439 + 0.744862I
a = 0.539798 + 0.182582I
b = 0
4.66906 + 2.82812I 2.80443 4.65175I
u = 0.754878
a = 3.07960
b = 0
0.531480 10.6089
5
III. I
u
3
=
hu
5
+5u
4
+7u
3
2u
2
8u+1, u
4
2u
3
u
2
+2b+u+1, u
4
+5u
3
+7u
2
+au7i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
8
=
u
u
3
+ u
a
10
=
u
4
5u
3
7u
2
+ u + 7
1
2
u
4
+ u
3
+
1
2
u
2
1
2
u
1
2
a
1
=
1
2
u
4
3u
3
+ ···
5
2
u +
15
2
1
2
u
4
+
5
2
u
2
3
2
u
1
2
a
3
=
3
2
u
4
+ 5u
3
+
7
2
u
2
7
2
u
5
2
9
2
u
4
+ 14u
3
+ ···
37
2
u +
5
2
a
6
=
u
3
5u
4
8u
3
+ 2u
2
+ 9u 1
a
9
=
3u
4
12u
3
10u
2
+ 13u + 4
5
2
u
4
8u
3
+ ··· +
37
2
u
5
2
a
2
=
17
2
u
4
29u
3
+ ··· +
75
2
u
3
2
29
2
u
4
53u
3
+ ··· +
175
2
u
21
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
4
10u
3
15u
2
2u + 10
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 2.19630
a = 0.740989
b = 4.04985
5.74119 1.44344
u = 1.88542 0.91135I
a = 0.455489 + 1.119161I
b = 2.39378 + 2.57393I
14.3433 7.3743I 1.72840 + 2.44716I
u = 1.88542 + 0.91135I
a = 0.455489 1.119161I
b = 2.39378 2.57393I
14.3433 + 7.3743I 1.72840 2.44716I
u = 0.122993
a = 7.00757
b = 0.551958
1.12640 9.50804
u = 0.844155
a = 0.659538
b = 0.289662
1.21003 9.40829
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
10
(u 1)
3
(u
2
+ u 1)(u
5
+ 6u
4
+ 11u
3
+ u
2
12u + 1)
c
2
u
2
(u
3
u
2
+ 2u 1)(u
5
+ u
4
4u
3
23u
2
+ 4u + 4)
c
3
u
3
(u
2
+ u 1)(u
5
+ u
4
7u
3
52u
2
12u + 8)
c
4
(u 1)
2
(u
3
+ u
2
1)(u
5
+ 5u
4
+ 7u
3
2u
2
8u + 1)
c
5
u
2
(u
3
+ u
2
+ 2u + 1)(u
5
+ u
4
4u
3
23u
2
+ 4u + 4)
c
6
(u 1)
2
(u
3
u
2
+ 2u 1)(u
5
+ 11u
4
+ 53u
3
+ 126u
2
+ 68u + 1)
c
7
(u + 1)
2
(u
3
u
2
+ 1)(u
5
+ 5u
4
+ 7u
3
2u
2
8u + 1)
c
8
(u + 1)
3
(u
2
u 1)(u
5
+ 6u
4
+ 11u
3
+ u
2
12u + 1)
c
9
u
3
(u
2
u 1)(u
5
+ u
4
7u
3
52u
2
12u + 8)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
, c
10
(y 1)
3
(y
2
3y + 1)(y
5
14y
4
+ 85y
3
277y
2
+ 142y 1)
c
2
, c
5
y
2
(y
3
+ 3y
2
+ 2y 1)(y
5
9y
4
+ 70y
3
569y
2
+ 200y 16)
c
3
, c
9
y
3
(y
2
3y + 1)(y
5
15y
4
+ 129y
3
2552y
2
+ 976y 64)
c
4
, c
7
(y 1)
2
(y
3
y
2
+ 2y 1)(y
5
11y
4
+ 53y
3
126y
2
+ 68y 1)
c
6
(y 1)
2
(y
3
+ 3y
2
+ 2y 1)(y
5
15y
4
+ ··· + 4372y 1)
9