10
143
(K10n
26
)
1
Arc Sequences
6 5 10 8 1 2 9 5 3 9
Solving Sequence
1,5
6 2
3,9
8 7 10 4
c
5
c
1
c
2
c
8
c
7
c
10
c
3
c
4
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= ha
3
a + 1, u 1, a
2
+ bi
I
u
2
= hu
2
2, b 1, 2a ui
I
u
3
= hu
13
+ 2u
12
3u
11
9u
10
u
9
+ 12u
8
+ 14u
7
+ 7u
6
11u
5
23u
4
12u
3
2u
2
+ 2,
u
12
5u
10
+ 9u
8
4u
6
7u
4
+ 2u
3
+ 6u
2
+ 4a 6u 2,
u
12
4u
10
+ 5u
8
+ u
6
+ 2u
5
6u
4
2u
3
2u
2
+ 4b 4ui
I
v
1
= hb 1, v 1, ai
There are 4 irreducible components with 19 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
= ha
3
a + 1, u 1, a
2
+ bi
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
6
=
1
1
a
2
=
0
1
a
3
=
0
1
a
9
=
a
a
2
a
8
=
a
a
2
a
a
7
=
1
0
a
10
=
a
a
2
a
a
4
=
a
2
a
2
a
(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
a = 1.32472
b = 1.75488
1.64493 6.00000
u = 1.00000
a = 0.662359 0.562280I
b = 0.122561 0.744862I
1.64493 6.00000
u = 1.00000
a = 0.662359 + 0.562280I
b = 0.122561 + 0.744862I
1.64493 6.00000
3
II. I
u
2
= hu
2
2, b 1, 2a ui
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
6
=
u
u
a
2
=
1
2
a
3
=
1
0
a
9
=
1
2
u
1
a
8
=
1
2
u
u + 1
a
7
=
0
u
a
10
=
1
2
u + 1
1
a
4
=
1
2
u
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.41421
a = 0.707107
b = 1.00000
4.93480 8.00000
u = 1.41421
a = 0.707107
b = 1.00000
4.93480 8.00000
5
III.
I
u
3
= hu
13
+2u
12
+· · ·−2u
2
+2, u
12
5u
10
+· · ·+4a2, u
12
4u
10
+· · ·+4b4ui
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
6
=
u
u
a
2
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
4
2u
2
a
9
=
1
4
u
12
+
5
4
u
10
+ ··· +
3
2
u +
1
2
1
4
u
12
+ u
10
+ ··· +
1
2
u
2
+ u
a
8
=
1
4
u
12
+
5
4
u
10
+ ··· +
3
2
u +
1
2
1
4
u
12
+
3
4
u
11
+ ··· +
1
2
u + 1
a
7
=
u
3
+ 2u
u
3
+ u
a
10
=
1
2
u
6
3
2
u
4
1
2
u
3
+ u
2
+ u + 1
1
4
u
11
+ u
9
+ ··· +
1
2
u
2
+
1
2
u
a
4
=
1
4
u
10
u
8
+ ···
1
2
u +
1
2
1
4
u
12
+ u
10
+ ···
3
2
u
2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 2u
12
10u
10
2u
9
+ 18u
8
+ 8u
7
4u
6
10u
5
26u
4
+ 20u
2
+ 2u + 8
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.38959
a = 0.0977488
b = 1.12566
6.53354 13.9760
u = 1.236958 0.573659I
a = 1.088727 0.567381I
b = 0.63064 1.61844I
6.78115 + 1.92961I 2.66803 0.98070I
u = 1.236958 + 0.573659I
a = 1.088727 + 0.567381I
b = 0.63064 + 1.61844I
6.78115 1.92961I 2.66803 + 0.98070I
u = 1.197108 0.332616I
a = 0.277849 + 0.753131I
b = 0.236552 + 1.226152I
1.92578 + 4.88678I 6.41460 5.91732I
u = 1.197108 + 0.332616I
a = 0.277849 0.753131I
b = 0.236552 1.226152I
1.92578 4.88678I 6.41460 + 5.91732I
u = 0.116060 1.025315I
a = 0.06204 + 1.64663I
b = 0.03703 + 2.02819I
10.21606 + 3.70097I 0.67358 2.50956I
u = 0.116060 + 1.025315I
a = 0.06204 1.64663I
b = 0.03703 2.02819I
10.21606 3.70097I 0.67358 + 2.50956I
u = 0.094132 0.586012I
a = 0.95207 1.19045I
b = 0.036425 0.507634I
1.38205 1.36942I 0.56235 + 3.09698I
u = 0.094132 + 0.586012I
a = 0.95207 + 1.19045I
b = 0.036425 + 0.507634I
1.38205 + 1.36942I 0.56235 3.09698I
u = 0.418617
a = 0.885586
b = 0.580187
0.992576 11.4261
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.40252 0.47847I
a = 0.974688 0.539119I
b = 0.58023 2.02564I
5.44762 9.07090I 4.16718 + 5.02365I
u = 1.40252 + 0.47847I
a = 0.974688 + 0.539119I
b = 0.58023 + 2.02564I
5.44762 + 9.07090I 4.16718 5.02365I
u = 1.45446
a = 0.759729
b = 0.0687730
3.37738 1.87576
8
IV. I
v
1
= hb 1, v 1, ai
(i) Arc colorings
a
1
=
1
0
a
5
=
1
0
a
6
=
1
0
a
2
=
1
0
a
3
=
1
0
a
9
=
0
1
a
8
=
1
1
a
7
=
1
0
a
10
=
1
1
a
4
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
10
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
(u)(u + 1)
3
(u
2
2)(u
13
2u
12
+ ··· + 2u
2
2)
c
2
u
4
(u
2
2)(u
13
+ 3u
12
+ ··· 92u + 46)
c
3
(u 1)(u + 1)
2
(u
3
u + 1)(u
13
2u
12
+ ··· 3u 1)
c
4
(u 1)(u + 1)
2
(u
3
u + 1)(u
13
+ 2u
12
+ ··· + 9u 1)
c
7
(u 1)(u + 1)
2
(u
3
+ 2u
2
+ u + 1)(u
13
+ 18u
12
+ ··· + 65u + 1)
c
8
(u 1)
2
(u + 1)(u
3
u + 1)(u
13
+ 2u
12
+ ··· + 9u 1)
c
9
(u + 1)
3
(u
3
u + 1)(u
13
2u
12
+ ··· 3u 1)
c
10
(u 1)(u + 1)
2
(u
3
+ 2u
2
+ u + 1)(u
13
+ 2u
12
+ ··· + 17u + 1)
11
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
(y)(y 2)
2
(y 1)
3
(y
13
10y
12
+ ··· + 8y 4)
c
2
y
4
(y 2)
2
(y
13
+ 23y
12
+ ··· + 7728y 2116)
c
3
, c
9
(y 1)
3
(y
3
2y
2
+ y 1)(y
13
2y
12
+ ··· + 17y 1)
c
4
, c
8
(y 1)
3
(y
3
2y
2
+ y 1)(y
13
18y
12
+ ··· + 65y 1)
c
7
(y 1)
3
(y
3
2y
2
3y 1)(y
13
42y
12
+ ··· + 2989y 1)
c
10
(y 1)
3
(y
3
2y
2
3y 1)(y
13
+ 22y
12
+ ··· + 205y 1)
12