10
129
(K10n
18
)
1
Arc Sequences
4 9 5 2 10 9 5 3 6 7
Solving Sequence
5,10 2,6
4 1 3 9 7 8
c
5
c
4
c
1
c
3
c
9
c
6
c
7
c
2
, c
8
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
3
a
2
+ 2a 1, u 1, b a + 1i
I
u
2
= hu
15
+ 4u
14
+ 7u
13
+ u
12
13u
11
20u
10
3u
9
+ 15u
8
+ 18u
7
+ 7u
6
+ 6u
5
5u
3
6u
2
3u 1,
u
14
+ 3u
13
+ 4u
12
3u
11
10u
10
10u
9
+ 7u
8
+ 8u
7
+ 10u
6
3u
5
+ 9u
4
5u
3
+ 4u
2
+ 4b 6u 1,
u
14
+ 3u
13
+ 4u
12
3u
11
10u
10
10u
9
+ 7u
8
+ 8u
7
+ 10u
6
3u
5
+ 9u
4
5u
3
+ 4u
2
+ 4a 10u 1i
There are 2 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
= ha
3
a
2
+ 2a 1, u 1, b a + 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
a
a 1
a
2
=
1
0
a
6
=
a
2
a
2
a + 1
a
4
=
1
1
a
1
=
0
1
a
3
=
1
0
a
9
=
a
2
a + 1
0
a
7
=
a
2
a + 1
a
2
a + 1
a
8
=
a
2
a + 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5a
2
4a + 4
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.215080 1.307141I
b = 0.78492 1.30714I
4.66906 2.82812I 5.17211 + 2.41717I
u = 1.00000
a = 0.215080 + 1.307141I
b = 0.78492 + 1.30714I
4.66906 + 2.82812I 5.17211 2.41717I
u = 1.00000
a = 0.569840
b = 0.430160
0.531480 3.34423
3
II.
I
u
2
= hu
15
+4u
14
+· · ·3u1, u
14
+3u
13
+· · ·+4b1, u
14
+3u
13
+· · ·+4a1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
4
u
14
3
4
u
13
+ ··· +
5
2
u +
1
4
1
4
u
14
3
4
u
13
+ ··· +
3
2
u +
1
4
a
2
=
1
0
a
6
=
3
2
u
14
5u
13
+ ··· +
7
2
u + 2
5
4
u
14
17
4
u
13
+ ··· + 4u +
7
4
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
u
3
+ u
a
9
=
1
4
u
14
+
1
4
u
13
+ ··· + 2u +
5
4
1
2
u
14
+
3
2
u
13
+ ··· u
2
+
1
2
a
7
=
1
4
u
14
+
9
4
u
13
+ ··· 4u
7
4
1
4
u
14
1
4
u
13
+ ··· +
1
2
u
2
1
4
a
8
=
1
4
u
14
+
9
4
u
13
+ ··· 4u
7
4
3
2
u
14
+
11
2
u
13
+ ··· 4u
3
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
14
+
33
2
u
13
+ 22u
12
15u
11
125
2
u
10
115
2
u
9
+
73
2
u
8
+ 72u
7
+
46u
6
8u
5
+
23
2
u
4
17u
3
43
2
u
2
35
2
u
3
2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.13244 0.99333I
a = 0.363842 + 0.863198I
b = 1.49628 + 1.85653I
6.09422 8.90152I 0.37309 + 5.02376I
u = 1.13244 + 0.99333I
a = 0.363842 0.863198I
b = 1.49628 1.85653I
6.09422 + 8.90152I 0.37309 5.02376I
u = 1.05231 1.07767I
a = 0.411245 + 0.477015I
b = 0.64106 + 1.55469I
10.46561 3.92970I 3.25200 + 2.37642I
u = 1.05231 + 1.07767I
a = 0.411245 0.477015I
b = 0.64106 1.55469I
10.46561 + 3.92970I 3.25200 2.37642I
u = 0.92821 1.13080I
a = 0.872830 0.284735I
b = 0.055378 + 0.846062I
6.78648 + 1.17157I 0.521469 0.840506I
u = 0.92821 + 1.13080I
a = 0.872830 + 0.284735I
b = 0.055378 0.846062I
6.78648 1.17157I 0.521469 + 0.840506I
u = 0.560305 0.345696I
a = 0.83599 1.92579I
b = 0.27569 1.58010I
3.65536 3.51330I 0.20706 + 4.67402I
u = 0.560305 + 0.345696I
a = 0.83599 + 1.92579I
b = 0.27569 + 1.58010I
3.65536 + 3.51330I 0.20706 4.67402I
u = 0.195944 0.500014I
a = 0.066829 1.182872I
b = 0.262773 0.682859I
1.37013 0.70150I 5.29100 + 2.23884I
u = 0.195944 + 0.500014I
a = 0.066829 + 1.182872I
b = 0.262773 + 0.682859I
1.37013 + 0.70150I 5.29100 2.23884I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.282237 0.716387I
a = 1.22334 0.86060I
b = 0.941101 0.144211I
1.32042 + 2.58137I 0.00443 4.00241I
u = 0.282237 + 0.716387I
a = 1.22334 + 0.86060I
b = 0.941101 + 0.144211I
1.32042 2.58137I 0.00443 + 4.00241I
u = 0.877160
a = 0.259666
b = 0.617494
1.26612 9.41313
u = 1.148384 0.278021I
a = 0.163777 + 0.959972I
b = 1.31216 + 1.23799I
4.30318 + 1.14653I 3.69630 + 0.14216I
u = 1.148384 + 0.278021I
a = 0.163777 0.959972I
b = 1.31216 1.23799I
4.30318 1.14653I 3.69630 0.14216I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
3
(u
15
+ 4u
14
+ ··· 3u 1)
c
2
, c
8
u
3
(u
15
+ u
14
+ ··· + 12u + 8)
c
3
(u 1)
3
(u
15
+ 2u
14
+ ··· 3u + 1)
c
4
(u + 1)
3
(u
15
+ 4u
14
+ ··· 3u 1)
c
5
, c
6
(u
3
+ u
2
+ 2u + 1)(u
15
+ 2u
14
+ ··· + 4u + 1)
c
7
(u
3
+ u
2
1)(u
15
+ 8u
14
+ ··· + 280u 49)
c
9
(u
3
u
2
+ 2u 1)(u
15
+ 2u
14
+ ··· + 4u + 1)
c
10
(u
3
+ u
2
1)(u
15
+ 2u
14
+ ··· + 2u 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
3
(y
15
2y
14
+ ··· 3y 1)
c
2
, c
8
y
3
(y
15
+ 21y
14
+ ··· 48y 64)
c
3
(y 1)
3
(y
15
+ 26y
14
+ ··· 3y 1)
c
5
, c
6
, c
9
(y
3
+ 3y
2
+ 2y 1)(y
15
+ 12y
14
+ ··· + 20y 1)
c
7
(y
3
y
2
+ 2y 1)(y
15
32y
14
+ ··· + 220108y 2401)
c
10
(y
3
y
2
+ 2y 1)(y
15
20y
14
+ ··· + 20y 1)
8