10
36
(K10a
5
)
1
Arc Sequences
8 7 6 9 4 3 10 1 5 2
Solving Sequence
1,9
8 2 10 7 3 6 4 5
c
8
c
1
c
10
c
7
c
2
c
6
c
3
c
5
c
4
, c
9
Representation Ideals
I = I
u
1
I
u
1
= hu
25
u
24
+ ··· + 3u 1i
There are 1 irreducible components with 25 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
25
u
24
+ · · · + 3u 1i
(i) Arc colorings
a
1
=
1
0
a
9
=
0
u
a
8
=
u
u
a
2
=
u
2
+ 1
u
2
a
10
=
u
4
+ u
2
+ 1
u
4
a
7
=
u
7
+ 2u
5
+ 2u
3
u
7
u
5
+ u
a
3
=
u
12
+ 3u
10
+ 5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1
u
12
2u
10
2u
8
+ u
4
a
6
=
u
17
+ 4u
15
+ 9u
13
+ 12u
11
+ 11u
9
+ 8u
7
+ 6u
5
+ 4u
3
+ u
u
17
3u
15
5u
13
4u
11
u
9
+ u
a
4
=
u
22
+ 5u
20
+ ··· + 2u
2
+ 1
u
22
4u
20
9u
18
12u
16
10u
14
6u
12
3u
10
2u
8
+ u
6
+ 2u
4
+ u
2
a
5
=
u
22
+ 5u
20
+ ··· + 2u
2
+ 1
u
24
+ 4u
22
+ ··· + 4u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
23
4u
22
+ 20u
21
20u
20
+ 56u
19
60u
18
+ 104u
17
116u
16
+
144u
15
164u
14
+ 168u
13
180u
12
+ 172u
11
168u
10
+ 156u
9
148u
8
+ 112u
7
108u
6
+ 68u
5
60u
4
+ 40u
3
20u
2
+ 16u 14
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.811783 0.502716I
11.15258 + 1.58500I 0.08176 2.23225I
u = 0.811783 + 0.502716I
11.15258 1.58500I 0.08176 + 2.23225I
u = 0.642433 1.078133I
9.42796 + 3.86019I 2.25009 2.37671I
u = 0.642433 + 1.078133I
9.42796 3.86019I 2.25009 + 2.37671I
u = 0.607687 0.546225I
2.49752 + 1.43161I 0.07046 3.44213I
u = 0.607687 + 0.546225I
2.49752 1.43161I 0.07046 + 3.44213I
u = 0.560222 1.000381I
1.18303 + 3.19832I 2.28028 2.80466I
u = 0.560222 + 1.000381I
1.18303 3.19832I 2.28028 + 2.80466I
u = 0.414965 0.875362I
0.32971 + 1.74239I 2.38307 3.79759I
u = 0.414965 + 0.875362I
0.32971 1.74239I 2.38307 + 3.79759I
u = 0.017130 1.141397I
5.28985 + 3.20690I 5.88987 2.45318I
u = 0.017130 + 1.141397I
5.28985 3.20690I 5.88987 + 2.45318I
u = 0.230766 0.989612I
2.08350 + 1.12769I 10.19939 3.41549I
u = 0.230766 + 0.989612I
2.08350 1.12769I 10.19939 + 3.41549I
u = 0.422134
1.05962 9.24235
u = 0.425744 1.036929I
3.47336 3.28459I 12.75115 + 5.14665I
u = 0.425744 + 1.036929I
3.47336 + 3.28459I 12.75115 5.14665I
u = 0.551126 1.064191I
0.01805 7.68313I 5.93165 + 8.92800I
u = 0.551126 + 1.064191I
0.01805 + 7.68313I 5.93165 8.92800I
u = 0.636918 1.090251I
9.20201 10.47619I 2.72320 + 7.02847I
u = 0.636918 + 1.090251I
9.20201 + 10.47619I 2.72320 7.02847I
u = 0.647056 0.395539I
1.88922 + 3.01264I 1.96862 4.46588I
u = 0.647056 + 0.395539I
1.88922 3.01264I 1.96862 + 4.46588I
u = 0.817283 0.481210I
11.02795 + 5.03718I 0.15373 2.54574I
u = 0.817283 + 0.481210I
11.02795 5.03718I 0.15373 + 2.54574I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
8
(u
25
+ u
24
+ ··· + 3u + 1)
c
2
, c
3
, c
5
c
6
(u
25
+ 5u
24
+ ··· + u + 1)
c
4
, c
9
(u
25
+ u
24
+ ··· + u 1)
c
7
(u
25
+ u
24
+ ··· 5u 2)
c
10
(u
25
+ 11u
24
+ ··· + u 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
(y
25
+ 11y
24
+ ··· + y 1)
c
2
, c
3
, c
5
c
6
(y
25
+ 31y
24
+ ··· + 5y 1)
c
4
, c
9
(y
25
5y
24
+ ··· + y 1)
c
7
(y
25
+ 3y
24
+ ··· 31y 4)
c
10
(y
25
+ 7y
24
+ ··· + 21y 1)
5