9
45
(K9n
2
)
1
Arc Sequences
6 9 1 7 1 4 6 3 8
Solving Sequence
3,9 2,6
1 4 8 7 5
c
2
c
1
c
3
c
8
c
7
c
4
c
5
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
2
b + 1, u + 1, b + a + 1i
I
u
2
= hu
13
3u
12
+ 13u
11
30u
10
+ 64u
9
113u
8
+ 152u
7
175u
6
+ 161u
5
84u
4
+ 51u
3
28u
2
+ 8u 1,
19816939u
12
55156282u
11
+ ··· + 33713672a + 29935399,
19816939u
12
55156282u
11
+ ··· + 33713672b + 63649071i
There are 2 irreducible components with 15 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
= hb
2
b + 1, u + 1, b + a + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
b 1
b
a
2
=
0
b 1
a
6
=
0
1
a
1
=
0
b 1
a
4
=
1
b
a
8
=
1
b
a
7
=
1
b 1
a
5
=
0
1
a
5
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b + 11
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.500000 0.866025I
b = 0.500000 0.866025I
1.64493 2.02988I 9.00000 + 3.46410I
u = 1.00000
a = 0.500000 + 0.866025I
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 9.00000 3.46410I
3
II.
I
u
2
= hu
13
3u
12
+ · · · + 8u 1, 1.98 × 10
7
u
12
5.52 × 10
7
u
11
+ · · · + 3.37 ×
10
7
a +2.99 ×10
7
, 1.98× 10
7
u
12
5.52× 10
7
u
11
+· · ·+ 3.37 ×10
7
b +6.36 ×10
7
i
(i) Arc colorings
a
3
=
1
0
a
9
=
0.587801u
12
+ 1.63602u
11
+ ··· + 13.0224u 0.887931
0.587801u
12
+ 1.63602u
11
+ ··· + 13.0224u 1.88793
a
2
=
0.985528u
12
+ 2.84002u
11
+ ··· + 23.9193u 3.39191
0.397727u
12
+ 1.20399u
11
+ ··· + 10.8970u 2.50398
a
6
=
0
u
a
1
=
0.985528u
12
+ 2.84002u
11
+ ··· + 23.9193u 3.39191
0.326785u
12
+ 1.05153u
11
+ ··· + 10.8440u 2.62055
a
4
=
1.09658u
12
2.63968u
11
+ ··· 10.3426u + 0.0632682
1.23787u
12
3.35601u
11
+ ··· 21.7317u + 2.98451
a
8
=
1
0.587801u
12
+ 1.63602u
11
+ ··· + 13.0224u 1.88793
a
7
=
1
0.587801u
12
+ 1.63602u
11
+ ··· + 13.0224u 1.88793
a
5
=
0.446517u
12
0.919696u
11
+ ··· 1.63319u 1.03331
0.650065u
12
1.71998u
11
+ ··· 10.7094u + 1.09658
a
5
=
0.446517u
12
0.919696u
11
+ ··· 1.63319u 1.03331
0.650065u
12
1.71998u
11
+ ··· 10.7094u + 1.09658
(ii) Obstruction class = 1
(iii) Cusp Shapes =
52897393
8428418
u
12
70999684
4214209
u
11
+ ···
924453227
8428418
u +
183056483
8428418
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.45450 1.78039I
a = 0.58696 1.30049I
b = 0.413040 1.300494I
8.78028 + 1.38297I 1.065751 0.716223I
u = 0.45450 + 1.78039I
a = 0.58696 + 1.30049I
b = 0.413040 + 1.300494I
8.78028 1.38297I 1.065751 + 0.716223I
u = 0.247334 0.568423I
a = 1.27822 1.06780I
b = 0.278216 1.067795I
1.60812 2.52293I 1.64572 + 4.38707I
u = 0.247334 + 0.568423I
a = 1.27822 + 1.06780I
b = 0.278216 + 1.067795I
1.60812 + 2.52293I 1.64572 4.38707I
u = 0.10517 1.68329I
a = 0.0642219 0.1085660I
b = 0.935778 0.108566I
4.36446 3.30324I 4.83610 + 2.39821I
u = 0.10517 + 1.68329I
a = 0.0642219 + 0.1085660I
b = 0.935778 + 0.108566I
4.36446 + 3.30324I 4.83610 2.39821I
u = 0.232023 0.173210I
a = 1.60376 0.84508I
b = 0.603756 0.845078I
0.60016 2.36301I 1.43513 + 4.19898I
u = 0.232023 + 0.173210I
a = 1.60376 + 0.84508I
b = 0.603756 + 0.845078I
0.60016 + 2.36301I 1.43513 4.19898I
u = 0.39449 2.02318I
a = 0.465618 + 1.261534I
b = 0.534382 + 1.261534I
7.87584 8.60203I 2.41458 + 5.32797I
u = 0.39449 + 2.02318I
a = 0.465618 1.261534I
b = 0.534382 1.261534I
7.87584 + 8.60203I 2.41458 5.32797I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.423793
a = 1.33378
b = 0.333779
1.00303 10.1182
u = 1.258261 0.460042I
a = 0.834339 + 0.841449I
b = 0.165661 + 0.841449I
0.965349 0.999086I 3.54362 0.58191I
u = 1.258261 + 0.460042I
a = 0.834339 0.841449I
b = 0.165661 0.841449I
0.965349 + 0.999086I 3.54362 + 0.58191I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
u
2
(u
13
+ u
12
+ ··· + 4u 4)
c
2
, c
8
(u
2
u + 1)(u
13
+ 2u
12
+ ··· + u 1)
c
3
(u
2
u + 1)(u
13
+ 2u
12
+ ··· + 3u + 1)
c
4
, c
6
(u + 1)
2
(u
13
+ 3u
12
+ ··· 2u 1)
c
7
(u + 1)
2
(u
13
+ 3u
12
+ ··· + 8u + 1)
c
9
(u
2
+ u + 1)(u
13
+ 8u
12
+ ··· + 5u 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
y
2
(y
13
+ 15y
12
+ ··· 56y 16)
c
2
, c
8
(y
2
+ y + 1)(y
13
+ 8y
12
+ ··· + 5y 1)
c
3
(y
2
+ y + 1)(y
13
16y
12
+ ··· + 5y 1)
c
4
, c
6
(y 1)
2
(y
13
3y
12
+ ··· + 8y 1)
c
7
(y 1)
2
(y
13
+ 17y
12
+ ··· + 8y 1)
c
9
(y
2
+ y + 1)(y
13
4y
12
+ ··· + 85y 1)
8