10
7
(K10a
65
)
1
Arc Sequences
7 8 10 9 1 2 6 5 4 3
Solving Sequence
1,7
2 6 8 3 5 10 4 9
c
1
c
6
c
7
c
2
c
5
c
10
c
3
c
9
c
4
, c
8
Representation Ideals
I = I
u
1
I
u
1
= hu
21
+ u
20
+ ··· + u + 1i
There are 1 irreducible components with 21 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
21
+ u
20
+ 6u
19
+ 5u
18
+ 17u
17
+ 13u
16
+ 28u
15
+ 20u
14
+ 28u
13
+
20u
12
+ 16u
11
+ 11u
10
+ 3u
9
+ u
8
2u
7
4u
6
u
5
u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
8
=
u
3
u
5
+ u
3
+ u
a
3
=
u
6
u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
5
=
u
3
u
3
+ u
a
10
=
u
14
+ 3u
12
+ 4u
10
+ u
8
2u
6
2u
4
+ 1
u
16
4u
14
8u
12
8u
10
4u
8
a
4
=
u
19
4u
17
8u
15
8u
13
5u
11
2u
9
2u
7
u
3
u
19
5u
17
12u
15
15u
13
9u
11
+ u
9
+ 4u
7
+ 2u
5
u
3
u
a
9
=
u
11
2u
9
2u
7
u
3
u
11
3u
9
4u
7
u
5
+ u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
19
4u
18
20u
17
20u
16
48u
15
52u
14
64u
13
76u
12
48u
11
64u
10
16u
9
16u
8
+ 4u
7
+ 16u
6
+ 8u
5
+ 16u
4
+ 4u
3
4u
2
4u 10
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.794642 0.241148I
10.29500 4.13640I 0.28719 + 2.17514I
u = 0.794642 + 0.241148I
10.29500 + 4.13640I 0.28719 2.17514I
u = 0.639263
1.31636 8.13788
u = 0.544516 1.163608I
7.57313 + 9.11591I 3.42568 5.67037I
u = 0.544516 + 1.163608I
7.57313 9.11591I 3.42568 + 5.67037I
u = 0.515219 0.758542I
3.44776 + 2.10610I 0.68965 4.22092I
u = 0.515219 + 0.758542I
3.44776 2.10610I 0.68965 + 4.22092I
u = 0.448707 1.150096I
4.43097 + 4.04104I 10.76568 4.27407I
u = 0.448707 + 1.150096I
4.43097 4.04104I 10.76568 + 4.27407I
u = 0.297476 1.182770I
5.89549 0.72644I 5.47305 0.34896I
u = 0.297476 + 1.182770I
5.89549 + 0.72644I 5.47305 + 0.34896I
u = 0.199725 0.739431I
0.474299 1.026505I 6.88729 + 6.49406I
u = 0.199725 + 0.739431I
0.474299 + 1.026505I 6.88729 6.49406I
u = 0.375476 1.140926I
2.38679 0.77154I 6.91276 0.81413I
u = 0.375476 + 1.140926I
2.38679 + 0.77154I 6.91276 + 0.81413I
u = 0.504141 1.153182I
1.47889 7.30035I 5.16109 + 7.23595I
u = 0.504141 + 1.153182I
1.47889 + 7.30035I 5.16109 7.23595I
u = 0.631235 0.777388I
12.97592 2.44340I 0.84460 + 3.15661I
u = 0.631235 + 0.777388I
12.97592 + 2.44340I 0.84460 3.15661I
u = 0.709616 0.181075I
1.31805 + 2.71325I 1.55258 3.99913I
u = 0.709616 + 0.181075I
1.31805 2.71325I 1.55258 + 3.99913I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
21
+ u
20
+ ··· + u + 1)
c
2
, c
5
(u
21
+ u
20
+ ··· + u 5)
c
3
, c
4
, c
8
c
9
, c
10
(u
21
+ u
20
+ ··· + u 1)
c
7
(u
21
+ 11u
20
+ ··· u 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
21
+ 11y
20
+ ··· y 1)
c
2
, c
5
(y
21
13y
20
+ ··· 69y 25)
c
3
, c
4
, c
8
c
9
, c
10
(y
21
+ 27y
20
+ ··· y 1)
c
7
(y
21
y
20
+ ··· + 11y 1)
5