9
22
(K9a
2
)
1
Arc Sequences
4 5 8 6 2 9 1 3 7
Solving Sequence
2,5
3
6,9
7 4 1 8
c
2
c
5
c
6
c
4
c
1
c
8
c
3
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, a u, b + 1i
I
u
2
= hu
23
2u
22
+ ··· 2u + 1, u
22
+ 5u
20
+ ··· + b + 1, 2u
22
2u
21
+ ··· + a + ui
There are 2 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
2
+ u + 1, a u, b + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u + 1
a
6
=
u
u
a
9
=
u
1
a
7
=
2u
u 1
a
4
=
1
u + 1
a
1
=
u
u
a
8
=
u
1
a
8
=
u
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u 1
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 1.00000
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 1.00000
1.64493 2.02988I 3.00000 + 3.46410I
3
II.
I
u
2
= hu
23
2u
22
+· · ·2u+1, u
22
+5u
20
+· · ·+b+1, 2u
22
2u
21
+· · ·+a+ui
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
6
=
u
u
a
9
=
2u
22
+ 2u
21
+ ··· + 2u
2
u
u
22
5u
20
+ ··· + u 1
a
7
=
u
22
u
21
+ ··· + u + 1
u
21
u
20
+ ··· u + 1
a
4
=
u
3
u
3
+ u
a
1
=
u
6
u
4
+ 1
u
6
2u
4
u
2
a
8
=
u
22
5u
20
+ ··· + 4u
3
1
2u
21
+ u
20
+ ··· + 2u 1
a
8
=
u
22
5u
20
+ ··· + 4u
3
1
2u
21
+ u
20
+ ··· + 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
= u
22
u
21
+ 8u
20
9u
19
+ 31u
18
35u
17
+ 74u
16
84u
15
+ 126u
14
138u
13
+ 160u
12
159u
11
+ 155u
10
129u
9
+ 104u
8
66u
7
+ 41u
6
15u
5
+ 2u
4
+ 13u
3
5u
2
+ 7u 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.730473 0.812317I
a = 0.81977 1.18006I
b = 1.61145 0.23019I
3.41052 + 2.74438I 6.00137 3.42075I
u = 0.730473 + 0.812317I
a = 0.81977 + 1.18006I
b = 1.61145 + 0.23019I
3.41052 2.74438I 6.00137 + 3.42075I
u = 0.616508
a = 2.24266
b = 0.117086
2.78711 2.32391
u = 0.479277 0.962679I
a = 0.376112 + 0.650926I
b = 0.260204 + 0.996761I
0.27712 + 2.59653I 1.46303 3.78636I
u = 0.479277 + 0.962679I
a = 0.376112 0.650926I
b = 0.260204 0.996761I
0.27712 2.59653I 1.46303 + 3.78636I
u = 0.476560 0.630579I
a = 1.207094 + 0.138559I
b = 0.783975 0.194295I
0.71923 + 1.37448I 2.70178 4.35124I
u = 0.476560 + 0.630579I
a = 1.207094 0.138559I
b = 0.783975 + 0.194295I
0.71923 1.37448I 2.70178 + 4.35124I
u = 0.449028 1.143790I
a = 0.791333 0.928166I
b = 1.14440 2.03951I
5.84331 + 3.99588I 6.60901 3.49800I
u = 0.449028 + 1.143790I
a = 0.791333 + 0.928166I
b = 1.14440 + 2.03951I
5.84331 3.99588I 6.60901 + 3.49800I
u = 0.179248 0.701899I
a = 1.24757 + 1.08672I
b = 1.289223 0.364913I
2.10210 0.88878I 6.39291 0.92577I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.179248 + 0.701899I
a = 1.24757 1.08672I
b = 1.289223 + 0.364913I
2.10210 + 0.88878I 6.39291 + 0.92577I
u = 0.282827 1.245841I
a = 0.296506 0.242999I
b = 0.108633 + 0.911739I
11.61978 + 1.64388I 9.30470 0.40272I
u = 0.282827 + 1.245841I
a = 0.296506 + 0.242999I
b = 0.108633 0.911739I
11.61978 1.64388I 9.30470 + 0.40272I
u = 0.386982 1.120878I
a = 0.784305 + 0.939860I
b = 0.051592 + 0.571060I
4.15124 1.33135I 7.15950 + 0.67575I
u = 0.386982 + 1.120878I
a = 0.784305 0.939860I
b = 0.051592 0.571060I
4.15124 + 1.33135I 7.15950 0.67575I
u = 0.501837 1.137101I
a = 1.53815 0.80154I
b = 1.07046 1.39770I
3.32060 6.47771I 4.77780 + 6.52194I
u = 0.501837 + 1.137101I
a = 1.53815 + 0.80154I
b = 1.07046 + 1.39770I
3.32060 + 6.47771I 4.77780 6.52194I
u = 0.565955 1.190510I
a = 1.84065 + 0.39307I
b = 2.12356 + 1.52738I
9.6533 10.6207I 7.02627 + 6.45650I
u = 0.565955 + 1.190510I
a = 1.84065 0.39307I
b = 2.12356 1.52738I
9.6533 + 10.6207I 7.02627 6.45650I
u = 0.657802 0.201077I
a = 0.771798 + 0.323505I
b = 0.080165 + 1.062697I
0.66432 + 2.00215I 1.23588 3.62705I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.657802 + 0.201077I
a = 0.771798 0.323505I
b = 0.080165 1.062697I
0.66432 2.00215I 1.23588 + 3.62705I
u = 0.868940 0.243856I
a = 0.99574 1.13587I
b = 0.657889 0.957400I
6.80889 + 5.35900I 4.49542 3.06793I
u = 0.868940 + 0.243856I
a = 0.99574 + 1.13587I
b = 0.657889 + 0.957400I
6.80889 5.35900I 4.49542 + 3.06793I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)(u
23
+ 2u
22
+ ··· + 18u + 9)
c
2
(u
2
+ u + 1)(u
23
+ 2u
22
+ ··· 2u 1)
c
3
, c
8
u
2
(u
23
+ u
22
+ ··· + 8u 4)
c
4
(u
2
u + 1)(u
23
+ 12u
22
+ ··· 2u 1)
c
5
(u
2
u + 1)(u
23
+ 2u
22
+ ··· 2u 1)
c
6
, c
7
(u 1)
2
(u
23
+ 3u
22
+ ··· u 1)
c
9
(u + 1)
2
(u
23
+ 3u
22
+ ··· u 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
2
+ y + 1)(y
23
12y
22
+ ··· 450y 81)
c
2
, c
5
(y
2
+ y + 1)(y
23
+ 12y
22
+ ··· 2y 1)
c
3
, c
8
y
2
(y
23
+ 15y
22
+ ··· 40y 16)
c
4
(y
2
+ y + 1)(y
23
+ 24y
21
+ ··· + 10y 1)
c
6
, c
7
, c
9
(y 1)
2
(y
23
23y
22
+ ··· 7y 1)
9