10
25
(K10a
61
)
1
Arc Sequences
6 8 7 10 1 3 4 2 5 9
Solving Sequence
5,10
4 9 1 6 2 8 7 3
c
4
c
9
c
10
c
5
c
1
c
8
c
7
c
3
c
2
, c
6
Representation Ideals
I = I
u
1
I
u
1
= hu
32
+ u
31
+ ··· 2u 1i
There are 1 irreducible components with 32 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
32
+ u
31
+ · · · 2u 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
9
=
u
u
a
1
=
u
3
u
3
+ u
a
6
=
u
6
u
4
+ 1
u
6
+ 2u
4
+ u
2
a
2
=
u
9
+ 2u
7
+ u
5
2u
3
u
u
9
3u
7
3u
5
+ u
a
8
=
u
17
4u
15
7u
13
4u
11
+ 3u
9
+ 6u
7
+ 2u
5
u
u
17
+ 5u
15
+ 11u
13
+ 12u
11
+ 5u
9
2u
7
2u
5
+ u
a
7
=
u
19
+ 4u
17
+ 8u
15
+ 8u
13
+ 5u
11
+ 2u
9
+ 2u
7
+ u
3
u
21
+ 5u
19
+ 13u
17
+ 20u
15
+ 20u
13
+ 11u
11
+ u
9
4u
7
u
5
+ u
3
+ u
a
3
=
u
25
+ 6u
23
+ ··· 2u
3
u
u
25
7u
23
+ ··· 5u
5
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
30
+ 4u
29
+ 32u
28
+ 32u
27
+ 124u
26
+ 128u
25
+ 292u
24
+ 316u
23
+ 448u
22
+ 516u
21
+
440u
20
+ 540u
19
+ 232u
18
+ 292u
17
20u
16
64u
15
140u
14
232u
13
108u
12
144u
11
24u
10
+ 16u
9
+ 28u
8
+ 64u
7
+ 24u
6
+ 28u
5
+ 8u
4
12u
3
8u
2
12u 14
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.792800 0.172177I
3.13584 7.01747I 7.66223 + 4.88322I
u = 0.792800 + 0.172177I
3.13584 + 7.01747I 7.66223 4.88322I
u = 0.649942 0.248644I
1.32933 + 0.52783I 5.59448 0.64788I
u = 0.649942 + 0.248644I
1.32933 0.52783I 5.59448 + 0.64788I
u = 0.605013
1.22821 8.26166
u = 0.561289 0.769750I
4.01456 + 2.24194I 0.65690 3.79727I
u = 0.561289 + 0.769750I
4.01456 2.24194I 0.65690 + 3.79727I
u = 0.521034 1.182057I
6.10646 + 11.87578I 10.77954 7.99531I
u = 0.521034 + 1.182057I
6.10646 11.87578I 10.77954 + 7.99531I
u = 0.492704 1.133861I
3.89830 + 3.89503I 9.35061 2.90091I
u = 0.492704 + 1.133861I
3.89830 3.89503I 9.35061 + 2.90091I
u = 0.433982 1.139381I
4.28206 + 3.88889I 10.89128 4.90467I
u = 0.433982 + 1.139381I
4.28206 3.88889I 10.89128 + 4.90467I
u = 0.357265 1.197706I
7.25067 3.23058I 12.64791 + 1.85611I
u = 0.357265 + 1.197706I
7.25067 + 3.23058I 12.64791 1.85611I
u = 0.180753 1.016976I
5.00599 + 2.81562I 13.51638 3.82546I
u = 0.180753 + 1.016976I
5.00599 2.81562I 13.51638 + 3.82546I
u = 0.192477 0.755088I
0.501058 1.034985I 7.18759 + 6.41402I
u = 0.192477 + 0.755088I
0.501058 + 1.034985I 7.18759 6.41402I
u = 0.362087 1.159287I
2.34434 0.39737I 7.83598 0.58140I
u = 0.362087 + 1.159287I
2.34434 + 0.39737I 7.83598 + 0.58140I
u = 0.450235 1.200345I
10.82667 4.39858I 14.8085 + 3.5355I
u = 0.450235 + 1.200345I
10.82667 + 4.39858I 14.8085 3.5355I
u = 0.514933 1.164397I
1.27472 7.88151I 6.19556 + 6.68910I
u = 0.514933 + 1.164397I
1.27472 + 7.88151I 6.19556 6.68910I
u = 0.565288 0.826638I
0.06115 6.17510I 5.73067 + 6.90538I
u = 0.565288 + 0.826638I
0.06115 + 6.17510I 5.73067 6.90538I
u = 0.570562 0.700867I
0.29651 + 1.65231I 4.59303 0.15309I
u = 0.570562 + 0.700867I
0.29651 1.65231I 4.59303 + 0.15309I
u = 0.747372 0.188735I
1.56622 + 3.15266I 2.67728 3.41480I
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.747372 + 0.188735I
1.56622 3.15266I 2.67728 + 3.41480I
u = 0.778647
7.31963 11.4825
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
32
+ u
31
+ ··· 14u 5)
c
2
, c
8
(u
32
+ 3u
31
+ ··· 4u
4
+ 1)
c
3
, c
6
, c
7
(u
32
+ u
31
+ ··· 2u 1)
c
4
, c
9
(u
32
+ u
31
+ ··· 2u 1)
c
10
(u
32
+ 17u
31
+ ··· 8u
2
+ 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
32
23y
31
+ ··· 296y + 25)
c
2
, c
8
(y
32
+ 17y
31
+ ··· 8y
2
+ 1)
c
3
, c
6
, c
7
(y
32
27y
31
+ ··· + 16y
2
+ 1)
c
4
, c
9
(y
32
+ 17y
31
+ ··· 8y
2
+ 1)
c
10
(y
32
3y
31
+ ··· 16y + 1)
5