10
18
(K10a
63
)
1
Arc Sequences
7 5 9 8 10 1 6 4 3 2
Solving Sequence
1,7
2 6 8 10 5 3 4 9
c
1
c
6
c
7
c
10
c
5
c
2
c
4
c
9
c
3
, c
8
Representation Ideals
I = I
u
1
I
u
1
= hu
27
u
26
+ ··· + u
2
+ 1i
There are 1 irreducible components with 27 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
27
u
26
+ · · · + u
2
+ 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
2
=
1
u
2
a
6
=
u
u
a
8
=
u
3
u
3
+ u
a
10
=
u
2
+ 1
u
4
a
5
=
u
7
+ 2u
5
2u
3
+ 2u
u
9
+ u
7
u
5
+ u
a
3
=
u
14
3u
12
+ 6u
10
9u
8
+ 8u
6
6u
4
+ 2u
2
+ 1
u
16
2u
14
+ 4u
12
4u
10
+ 2u
8
2u
4
+ 2u
2
a
4
=
u
15
2u
13
+ 4u
11
4u
9
+ 2u
7
2u
3
+ 2u
u
15
3u
13
+ 6u
11
9u
9
+ 8u
7
6u
5
+ 2u
3
+ u
a
9
=
u
26
+ 5u
24
+ ··· + u
2
+ 1
u
26
+ u
25
+ ··· + u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
25
16u
23
+ 4u
22
+ 48u
21
12u
20
96u
19
+ 36u
18
+ 152u
17
64u
16
188u
15
+ 100u
14
+ 184u
13
120u
12
140u
11
+ 116u
10
+ 76u
9
96u
8
20u
7
+
56u
6
8u
5
28u
4
+ 12u
3
+ 8u
2
4u 2
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.074428 0.112584I
11.51840 4.62424I 10.86711 + 3.60523I
u = 1.074428 + 0.112584I
11.51840 + 4.62424I 10.86711 3.60523I
u = 0.995157 0.708453I
1.37783 8.19998I 2.79147 + 8.55054I
u = 0.995157 + 0.708453I
1.37783 + 8.19998I 2.79147 8.55054I
u = 0.904725
1.66811 4.57266
u = 0.899171 0.605220I
0.95481 2.34352I 6.62935 + 2.39389I
u = 0.899171 + 0.605220I
0.95481 + 2.34352I 6.62935 2.39389I
u = 0.863222 0.756020I
1.89158 2.85128I 2.36117 + 2.96428I
u = 0.863222 + 0.756020I
1.89158 + 2.85128I 2.36117 2.96428I
u = 0.700843 0.775200I
2.26803 + 2.57835I 0.81917 3.65038I
u = 0.700843 + 0.775200I
2.26803 2.57835I 0.81917 + 3.65038I
u = 0.173757 0.447464I
0.041447 1.170264I 0.65568 + 5.80154I
u = 0.173757 + 0.447464I
0.041447 + 1.170264I 0.65568 5.80154I
u = 0.270002 0.632457I
7.20164 + 2.51533I 4.12254 2.69602I
u = 0.270002 + 0.632457I
7.20164 2.51533I 4.12254 + 2.69602I
u = 0.660859 0.808474I
5.18836 4.92710I 3.80267 + 2.17668I
u = 0.660859 + 0.808474I
5.18836 + 4.92710I 3.80267 2.17668I
u = 0.759347 0.744821I
3.27525 + 1.04588I 2.08117 3.01333I
u = 0.759347 + 0.744821I
3.27525 1.04588I 2.08117 + 3.01333I
u = 0.955694 0.704714I
2.67334 + 4.47788I 0.69991 3.02325I
u = 0.955694 + 0.704714I
2.67334 4.47788I 0.69991 + 3.02325I
u = 0.984154 0.531838I
9.06338 + 1.66777I 8.35861 2.79123I
u = 0.984154 + 0.531838I
9.06338 1.66777I 8.35861 + 2.79123I
u = 1.005609 0.098775I
3.62827 + 2.81912I 9.45302 5.56399I
u = 1.005609 + 0.098775I
3.62827 2.81912I 9.45302 + 5.56399I
u = 1.023276 0.710146I
6.28352 + 10.63984I 5.63394 6.90100I
u = 1.023276 + 0.710146I
6.28352 10.63984I 5.63394 + 6.90100I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
27
+ u
26
+ ··· u
2
1)
c
2
(u
27
+ 7u
26
+ ··· + 8u + 1)
c
3
, c
4
, c
8
c
9
(u
27
+ u
26
+ ··· + 2u + 1)
c
5
(u
27
+ u
26
+ ··· + 8u + 4)
c
7
, c
10
(u
27
+ 9u
26
+ ··· 2u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
27
9y
26
+ ··· 2y 1)
c
2
(y
27
y
26
+ ··· 34y 1)
c
3
, c
4
, c
8
c
9
(y
27
+ 31y
26
+ ··· 2y 1)
c
5
(y
27
5y
26
+ ··· + 56y 16)
c
7
, c
10
(y
27
+ 19y
26
+ ··· 2y 1)
5