10
161
(K10n
31
)
1
Arc Sequences
7 9 7 2 9 1 3 6 4 2
Solving Sequence
2,9 3,7
4 10 1 6 5 8
c
2
c
3
c
9
c
1
c
6
c
5
c
8
c
4
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
4
2u
2
+ u + 1, u
2
+ a 2, u
3
+ b u + 1i
I
u
2
= hu
6
+ 6u
5
+ 16u
4
+ 21u
3
+ 11u
2
2u 4, u
4
4u
3
6u
2
+ 2a u + 3,
u
5
6u
4
14u
3
13u
2
+ 2b u + 4i
There are 2 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
= hu
4
2u
2
+ u + 1, u
2
+ a 2, u
3
+ b u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
u
2
+ 2
u
3
+ u 1
a
3
=
u
3
+ u 1
u
3
2u + 1
a
7
=
0
u
a
4
=
u
3
+ u 1
u + 1
a
10
=
u
2
+ 1
u
2
a
1
=
1
u
2
a
6
=
u
u
3
+ u
a
5
=
u
3
+ 2u 2
u + 1
a
8
=
1
u
3
+ 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
3
2u
2
+ 3u 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.49022
a = 0.220744
b = 0.819173
8.36260 21.5309
u = 0.524889
a = 1.72449
b = 1.38028
4.29983 8.41490
u = 1.007552 0.513116I
a = 1.24813 + 1.03398I
b = 0.219447 + 0.914474I
3.04135 + 1.96274I 4.02709 2.32656I
u = 1.007552 + 0.513116I
a = 1.24813 1.03398I
b = 0.219447 0.914474I
3.04135 1.96274I 4.02709 + 2.32656I
3
II. I
u
2
= hu
6
+ 6u
5
+ 16u
4
+ 21u
3
+ 11u
2
2u 4, u
4
4u
3
6u
2
+ 2a
u + 3, u
5
6u
4
14u
3
13u
2
+ 2b u + 4i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
2
u
4
+ 2u
3
+ 3u
2
+
1
2
u
3
2
1
2
u
5
+ 3u
4
+ 7u
3
+
13
2
u
2
+
1
2
u 2
a
3
=
1
4
u
5
u
4
2u
3
5
4
u
2
+
3
4
u + 1
1
2
u
5
+ 2u
4
+ 3u
3
+
5
2
u
2
+
1
2
u 1
a
7
=
0
u
a
4
=
1
4
u
5
u
4
2u
3
5
4
u
2
+
3
4
u + 1
1
2
u
5
2u
4
4u
3
5
2
u
2
+
1
2
u + 1
a
10
=
u
2
1
u
2
a
1
=
1
u
2
a
6
=
u
u
3
+ u
a
5
=
1
4
u
5
+ u
4
+ 2u
3
+
5
4
u
2
+
1
4
u
1
2
u
5
2u
4
4u
3
5
2
u
2
+
1
2
u + 1
a
8
=
1
2
u
4
2u
3
2u
2
1
2
u +
1
2
3
2
u
5
9u
4
17u
3
23
2
u
2
+
3
2
u + 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
5
+ 6u
4
+ 14u
3
+ 13u
2
4u 18
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.58486
a = 0.435792
b = 0.403945
7.78420 6.88359
u = 1.55395 1.43504I
a = 0.602692 0.633679I
b = 1.19447 2.58259I
13.6396 5.6388I 8.61921 + 2.01004I
u = 1.55395 + 1.43504I
a = 0.602692 + 0.633679I
b = 1.19447 + 2.58259I
13.6396 + 5.6388I 8.61921 2.01004I
u = 0.878332 0.695514I
a = 0.586872 + 1.122636I
b = 0.244201 + 0.971888I
2.08576 2.67800I 9.11994 + 5.42135I
u = 0.878332 + 0.695514I
a = 0.586872 1.122636I
b = 0.244201 0.971888I
2.08576 + 2.67800I 9.11994 5.42135I
u = 0.449415
a = 0.467432
b = 0.304480
0.637429 15.6381
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
4
2u
2
+ u + 1)(u
6
+ 6u
5
+ 16u
4
+ 21u
3
+ 11u
2
2u 4)
c
2
, c
8
(u
4
+ u
3
1)(u
6
+ 2u
5
+ 8u
4
u
3
+ 7u
2
+ u 1)
c
3
, c
9
(u
4
+ u 1)(u
6
+ u
5
+ 9u
4
11u
3
4u
2
2u 1)
c
4
(u
4
+ 4u
3
+ 4u
2
+ u + 1)(u
6
+ 3u
5
3u
4
15u
3
10u
2
+ 1)
c
5
(u
4
u
3
1)(u
6
+ 2u
5
+ 8u
4
u
3
+ 7u
2
+ u 1)
c
6
(u
4
2u
2
u + 1)(u
6
+ 6u
5
+ 16u
4
+ 21u
3
+ 11u
2
2u 4)
c
7
(u
4
u 1)(u
6
+ u
5
+ 9u
4
11u
3
4u
2
2u 1)
c
10
(u
4
4u
3
+ ··· 5u + 1)(u
6
+ 4u
5
+ ··· + 92u + 16)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
4
4y
3
+ ··· 5y + 1)(y
6
4y
5
+ ··· 92y + 16)
c
2
, c
5
, c
8
(y
4
y
3
2y
2
+ 1)(y
6
+ 12y
5
+ 82y
4
+ 105y
3
+ 35y
2
15y + 1)
c
3
, c
7
, c
9
(y
4
2y
2
y + 1)(y
6
+ 17y
5
+ 95y
4
191y
3
46y
2
+ 4y + 1)
c
4
(y
4
8y
3
+ ··· + 7y + 1)(y
6
15y
5
+ ··· 20y + 1)
c
10
(y
4
4y
3
2y
2
13y + 1)
(y
6
+ 36y
5
+ 246y
4
2029y
3
6671y
2
6000y + 256)
7