9
16
(K9a
25
)
1
Arc Sequences
5 6 8 3 2 9 1 4 7
Solving Sequence
6,9
7
1,4
8 3 2 5
c
6
c
9
c
8
c
3
c
2
c
5
c
1
, c
4
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu 1, b, a + 1i
I
u
2
= hu
8
u
7
+ 2u
6
+ 5u
5
+ u
4
u
3
+ 7u
2
+ 3u 1,
3u
7
+ 12u
6
26u
5
+ 15u
4
+ 16u
3
21u
2
+ 40a 46u + 73,
3u
7
12u
6
+ 26u
5
15u
4
16u
3
+ 21u
2
+ 40b + 46u 33i
I
u
3
= hu
12
5u
11
+ 10u
10
12u
9
+ 29u
8
89u
7
+ 159u
6
147u
5
+ 53u
4
+ 7u
3
+ 4u
2
18u + 9,
903311u
11
4195622u
10
+ ··· + 8975451b 5924742,
6114722u
11
23293684u
10
+ ··· + 26926353a 34182915i
There are 3 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 1, b, a + 1i
(i) Arc colorings
a
6
=
0
1
a
9
=
1
0
a
7
=
1
1
a
1
=
0
1
a
4
=
1
0
a
8
=
1
0
a
3
=
1
0
a
2
=
1
1
a
5
=
1
0
a
5
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
3.28987 12.0000
3
II. I
u
2
= hu
8
u
7
+ 2u
6
+ 5u
5
+ u
4
u
3
+ 7u
2
+ 3u 1, 3u
7
+ 12u
6
+
· · · + 40a + 73, 3u
7
12u
6
+ · · · + 40b 33i
(i) Arc colorings
a
6
=
0
u
a
9
=
0.0750000u
7
0.300000u
6
+ ··· + 1.15000u 1.82500
0.0750000u
7
+ 0.300000u
6
+ ··· 1.15000u + 0.825000
a
7
=
3
4
u
7
+
6
5
u
6
+ ···
47
10
u +
9
20
21
40
u
7
7
10
u
6
+ ··· +
73
20
u
3
8
a
1
=
7
10
u
7
+ u
6
+ ···
28
5
u
1
10
21
40
u
7
7
10
u
6
+ ··· +
73
20
u
3
8
a
4
=
1
0
a
8
=
1
0.0750000u
7
+ 0.300000u
6
+ ··· 1.15000u + 0.825000
a
3
=
0.0750000u
7
+ 0.300000u
6
+ ··· 1.15000u + 1.82500
3
10
u
7
3
5
u
6
+ ··· + 2u
7
10
a
2
=
0.0750000u
7
+ 0.300000u
6
+ ··· 1.15000u + 1.82500
23
40
u
7
9
10
u
6
+ ··· +
11
4
u
37
40
a
5
=
3
4
u
7
6
5
u
6
+ ··· +
47
10
u
9
20
3
5
u
7
+ u
6
+ ···
24
5
u +
6
5
a
5
=
3
4
u
7
6
5
u
6
+ ··· +
47
10
u
9
20
3
5
u
7
+ u
6
+ ···
24
5
u +
6
5
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
7
2u
6
+ 4u
5
+ u
4
u
2
+ 4u 15
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.003721 0.582684I
a = 0.178387 0.567011I
b = 0.821613 + 0.567011I
1.48505 + 2.26376I 5.94128 4.53378I
u = 1.003721 + 0.582684I
a = 0.178387 + 0.567011I
b = 0.821613 0.567011I
1.48505 2.26376I 5.94128 + 4.53378I
u = 0.631839
a = 2.38845
b = 1.38845
13.4445 18.3375
u = 0.221111
a = 1.54996
b = 0.549965
0.719034 14.1601
u = 0.669498 0.907562I
a = 0.567656 1.079148I
b = 0.432344 + 1.079148I
6.22518 3.55755I 14.5274 + 2.6249I
u = 0.669498 + 0.907562I
a = 0.567656 + 1.079148I
b = 0.432344 1.079148I
6.22518 + 3.55755I 14.5274 2.6249I
u = 1.03959 1.75991I
a = 0.215252 + 0.684012I
b = 1.215252 0.684012I
8.73978 9.88301I 15.2825 + 6.0696I
u = 1.03959 + 1.75991I
a = 0.215252 0.684012I
b = 1.215252 + 0.684012I
8.73978 + 9.88301I 15.2825 6.0696I
5
III.
I
u
3
= hu
12
5u
11
+ · · · 18u + 9 , 9.03 × 10
5
u
11
4.20 × 10
6
u
10
+ · · · + 8.98 ×
10
6
b 5.92 ×10
6
, 6.11× 10
6
u
11
2.33× 10
7
u
10
+· · ·+ 2 .69 ×10
7
a 3.42 ×10
7
i
(i) Arc colorings
a
6
=
0
u
a
9
=
0.227091u
11
+ 0.865089u
10
+ ··· 3.11196u + 1.26950
0.100642u
11
+ 0.467455u
10
+ ··· 2.36989u + 0.660105
a
7
=
0.214190u
11
+ 0.914563u
10
+ ··· 3.09508u + 2.07366
0.0943647u
11
+ 0.434381u
10
+ ··· 1.57801u + 0.800130
a
1
=
0.0785567u
11
+ 0.165798u
10
+ ··· 1.03587u 0.914502
0.0943647u
11
+ 0.434381u
10
+ ··· 1.57801u + 0.800130
a
4
=
1
0
a
8
=
0.327733u
11
+ 1.33254u
10
+ ··· 5.48185u + 1.92960
0.100642u
11
+ 0.467455u
10
+ ··· 2.36989u + 0.660105
a
3
=
0.315889u
11
+ 1.16664u
10
+ ··· 2.61859u + 0.729262
0.226986u
11
+ 0.816487u
10
+ ··· 2.32852u + 0.707010
a
2
=
0.315889u
11
+ 1.16664u
10
+ ··· 2.61859u + 0.729262
0.151084u
11
0.801806u
10
+ ··· + 2.25899u 3.00825
a
5
=
0.191123u
11
+ 0.894945u
10
+ ··· 1.91215u + 4.12370
0.0252206u
11
+ 0.167175u
10
+ ··· + 0.0554509u + 1.17407
a
5
=
0.191123u
11
+ 0.894945u
10
+ ··· 1.91215u + 4.12370
0.0252206u
11
+ 0.167175u
10
+ ··· + 0.0554509u + 1.17407
(ii) Obstruction class = 1
(iii) Cusp Shapes =
4222636
2991817
u
11
+
52097752
8975451
u
10
+ ···
55563596
2991817
u +
492894
2991817
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.36462 1.50351I
a = 0.350390 + 0.583945I
b = 1.073950 0.558752I
3.28987 + 5.69302I 12.00000 5.51057I
u = 1.36462 + 1.50351I
a = 0.350390 0.583945I
b = 1.073950 + 0.558752I
3.28987 5.69302I 12.00000 + 5.51057I
u = 0.452270 0.361855I
a = 1.35006 1.50455I
b = 0.428243 + 0.664531I
1.39926 + 0.92430I 8.28328 0.79423I
u = 0.452270 + 0.361855I
a = 1.35006 + 1.50455I
b = 0.428243 0.664531I
1.39926 0.92430I 8.28328 + 0.79423I
u = 0.601198 0.542039I
a = 1.255408 0.259181I
b = 1.002193 0.295542I
5.18047 0.92430I 15.7167 + 0.7942I
u = 0.601198 + 0.542039I
a = 1.255408 + 0.259181I
b = 1.002193 + 0.295542I
5.18047 + 0.92430I 15.7167 0.7942I
u = 0.94950 1.12226I
a = 0.306433 0.510689I
b = 1.073950 + 0.558752I
3.28987 5.69302I 12.00000 + 5.51057I
u = 0.94950 + 1.12226I
a = 0.306433 + 0.510689I
b = 1.073950 0.558752I
3.28987 + 5.69302I 12.00000 5.51057I
u = 1.108238 0.263574I
a = 0.031505 + 0.447360I
b = 0.428243 0.664531I
1.39926 0.92430I 8.28328 + 0.79423I
u = 1.108238 + 0.263574I
a = 0.031505 0.447360I
b = 0.428243 + 0.664531I
1.39926 + 0.92430I 8.28328 0.79423I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.65795 0.89021I
a = 0.584465 + 0.398183I
b = 1.002193 0.295542I
5.18047 0.92430I 15.7167 + 0.7942I
u = 1.65795 + 0.89021I
a = 0.584465 0.398183I
b = 1.002193 + 0.295542I
5.18047 + 0.92430I 15.7167 0.7942I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
6
c
7
(u 1)(u
8
+ u
7
4u
6
3u
5
+ 5u
4
+ u
3
u
2
+ 3u 1)
(u
12
+ u
11
4u
10
2u
9
+ 7u
8
u
7
5u
6
+ 5u
5
u
4
3u
3
+ 2u
2
+ 1)
c
3
, c
8
u(u
6
u
5
u
4
+ 2u
3
u + 1)
2
(u
8
+ 3u
7
+ 3u
6
2u
5
8u
4
9u
3
3u
2
+ 2u + 2)
c
4
u(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
2
(u
8
+ 3u
7
+ 5u
6
+ 4u
5
+ 2u
4
+ 13u
3
+ 13u
2
+ 16u + 4)
c
5
, c
9
(u + 1)(u
8
+ u
7
4u
6
3u
5
+ 5u
4
+ u
3
u
2
+ 3u 1)
(u
12
+ u
11
4u
10
2u
9
+ 7u
8
u
7
5u
6
+ 5u
5
u
4
3u
3
+ 2u
2
+ 1)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
5
c
6
, c
7
, c
9
(y 1)(y
8
9y
7
+ 32y
6
53y
5
+ 31y
4
+ 15y
3
15y
2
7y + 1)
(y
12
9y
11
+ ··· + 4y + 1)
c
3
, c
8
y(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
2
(y
8
3y
7
+ 5y
6
4y
5
+ 2y
4
13y
3
+ 13y
2
16y + 4)
c
4
y(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
(y
8
+ y
7
+ 5y
6
48y
5
58y
4
205y
3
231y
2
152y + 16)
10