10
6
(K10a
70
)
1
Arc Sequences
6 8 9 10 7 1 5 3 4 2
Solving Sequence
2,6
1 7 5 8 3 10 4 9
c
1
c
6
c
5
c
7
c
2
c
10
c
4
c
9
c
3
, c
8
Representation Ideals
I = I
u
1
I
u
1
= hu
18
+ u
17
+ ··· u 1i
There are 1 irreducible components with 18 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
18
+ u
17
+ 3u
16
+ 2u
15
+ 8u
14
+ 5u
13
+ 13u
12
+ 6u
11
+ 17u
10
+
5u
9
+ 15u
8
+ 2u
7
+ 10u
6
2u
5
+ 2u
4
4u
3
u
2
u 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
u
3
+ u
a
5
=
u
3
u
5
+ u
3
+ u
a
8
=
u
5
+ u
u
7
+ u
5
+ 2u
3
+ u
a
3
=
u
12
+ u
10
+ 3u
8
+ 2u
6
+ 2u
4
+ u
2
+ 1
u
14
+ 2u
12
+ 5u
10
+ 6u
8
+ 6u
6
+ 4u
4
+ u
2
a
10
=
u
2
+ 1
u
2
a
4
=
u
9
+ 2u
7
+ 3u
5
+ 4u
3
+ u
u
9
+ u
7
+ 3u
5
+ 2u
3
+ u
a
9
=
u
16
3u
14
7u
12
12u
10
13u
8
12u
6
6u
4
+ 1
u
16
2u
14
6u
12
8u
10
10u
8
8u
6
4u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
16
+ 4u
15
+ 8u
14
+ 8u
13
+ 24u
12
+ 20u
11
+ 28u
10
+ 24u
9
+
36u
8
+ 20u
7
+ 20u
6
+ 12u
5
+ 8u
4
4u
3
8u
2
8u 10
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.855326 0.759946I
5.70958 2.31893I 7.86761 + 0.27178I
u = 0.855326 + 0.759946I
5.70958 + 2.31893I 7.86761 0.27178I
u = 0.798203 0.890045I
5.14256 + 2.99347I 2.16456 2.96884I
u = 0.798203 + 0.890045I
5.14256 2.99347I 2.16456 + 2.96884I
u = 0.774589 0.997585I
6.44242 + 8.39094I 9.04735 5.13904I
u = 0.774589 + 0.997585I
6.44242 8.39094I 9.04735 + 5.13904I
u = 0.479029
1.27899 7.23654
u = 0.261770 0.920605I
3.83985 + 2.54428I 13.6710 5.1939I
u = 0.261770 + 0.920605I
3.83985 2.54428I 13.6710 + 5.1939I
u = 0.211837 0.649664I
0.370697 0.965885I 6.45922 + 6.93392I
u = 0.211837 + 0.649664I
0.370697 + 0.965885I 6.45922 6.93392I
u = 0.272828 1.039357I
13.13097 3.24976I 13.8187 + 3.4932I
u = 0.272828 + 1.039357I
13.13097 + 3.24976I 13.8187 3.4932I
u = 0.703368
9.78395 7.88603
u = 0.779702 0.947695I
2.41083 6.44838I 7.16819 + 6.55335I
u = 0.779702 + 0.947695I
2.41083 + 6.44838I 7.16819 6.55335I
u = 0.813352 0.821748I
2.79760 + 0.47412I 6.24213 1.46151I
u = 0.813352 + 0.821748I
2.79760 0.47412I 6.24213 + 1.46151I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
18
+ u
17
+ ··· u 1)
c
2
, c
3
, c
4
c
8
, c
9
(u
18
+ u
17
+ ··· 3u 1)
c
5
, c
7
, c
10
(u
18
+ 5u
17
+ ··· + u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
18
+ 5y
17
+ ··· + y + 1)
c
2
, c
3
, c
4
c
8
, c
9
(y
18
23y
17
+ ··· + y + 1)
c
5
, c
7
, c
10
(y
18
+ 17y
17
+ ··· 23y + 1)
5