10
17
(K10a
107
)
1
Arc Sequences
6 7 1 9 10 2 3 4 5 8
Solving Sequence
2,7
3 8 6 1 4 9 10 5
c
2
c
7
c
6
c
1
c
3
c
8
c
10
c
5
c
4
, c
9
Representation Ideals
I = I
u
1
I
u
1
= hu
20
+ u
19
+ ··· u
2
+ 1i
There are 1 irreducible components with 20 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
20
+ u
19
11u
18
10u
17
+ 49u
16
+ 38u
15
114u
14
66u
13
+ 152u
12
+
47u
11
125u
10
4u
9
+ 67u
8
8u
7
20u
6
+ 10u
5
+ 5u
4
3u
3
u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
8
=
u
u
3
+ u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
u
6
+ 3u
4
2u
2
+ 1
u
6
+ 2u
4
+ u
2
a
9
=
u
15
+ 8u
13
24u
11
+ 34u
9
26u
7
+ 14u
5
4u
3
u
15
+ 7u
13
16u
11
+ 11u
9
+ 2u
7
+ u
a
10
=
u
6
+ 3u
4
2u
2
+ 1
u
8
+ 4u
6
4u
4
a
5
=
u
15
8u
13
+ 24u
11
34u
9
+ 26u
7
14u
5
+ 4u
3
u
17
9u
15
+ 31u
13
50u
11
+ 37u
9
12u
7
+ 4u
5
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
18
44u
16
+ 192u
14
4u
13
420u
12
+ 32u
11
+ 484u
10
92u
9
296u
8
+ 112u
7
+ 100u
6
56u
5
4u
4
+ 20u
3
4u
2
4u 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.58631 0.15748I
8.53676I 4.57594I
u = 1.58631 + 0.15748I
8.53676I 4.57594I
u = 1.58303 0.08477I
8.68051 2.11373I 5.79765 + 0.04379I
u = 1.58303 + 0.08477I
8.68051 + 2.11373I 5.79765 0.04379I
u = 1.38695
3.92816 1.96122
u = 0.886444
4.43265 0.716387
u = 0.638615 0.441759I
3.91005I 8.23335I
u = 0.638615 + 0.441759I
3.91005I 8.23335I
u = 0.232031 0.442395I
1.152208 + 0.756271I 5.04397 1.60900I
u = 0.232031 + 0.442395I
1.152208 0.756271I 5.04397 + 1.60900I
u = 0.265798 0.599404I
8.68051 2.11373I 5.79765 + 0.04379I
u = 0.265798 + 0.599404I
8.68051 + 2.11373I 5.79765 0.04379I
u = 0.613121 0.271451I
1.152208 + 0.756271I 5.04397 1.60900I
u = 0.613121 + 0.271451I
1.152208 0.756271I 5.04397 + 1.60900I
u = 0.653943 0.534643I
7.54354 + 5.98288I 2.92800 5.90364I
u = 0.653943 + 0.534643I
7.54354 5.98288I 2.92800 + 5.90364I
u = 1.51222
4.43265 0.716387
u = 1.58517 0.12489I
7.54354 + 5.98288I 2.92800 5.90364I
u = 1.58517 + 0.12489I
7.54354 5.98288I 2.92800 + 5.90364I
u = 1.60509
3.92816 1.96122
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
8
, c
9
(u
20
+ u
19
+ ··· u
2
+ 1)
c
3
, c
10
(u
20
+ 5u
19
+ ··· 4u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
8
, c
9
(y
20
23y
19
+ ··· 2y + 1)
c
3
, c
10
(y
20
+ y
19
+ ··· 46y + 1)
5