10
136
(K10n
3
)
1
Arc Sequences
9 1 7 3 10 4 5 2 5 3
Solving Sequence
5,10 3,6
1 2 4 7 8 9
c
5
c
10
c
2
c
4
c
6
c
7
c
9
c
1
, c
3
, c
8
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu
5
u
4
+ 5u
3
3u
2
u + 1, b + u, u
4
+ 4u
2
+ 2a + 3u 4i
I
u
2
= hu
6
+ 6u
4
u
3
+ 7u
2
+ 3u + 1, 11u
5
3u
4
+ 71u
3
22u
2
+ 23b + 83u + 25,
18u
5
7u
4
+ 112u
3
59u
2
+ 23a + 140u 3i
I
u
3
= hu
2
u + 1, a, b + ui
I
u
4
= hu
2
u + 1, a, b u + 1i
There are 4 irreducible components with 15 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
5
u
4
+ 5u
3
3u
2
u + 1, b + u, u
4
+ 4u
2
+ 2a + 3u 4i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
2
u
4
2u
2
3
2
u + 2
u
a
3
=
1
0
a
6
=
u
4
+ u
3
5u
2
+ 2u
1
2
u
3
+ u
1
2
a
1
=
1
2
u
4
2u
2
1
2
u + 2
u
a
2
=
1
2
u
4
+
1
2
u
3
+ ··· +
3
2
u +
3
2
u
2
a
4
=
u
u
a
7
=
1
2
u
4
+
1
2
u
3
+ ··· +
3
2
u
1
2
1
2
u
4
+ 2u
2
+
1
2
u 1
a
8
=
1
2
u
4
+
1
2
u
3
+ ··· +
3
2
u
1
2
u
2
+ u 1
a
9
=
1
2
u
4
2u
2
3
2
u + 2
1
2
u
3
u +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
4
4u
3
+ 24u
2
9u 4
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.504717
a = 2.21515
b = 0.504717
1.42879 7.49493
u = 0.17097 2.22112I
a = 0.185053 + 1.125980I
b = 0.17097 + 2.22112I
14.8579 + 7.7463I 4.23230 4.04224I
u = 0.17097 + 2.22112I
a = 0.185053 1.125980I
b = 0.17097 2.22112I
14.8579 7.7463I 4.23230 + 4.04224I
u = 0.581386 0.247464I
a = 0.577477 + 1.026324I
b = 0.581386 + 0.247464I
1.59034 + 1.66520I 2.97976 4.53195I
u = 0.581386 + 0.247464I
a = 0.577477 1.026324I
b = 0.581386 0.247464I
1.59034 1.66520I 2.97976 + 4.53195I
3
II. I
u
2
= hu
6
+ 6u
4
u
3
+ 7u
2
+ 3u + 1, 11u
5
3u
4
+ 71u
3
22u
2
+ 23b +
83u + 25, 18u
5
7u
4
+ · · · + 23a 3i
(i) Arc colorings
a
5
=
0
u
a
10
=
0.782609u
5
+ 0.304348u
4
+ ··· 6.08696u + 0.130435
0.478261u
5
+ 0.130435u
4
+ ··· 3.60870u 1.08696
a
3
=
1
0
a
6
=
0.173913u
5
+ 0.0434783u
4
+ ··· + 1.13043u + 2.30435
0.347826u
5
0.0869565u
4
+ ··· 1.26087u + 0.391304
a
1
=
0.304348u
5
+ 0.173913u
4
+ ··· 2.47826u + 1.21739
0.478261u
5
+ 0.130435u
4
+ ··· 3.60870u 1.08696
a
2
=
0.826087u
5
0.0434783u
4
+ ··· + 4.86957u + 3.69565
0.434783u
5
+ 0.391304u
4
+ ··· 2.82609u + 0.739130
a
4
=
u
u
a
7
=
0.391304u
5
+ 0.347826u
4
+ ··· + 2.04348u + 2.43478
0.130435u
5
+ 0.217391u
4
+ ··· 0.347826u + 0.521739
a
8
=
0.391304u
5
+ 0.347826u
4
+ ··· + 2.04348u + 2.43478
0.0434783u
5
0.260870u
4
+ ··· 1.78261u + 0.173913
a
9
=
0.782609u
5
+ 0.304348u
4
+ ··· 6.08696u + 0.130435
0.652174u
5
+ 0.0869565u
4
+ ··· 3.73913u 1.39130
(ii) Obstruction class = 1
(iii) Cusp Shapes =
15
23
u
5
2
23
u
4
+
101
23
u
3
30
23
u
2
+
155
23
u +
101
23
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.22670 2.19389I
a = 0.123914 1.199184I
b = 0.22670 2.19389I
15.2333 4.75217
u = 0.22670 + 2.19389I
a = 0.123914 + 1.199184I
b = 0.22670 + 2.19389I
15.2333 4.75217
u = 0.221168 0.280722I
a = 1.19276 + 2.11342I
b = 0.447867 + 1.186989I
0.60803 + 2.56897I 3.12391 2.13317I
u = 0.221168 + 0.280722I
a = 1.19276 2.11342I
b = 0.447867 1.186989I
0.60803 2.56897I 3.12391 + 2.13317I
u = 0.447867 1.186989I
a = 0.683326 + 0.019751I
b = 0.221168 + 0.280722I
0.60803 + 2.56897I 3.12391 2.13317I
u = 0.447867 + 1.186989I
a = 0.683326 0.019751I
b = 0.221168 0.280722I
0.60803 2.56897I 3.12391 + 2.13317I
5
III. I
u
3
= hu
2
u + 1, a, b + ui
(i) Arc colorings
a
5
=
0
u
a
10
=
0
u
a
3
=
1
0
a
6
=
0
u
a
1
=
u
u
a
2
=
u
u + 1
a
4
=
u
u
a
7
=
1
u 1
a
8
=
1
0
a
9
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u 4
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0
b = 0.500000 + 0.866025I
4.05977I 6.92820I
u = 0.500000 + 0.866025I
a = 0
b = 0.500000 0.866025I
4.05977I 6.92820I
7
IV. I
u
4
= hu
2
u + 1, a, b u + 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
0
u 1
a
3
=
1
0
a
6
=
0
u
a
1
=
u + 1
u 1
a
2
=
u + 1
u
a
4
=
u
u
a
7
=
1
u 1
a
8
=
1
0
a
9
=
0
u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0
b = 0.500000 0.866025I
0 3.00000
u = 0.500000 + 0.866025I
a = 0
b = 0.500000 + 0.866025I
0 3.00000
9
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
6
(u
2
u + 1)
2
(u
5
+ u
4
+ u
3
u
2
+ u 1)
(u
6
+ 2u
5
+ 2u
4
+ u
3
+ 3u
2
+ 3u + 1)
c
2
, c
4
, c
10
(u
2
+ u + 1)
2
(u
5
+ u
4
+ ··· u 1)(u
6
+ 6u
4
+ ··· 3u + 1)
c
5
, c
9
u
4
(u
3
2u
2
u 2)
2
(u
5
+ 5u
4
+ 6u
3
4u
2
8u 4)
c
7
(u
2
u + 1)
2
(u
5
+ 4u
4
+ 15u
3
+ 10u
2
u + 2)
(u
6
u
5
+ 14u
4
33u
3
+ 58u
2
45u + 17)
c
8
(u
2
+ u + 1)
2
(u
5
+ u
4
+ u
3
u
2
+ u 1)
(u
6
+ 2u
5
+ 2u
4
+ u
3
+ 3u
2
+ 3u + 1)
10
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
6
c
8
(y
2
+ y + 1)
2
(y
5
+ y
4
+ ··· y 1)(y
6
+ 6y
4
+ ··· 3y + 1)
c
2
, c
4
, c
10
(y
2
+ y + 1)
2
(y
5
+ 9y
4
+ 17y
3
17y
2
+ 7y 1)
(y
6
+ 12y
5
+ 50y
4
+ 85y
3
+ 67y
2
+ 5y + 1)
c
5
, c
9
y
4
(y
3
6y
2
7y 4)
2
(y
5
13y
4
+ 60y
3
72y
2
+ 32y 16)
c
7
(y
2
+ y + 1)
2
(y
5
+ 14y
4
+ 143y
3
146y
2
39y 4)
(y
6
+ 27y
5
+ 246y
4
+ 479y
3
+ 870y
2
53y + 289)
11