10
9
(K10a
110
)
1
Arc Sequences
7 9 8 2 10 1 3 4 5 6
Solving Sequence
3,8
4 9 2 5 10 7 1 6
c
3
c
8
c
2
c
4
c
9
c
7
c
1
c
6
c
5
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu 1i
I
u
2
= hu
18
8u
16
+ ··· + u + 1i
There are 2 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
= hu 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
1
a
4
=
1
1
a
9
=
1
0
a
2
=
1
0
a
5
=
2
1
a
10
=
1
1
a
7
=
1
1
a
1
=
2
1
a
6
=
1
0
(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
1.64493 6.00000
3
II. I
u
2
= hu
18
8u
16
u
15
+ 25u
14
+ 7u
13
35u
12
18u
11
+ 13u
10
+ 18u
9
+
16u
8
u
7
9u
6
5u
5
7u
4
3u
3
+ 2u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
2
=
u
4
+ u
2
+ 1
u
6
2u
4
+ u
2
a
5
=
u
8
3u
6
+ u
4
+ 2u
2
+ 1
u
10
+ 4u
8
5u
6
+ 2u
4
u
2
a
10
=
u
15
+ 6u
13
12u
11
+ 6u
9
+ 6u
7
2u
5
4u
3
u
17
7u
15
+ 19u
13
24u
11
+ 13u
9
2u
7
+ u
a
7
=
u
u
a
1
=
u
8
3u
6
+ u
4
+ 2u
2
+ 1
u
8
+ 4u
6
4u
4
a
6
=
u
15
6u
13
+ 12u
11
6u
9
6u
7
+ 2u
5
+ 4u
3
u
15
+ 7u
13
18u
11
+ 19u
9
4u
7
4u
5
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
17
32u
15
4u
14
+ 100u
13
+ 28u
12
144u
11
72u
10
+ 72u
9
+
76u
8
+ 32u
7
20u
6
28u
5
4u
4
12u
3
8u
2
+ 8u 2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.377459 0.269730I
1.32984 + 6.64718I 3.24506 6.19689I
u = 1.377459 + 0.269730I
1.32984 6.64718I 3.24506 + 6.19689I
u = 1.336420 0.120409I
3.49531 + 0.56492I 0.70794 + 1.84066I
u = 1.336420 + 0.120409I
3.49531 0.56492I 0.70794 1.84066I
u = 0.884053 0.346992I
11.15466 0.27346I 6.21894 1.07083I
u = 0.884053 + 0.346992I
11.15466 + 0.27346I 6.21894 + 1.07083I
u = 0.226840 0.766905I
13.25299 + 4.38839I 8.97609 3.55329I
u = 0.226840 + 0.766905I
13.25299 4.38839I 8.97609 + 3.55329I
u = 0.181355 0.488140I
0.150453 + 1.027515I 2.68106 6.45577I
u = 0.181355 + 0.488140I
0.150453 1.027515I 2.68106 + 6.45577I
u = 0.209503 0.678973I
3.70552 3.19755I 8.61366 + 5.32391I
u = 0.209503 + 0.678973I
3.70552 + 3.19755I 8.61366 5.32391I
u = 0.638700
1.71487 4.98727
u = 1.366895 0.206727I
4.78286 3.66002I 2.48971 + 4.64953I
u = 1.366895 + 0.206727I
4.78286 + 3.66002I 2.48971 4.64953I
u = 1.39250 0.31226I
8.11334 8.29410I 4.53964 + 4.66449I
u = 1.39250 + 0.31226I
8.11334 + 8.29410I 4.53964 4.66449I
u = 1.43575
3.96483 2.02739
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
c
9
, c
10
(u 1)(u
18
+ 2u
17
+ ··· + u + 1)
c
2
(u)(u
18
+ 3u
17
+ ··· 3u 3)
c
3
, c
7
, c
8
(u + 1)(u
18
8u
16
+ ··· u + 1)
c
4
(u + 1)(u
18
+ 4u
17
+ ··· + 5u 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
c
9
, c
10
(y 1)(y
18
24y
17
+ ··· + 3y + 1)
c
2
(y)(y
18
+ 3y
17
+ ··· 39y + 9)
c
3
, c
7
, c
8
(y 1)(y
18
16y
17
+ ··· + 3y + 1)
c
4
(y 1)(y
18
+ 22y
16
+ ··· 65y + 1)
7