10
152
(K10n
36
)
1
Arc Sequences
4 5 10 2 8 10 5 6 3 7
Solving Sequence
1,4
2
5,7
10 3 6 9 8
c
1
c
4
c
10
c
3
c
6
c
9
c
8
c
2
, c
5
, c
7
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu
2
+ u 1, b, a + u + 2i
I
u
2
= hb
2
b 1, u 1, b + a + 1i
I
u
3
= hu
4
+ u
3
+ 2u
2
u + 1, u
3
2u
2
+ 2b + 1, u
3
+ 2u
2
+ 2a + 2u 1i
I
u
4
= hu
5
+ 3u
4
+ 2u
3
3u
2
3u + 1, u
2
+ a 2u 1, u
4
2u
3
+ b + 2ui
There are 4 irreducible components with 13 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
2
+ u 1, b, a + u + 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u + 1
a
5
=
u
u + 1
a
7
=
u 2
0
a
10
=
1
0
a
3
=
u
u
a
6
=
u 2
0
a
9
=
u
u 1
a
8
=
2
u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 11
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.61803
a = 0.381966
b = 0
10.5276 11.0000
u = 0.618034
a = 2.61803
b = 0
2.63189 11.0000
3
II. I
u
2
= hb
2
b 1, u 1, b + a + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
1
0
a
7
=
b 1
b
a
10
=
0
b 1
a
3
=
0
1
a
6
=
b 1
b 1
a
9
=
0
b 1
a
8
=
1
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 11
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.61803
b = 0.618034
2.63189 11.0000
u = 1.00000
a = 0.618034
b = 1.61803
10.5276 11.0000
5
III. I
u
3
= hu
4
+ u
3
+ 2u
2
u + 1, u
3
2u
2
+ 2b + 1, u
3
+ 2u
2
+ 2a + 2u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
7
=
1
2
u
3
u
2
u +
1
2
1
2
u
3
+ u
2
1
2
a
10
=
u
3
u
2
u + 1
3
2
u
3
+ u
1
2
a
3
=
u
2
+ 1
u
3
+ 4u
2
u + 1
a
6
=
1
2
u
3
u
2
+ u
1
2
3
2
u
3
+ 4u
2
3u +
3
2
a
9
=
2u
3
3u
2
1
13
2
u
3
+
3
2
a
8
=
u
3
u
2
u
3
2
u
3
u
2
+ u
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 11
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.80902 1.40126I
a = 0.500000 0.866025I
b = 0.30902 + 2.26728I
7.23771 11.0000
u = 0.80902 + 1.40126I
a = 0.500000 + 0.866025I
b = 0.30902 2.26728I
7.23771 11.0000
u = 0.309017 0.535233I
a = 0.500000 + 0.866025I
b = 0.809017 0.330792I
0.657974 11.0000
u = 0.309017 + 0.535233I
a = 0.500000 0.866025I
b = 0.809017 + 0.330792I
0.657974 11.0000
7
IV.
I
u
4
= hu
5
+ 3u
4
+ 2u
3
3u
2
3u + 1, u
2
+ a 2u 1, u
4
2u
3
+ b + 2ui
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
7
=
u
2
+ 2u + 1
u
4
+ 2u
3
2u
a
10
=
u
3
2u
2
+ 2
u
3
u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
2u
2
+ 2u
2u
4
+ 2u
3
2u
2
u
a
9
=
2u
3
2u
2
+ 2u + 1
4u
4
+ 8u
3
4u
2
8u + 2
a
8
=
2u + 1
2u
3
+ u
2
2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
4
24u
3
24u
2
10
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.39814 0.93867I
a = 0.722590 + 0.747455I
b = 1.01518 1.84157I
5.12323 8.53607I 12.97824 + 4.17771I
u = 1.39814 + 0.93867I
a = 0.722590 0.747455I
b = 1.01518 + 1.84157I
5.12323 + 8.53607I 12.97824 4.17771I
u = 1.39373
a = 0.155021
b = 1.14610
11.4408 21.8304
u = 0.277157
a = 1.63113
b = 0.505833
0.775637 12.4017
u = 0.912859
a = 3.65903
b = 0.390081
2.96486 53.8114
9
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
5
(u 1)
2
(u
2
+ u 1)(u
4
+ u
3
+ ··· u + 1)(u
5
+ 3u
4
+ ··· 3u + 1)
c
3
, c
10
u
2
(u
2
u 1)(u
4
+ u
3
+ ··· + 8u + 4)(u
5
+ u
4
+ ··· u + 1)
c
4
, c
7
, c
8
(u + 1)
2
(u
2
u 1)(u
4
+ u
3
+ ··· u + 1)(u
5
+ 3u
4
+ ··· 3u + 1)
c
6
, c
9
u
2
(u
2
+ u 1)(u
4
+ u
3
+ ··· + 8u + 4)(u
5
+ u
4
+ ··· u + 1)
10
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
4
c
5
, c
7
, c
8
(y 1)
2
(y
2
3y + 1)(y
4
+ 3y
3
+ 8y
2
+ 3y + 1)
(y
5
5y
4
+ 16y
3
27y
2
+ 15y 1)
c
3
, c
6
, c
9
c
10
y
2
(y
2
3y + 1)(y
4
+ 9y
3
+ 17y
2
24y + 16)
(y
5
+ 3y
4
+ 12y
3
31y
2
+ 11y 1)
11